[{"data":1,"prerenderedAt":72980},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0016-yu-zhu-de-fei-xu":3,"i-ci:tag":72974,"i-mingcute:time-line":72978},{"post":4,"nextPost":195,"prevPost":205},{"id":5,"title":6,"body":7,"description":13,"draft":192,"enableCodeHighlight":192,"enableCodeHightlight":192,"enableComment":193,"enableD2":192,"enableKatex":192,"extension":194,"image":19,"important":192,"location":195,"meta":196,"navigation":193,"ogImage":195,"path":197,"seo":198,"stem":199,"summary":200,"tags":201,"time":203,"__hash__":204},"blog\u002Fblog\u002F2026\u002F0016-yu-zhu-de-fei-xu.md","预筑的废墟：苏联美学里永恒与必朽",{"type":8,"value":9,"toc":189},"minimark",[10,14,21,24,27,30,33,36,41,44,47,50,53,59,62,65,68,71,74,80,83,88,91,94,99,102,105,108,111,114,117,120,123,126,129,132,135,140,143,146,149,151,154,157,167,173,176,179,184],[11,12,13],"p",{},"苏联美学，永远具有一种冷冽的悲伤。它最擅长制造宏大的未来，也最清楚未来无法归还什么。它用巨像对抗死亡，用纪念碑对抗遗忘，用胜利解释牺牲，用不断向上的线条告诉所有人，继续向前。这种美学之所以永远带着寒意，是因为它最擅长的事情，恰恰是用永恒的尺度去丈量那些注定消逝的生命。",[11,15,16],{},[17,18],"img",{"alt":19,"src":20},"","https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112205632.png",[11,22,23],{},"尤其是构成主义与砼核风格，痴迷于巨大。那些混凝土浇筑的几何体，直插云霄的纪念碑，压迫感十足的工业建筑群，本质上是一种对抗遗忘的视觉宣言。列宁时期的纪念碑宣传计划，就明确将雕塑与建筑视为向大众传达政治理念的沟通工具，视觉符号在某种程度上被当作现实的替代品，用以确认理想已经实现。",[11,25,26],{},"但这里的悲剧性在于，巨像越是庞大，站在它面前的人就越显得渺小与短暂。在《雁南飞》等电影的构图中，人物常常被置于宏伟建筑之下，成为景观的附属品。冰冷的钢筋水泥占据了画面的绝大部分，传递出一种个人在集体面前的无能与身不由己。",[11,28,29],{},"巨物的压迫感和崇高感本就是一体。人站在纪念碑前，被收进宏大叙事里，你的牺牲、痛苦、失去，会被打磨成勋章，放进集体通往未来的序列。在这套语法里，个体的消逝不再是纯粹的湮灭，而是通向那个光明前途的耗材。肖斯塔科维奇第五交响曲那个著名的分歧刚好是绝佳注脚：外人听见胜利的凯歌，亲历者听见的是强行压下去的哀恸，是不得不乐观。曲调明亮，底色是吞下去的苦难。",[11,31,32],{},"这种美学在赋予观者崇高的使命感的同时，也让他们感受到了冰冷的压迫感。悲伤，就藏在那种你为之奉献一切的东西，比你活得更久的预感里。​用胜利解释牺牲，而解释本身需要反复讲述。",[11,34,35],{},"苏联美学最擅长将苦难转化为勋章。卫国战争题材的电影总是悲壮的，因为那场战争本身就是悲壮的，数千万人的伤亡，每一场胜利的代价都过于艰涩。但宏大的叙事要求它必须壮：烈士的血不能白流，复仇必须理直气壮。",[11,37,38],{},[17,39],{"alt":19,"src":40},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112147906.png",[11,42,43],{},"于是，艺术在这里承担了一个近乎残酷的职责，为无法挽回的损失赋予意义。肖斯塔科维奇的音乐就体现了这种撕裂感。他的第五交响曲末乐章，美国指挥家往往演奏得激昂欢腾，而苏联指挥家穆拉文斯基却处理得更慢更沉重。",[11,45,46],{},"那种拖滞感，像是在漫长而不安宁的等待中，被迫咽下所有说不出口的东西。音乐里的乐观与悲观同时存在，就像一个人用最阳光的曲调唱着最凄冷的歌词。",[11,48,49],{},"​那些不断向上的线条，例如，尖塔、直插天际的楼宇，仰望星空的人物姿态，本质上是一种方向性的命令。它们宣告未来是向上的、进步的，必然到来的。",[11,51,52],{},"尤其是苏联航天纪念碑。它们有一种非常迷人的矛盾：人类终于离开地球，向宇宙进发；但承载这种理想的，却是沉重、冰冷、几乎不朽的钢铁和混凝土。仿佛人类试图用最坚硬的物质，表达最遥远的梦想。",[11,54,55],{},[17,56],{"alt":57,"src":58},"苏联航天纪念碑","https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112146379.png",[11,60,61],{},"而今天，当我们重新观看这些建筑时，它们又有了另一层意义：它们证明了人类曾经如此认真地相信未来。",[11,63,64],{},"然而，悲剧的种子就埋在这个必然的种子里。\n当未来的必然性一旦失去，所有的向上线条都变成了悬在半空的讽刺。就像那首《最美好的前途》，曲调中满是经典的俄罗斯式忧伤，歌词却期盼着最美好的前途。可这首歌诞生后仅仅几年，苏联就解体了。",[11,66,67],{},"曲调是斯拉夫骨血里的小调忧伤，歌词是滚烫，笃定的未来许诺。它诞生的时候，所有人认真地相信那个前途。没过多久，许诺的体系崩塌。于是旋律和歌词之间的裂痕，不再只是艺术手法，变成历史本身的反讽。它提前歌颂的未来，没有如约而至。那些向上的线条，一下子悬在半空，不再指向天国一样的明天，只剩下孤零零的、凝固的期待。",[11,69,70],{},"曾经从零点起步奔向光明未来的承诺，变成了对个人命运无可奈何的隐喻。用最凄冷的曲调歌唱最阳光的歌词，这种形式与内容的错位，本身就是一种悲剧。苏联美学最擅长将苦难转化为勋章，却忘了勋章本身也是冰冷的。",[11,72,73],{},"所以，冷冽，它来自地域与历史的双重寒冷。斯拉夫民族的音乐本就多用自然小调，寒带的广袤与严酷塑造了沉郁的底色。\n更深一层，它来自一个把未来揉和了东正传统宗教来信奉的文明，最终不得不直面未来并未如期归还的结局。苏联美学用巨像对抗死亡，但巨像终将沦为废墟。用纪念碑对抗遗忘。",[11,75,76],{},[17,77],{"alt":78,"src":79},"切博克萨雷的楚瓦什国立歌剧芭蕾舞剧院（Чувашский государственный театр оперы и балета）","https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112206635.png",[11,81,82],{},"普通的崇高，是人和宏大之物一起奔赴希望。而这里的崇高，暗含一条无声的伏笔。它用永恒的尺度包裹转瞬即逝的生命，许诺你的不朽，是消融在集体叙事里的不朽。所以站在那些砼核墙体，巨大雕塑之下，压迫感和感动是一体两面，被赋予使命，同时渺小，痛苦、无法被补偿的失去，被宏大叙事收编，变成勋章。",[11,84,85],{},[17,86],{"alt":19,"src":87},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112213917.png",[11,89,90],{},"但纪念碑的主人已被遗忘。用胜利解释牺牲，但牺牲者带走了解释权。",[11,92,93],{},"​​用永恒的尺度去丈量那些注定消逝的生命，是苏联美学最残酷也最迷人的地方。斯拉夫的寒带，东正的传统，解体的结局，让那些直插云霄的混凝土几何体，成了对抗遗忘的视觉宣言，也成了个人命运无可奈何的隐喻。",[11,95,96],{},[17,97],{"alt":19,"src":98},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112146414.png",[11,100,101],{},"这本书《二手时间》写的真很棒，虽然我是 00 后，从未见识过苏联时代。但对苏联的感受，不像是二手怀旧而更像共情。\n令人扼腕的是，那种用永恒丈量必朽的动作本身，在别处也见过这个动作，甚至可能被这个动作丈量过。曾经相信过的人，后来怎么面对信念的消散。宏大巨像崩塌之后，留下来的是一个个被抛下的人，他们曾经把自己的生命押给那个未来。因为这套语法后来被无数地方，无数机构，无数广告反复使用，用到它变成一种通用的修辞外壳。",[11,103,104],{},"《二手时间》里那些老兵家属、理想主义者的后代，很多人不是在恨苏联，而是在为\"那个值得牺牲的东西究竟存不存在\"而痛苦——这比单纯的批判深刻得多。这本书里最残酷的地方，在于它没有简单地把苏联解体写成一个政治制度的终结。阿列克谢耶维奇写的根本不只是政权解体，而是一套意义语言的死亡。",[11,106,107],{},"它写的是人如何失去自己赖以理解世界的语言。",[11,109,110],{},"对于很多经历过苏联时代的人来说，苏联不只是一个国家，也不只是一个政治制度。",[11,112,113],{},"它可能是：父母年轻时相信的东西、一生奉献的事业、战争中牺牲的亲人的意义、对公平、平等和人类共同未来的想象、一套关于什么值得追求、什么值得牺牲的价值体系。",[11,115,116],{},"当这个体系消失时，真正令人痛苦的问题并不是谁赢了、谁输了。而是：如果我曾经为之奉献一生的东西最终没有实现，那么我的一生究竟意味着什么？",[11,118,119],{},"通常我们理解废墟，是因为某种已经存在的文明走向衰亡。罗马遗迹，古希腊神庙，被遗弃的工业设施，都属于这种工业风废土审美。",[11,121,122],{},"它和古罗马古希腊废墟不一样。古典废墟是文明的余烬，是曾经有过。苏联的废墟，是一场没能兑现的预演。它建造的时候，就已经在想象胜利的终点；最后留给后世的，是一群认真相信过终点的人，以及一堆静静矗立，等待未来到来，却再也等不到的混凝土巨像。",[11,124,125],{},"但苏联美学有一种非常特殊的时间错位。许多建筑、雕塑和艺术作品，本就为了纪念一个尚未完全到来的未来而建造的。不是在怀念过去，而是在提前纪念未来，而且它曾经如此认真地相信未来。",[11,127,128],{},"建筑可以保存，制度可以更替，口号可以遗忘，但那些曾经相信过它们的人，却只有一次生命。",[11,130,131],{},"而且，他们无法提前知道自己所相信的未来最终会变成什么。",[11,133,134],{},"有些废墟之所以令人悲伤，不是因为它们曾经辉煌，而是因为它们曾经如此确信自己不会成为废墟。",[11,136,137],{},[17,138],{"alt":19,"src":139},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112148277.png",[11,141,142],{},"人类文明似乎一直在重复同一种行为：我们建造比自己寿命更长的东西，试图让自己的信念获得永恒。金字塔、神庙、帝国大道、革命纪念碑、摩天大楼，乃至今天各种代表技术进步的宏伟工程，都有这种冲动。",[11,144,145],{},"人类知道自己会死，于是希望自己的劳动、信仰和理想能够活得更久。这本身并没有错。",[11,147,148],{},"真正悲伤的是，建筑可以保存，制度可以更替，口号可以遗忘，但那些曾经相信过它们的人，却只有一次生命。",[11,150,131],{},[11,152,153],{},"这大概就是苏联美学最独特的时间感。",[11,155,156],{},"用宏大、恒久的载体，收纳个体短暂的付出，许诺一个更远的、更好的明天”，这个冲动是人类共通的。我们总能在别的地方再次认出它。所以我们共情，不只是隔着历史看另一个遥远国度的废墟，也是看见人永恒的困境：",[158,159,160],"blockquote",{},[11,161,162,163,166],{},"我们总想造出比自己活得更久的东西，想让自己短暂的存在，借由某种宏大之物获得不朽。  ",[164,165],"br",{},"\n可永恒的尺度，永远会温柔又残酷地丈量所有必朽的生命。",[11,168,169],{},[17,170],{"alt":171,"src":172},"《再见，列宁》剧照","https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112152351.png",[11,174,175],{},"它的悲伤不是简单的怀旧，也不是对某个政治时代的浪漫化。",[11,177,178],{},"而是站在一个已经知道结局的时代，回望那些尚不知道结局的人。他们仰望天空。他们建造纪念碑。他们歌唱最美好的前途。而我们知道，未来并没有按照他们想象的方式到来。但我不认为这就意味着他们的相信毫无价值。",[158,180,181],{},[11,182,183],{},"一个未来最终没有实现，并不意味着曾经为它而活的人生，从来没有意义。",[11,185,186],{},[17,187],{"alt":19,"src":188},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112213350.png",{"title":19,"searchDepth":190,"depth":190,"links":191},2,[],false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0016-yu-zhu-de-fei-xu",{"title":6,"description":13},"blog\u002F2026\u002F0016-yu-zhu-de-fei-xu","苏联美学最核心的那层悲剧，不是衰败之后的挽歌，而是预先写给一个落空未来的纪念碑。那些向上收束的线条、厚重的混凝土体量、巨人般的雕塑，不是追忆过往荣光，而是以物质的永恒，为尚未兑现的理想确权。",[202],"LHPA","2026-11-11T22:15:41+08:00","vnX5uH_SGvoL6cAq_8jfaZHozYxtlD19uK94W97o_P4",{"id":206,"title":207,"body":208,"description":212,"draft":192,"enableCodeHighlight":192,"enableCodeHightlight":192,"enableComment":193,"enableD2":192,"enableKatex":193,"extension":194,"image":19,"important":192,"location":195,"meta":72964,"navigation":193,"ogImage":195,"path":72965,"seo":72966,"stem":72967,"summary":72968,"tags":72969,"time":72972,"__hash__":72973},"blog\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties.md","贝尔曼方程及其数学性质",{"type":8,"value":209,"toc":72918},[210,213,216,219,224,229,1807,2276,2783,2787,3653,4358,5211,5790,5796,6558,6682,6685,6689,7613,7836,8590,8955,10112,10115,10235,10239,10243,10251,11022,11029,11880,11883,11945,11949,12016,12694,12697,12851,13125,13129,13549,14370,14553,14746,15393,16017,16321,16375,16620,16835,17384,17386,17527,17531,17610,17784,17973,18187,18190,18307,18310,18584,18587,18937,18940,19316,19319,19853,19940,19944,19947,20248,20339,20768,21029,21285,21428,21546,21550,21554,21631,21841,22003,22289,22814,22876,23675,23874,23878,24834,24987,25344,25872,25875,26640,26766,26828,26832,27645,27857,28113,28120,28401,28432,28795,28826,29115,29593,29597,30662,30786,31146,31450,31750,32761,33107,33791,33794,34277,34279,34283,34521,34719,34959,34962,35805,35971,36286,36615,36732,36736,36740,37445,37546,37550,37614,38133,38254,39765,40057,40321,40569,41348,41626,42193,42866,43358,43836,44247,44309,44431,45223,45543,45547,45717,46083,46749,46752,47300,47365,47369,48265,48611,48614,48618,48734,49037,49251,49340,49556,49616,49826,50019,50256,50488,50490,50617,50853,51155,51225,51229,51233,51344,51580,51728,52667,52672,53340,53371,53563,53567,53644,54943,54948,56091,56094,56483,56486,57062,57065,57636,57639,58065,58068,58123,58571,58780,58784,58903,59288,59291,59998,60053,60582,60637,61035,61152,61363,61632,61798,61802,61906,62076,62235,62342,62374,62580,62627,62830,62884,62888,62892,63255,63964,64212,64819,65161,65752,66047,67324,67906,68206,68210,68275,68494,68498,68501,68921,68994,69099,69102,69106,69110,69157,69219,69843,69934,70095,70099,70213,70801,71295,71400,71404,71580,71750,72281,72338,72373,72544,72547,72550,72826,72915],[11,211,212],{},"贝尔曼方程（Bellman Equation）是动态规划（Dynamic Programming）和强化学习（Reinforcement Learning）中的核心数学方程，用于描述一个系统在当前状态下的最优价值，如何由当前收益和未来状态的价值共同决定。",[11,214,215],{},"简单来说：当前状态的价值 = 当前获得的收益 + 未来可能获得的折扣收益。它的核心思想是：一个复杂的长期决策问题，可以分解为一系列相互关联的短期决策问题。",[11,217,218],{},"它不是某一个孤立的等式，而是一族把值函数与自身联系起来的泛函方程。它的数学内容可以概括为三个层次。第一，折扣回报在时间上的递归分解。第二，一个定义在有界函数空间上的算子。第三，该算子的不动点。动态规划与强化学习中的许多结论，本质上都是这三个层次的不同推论。",[220,221,223],"h2",{"id":222},"一形式化","一、形式化",[225,226,228],"h3",{"id":227},"_11-折扣马尔可夫决策过程","1.1 折扣马尔可夫决策过程",[230,231,234,942,1684],"theorem",{"id":232,"title":233},"definition-mdp","定义 1.1（折扣马尔可夫决策过程）",[11,235,236,237,285,286,315,316,463,464,493,494,522,523,585,586,715,716,798,799,882,883,912,913,941],{},"设 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为状态集，",[238,287,289,303],{"className":288,"translate":242},[241],[238,290,292],{"className":291},[246],[248,293,294],{"xmlns":250},[252,295,296,301],{},[255,297,298],{},[258,299,300],{},"A",[262,302,300],{"encoding":264},[238,304,306],{"className":305,"ariaHidden":269},[268],[238,307,309,312],{"className":308},[273],[238,310],{"className":311,"style":278},[277],[238,313,300],{"className":314},[282,283]," 为动作集，",[238,317,319,364],{"className":318,"translate":242},[241],[238,320,322],{"className":321},[246],[248,323,324],{"xmlns":250},[252,325,326,361],{},[255,327,328,331,336,347,350,352,355,358],{},[258,329,330],{},"P",[332,333,335],"mo",{"stretchy":334},"false","(",[337,338,339,342],"msup",{},[258,340,341],{},"s",[332,343,346],{"mathvariant":344,"lspace":345,"rspace":345},"normal","0em","′",[332,348,349],{},"∣",[258,351,341],{},[332,353,354],{"separator":269},",",[258,356,357],{},"a",[332,359,360],{"stretchy":334},")",[262,362,363],{"encoding":264},"P(s'\\mid s,a)",[238,365,367,438],{"className":366,"ariaHidden":269},[268],[238,368,370,374,378,382,426,431,435],{"className":369},[273],[238,371],{"className":372,"style":373},[277],"height:1.0019em;vertical-align:-0.25em;",[238,375,330],{"className":376,"style":377},[282,283],"margin-right:0.1389em;",[238,379,335],{"className":380},[381],"mopen",[238,383,385,388],{"className":384},[282],[238,386,341],{"className":387},[282,283],[238,389,392],{"className":390},[391],"msupsub",[238,393,396],{"className":394},[395],"vlist-t",[238,397,400],{"className":398},[399],"vlist-r",[238,401,405],{"className":402,"style":404},[403],"vlist","height:0.7519em;",[238,406,408,413],{"style":407},"top:-3.063em;margin-right:0.05em;",[238,409],{"className":410,"style":412},[411],"pstrut","height:2.7em;",[238,414,420],{"className":415},[416,417,418,419],"sizing","reset-size6","size3","mtight",[238,421,423],{"className":422},[282,419],[238,424,346],{"className":425},[282,419],[238,427],{"className":428,"style":430},[429],"mspace","margin-right:0.2778em;",[238,432,349],{"className":433},[434],"mrel",[238,436],{"className":437,"style":430},[429],[238,439,441,445,448,452,456,459],{"className":440},[273],[238,442],{"className":443,"style":444},[277],"height:1em;vertical-align:-0.25em;",[238,446,341],{"className":447},[282,283],[238,449,354],{"className":450},[451],"mpunct",[238,453],{"className":454,"style":455},[429],"margin-right:0.1667em;",[238,457,357],{"className":458},[282,283],[238,460,360],{"className":461},[462],"mclose"," 为在状态 ",[238,465,467,480],{"className":466,"translate":242},[241],[238,468,470],{"className":469},[246],[248,471,472],{"xmlns":250},[252,473,474,478],{},[255,475,476],{},[258,477,341],{},[262,479,341],{"encoding":264},[238,481,483],{"className":482,"ariaHidden":269},[268],[238,484,486,490],{"className":485},[273],[238,487],{"className":488,"style":489},[277],"height:0.4306em;",[238,491,341],{"className":492},[282,283]," 执行动作 ",[238,495,497,510],{"className":496,"translate":242},[241],[238,498,500],{"className":499},[246],[248,501,502],{"xmlns":250},[252,503,504,508],{},[255,505,506],{},[258,507,357],{},[262,509,357],{"encoding":264},[238,511,513],{"className":512,"ariaHidden":269},[268],[238,514,516,519],{"className":515},[273],[238,517],{"className":518,"style":489},[277],[238,520,357],{"className":521},[282,283]," 后转移到状态 ",[238,524,526,544],{"className":525,"translate":242},[241],[238,527,529],{"className":528},[246],[248,530,531],{"xmlns":250},[252,532,533,541],{},[255,534,535],{},[337,536,537,539],{},[258,538,341],{},[332,540,346],{"mathvariant":344,"lspace":345,"rspace":345},[262,542,543],{"encoding":264},"s'",[238,545,547],{"className":546,"ariaHidden":269},[268],[238,548,550,553],{"className":549},[273],[238,551],{"className":552,"style":404},[277],[238,554,556,559],{"className":555},[282],[238,557,341],{"className":558},[282,283],[238,560,562],{"className":561},[391],[238,563,565],{"className":564},[395],[238,566,568],{"className":567},[399],[238,569,571],{"className":570,"style":404},[403],[238,572,573,576],{"style":407},[238,574],{"className":575,"style":412},[411],[238,577,579],{"className":578},[416,417,418,419],[238,580,582],{"className":581},[282,419],[238,583,346],{"className":584},[282,419]," 的概率，",[238,587,589,625],{"className":588,"translate":242},[241],[238,590,592],{"className":591},[246],[248,593,594],{"xmlns":250},[252,595,596,622],{},[255,597,598,601,604,606,609,611,613,615,618],{},[258,599,600],{},"r",[332,602,603],{},":",[258,605,260],{},[332,607,608],{},"×",[258,610,300],{},[332,612,608],{},[258,614,260],{},[332,616,617],{},"→",[258,619,621],{"mathvariant":620},"double-struck","R",[262,623,624],{"encoding":264},"r:S\\times A\\times S\\to\\mathbb{R}",[238,626,628,647,668,686,704],{"className":627,"ariaHidden":269},[268],[238,629,631,634,638,641,644],{"className":630},[273],[238,632],{"className":633,"style":489},[277],[238,635,600],{"className":636,"style":637},[282,283],"margin-right:0.0278em;",[238,639],{"className":640,"style":430},[429],[238,642,603],{"className":643},[434],[238,645],{"className":646,"style":430},[429],[238,648,650,654,657,661,665],{"className":649},[273],[238,651],{"className":652,"style":653},[277],"height:0.7667em;vertical-align:-0.0833em;",[238,655,260],{"className":656,"style":284},[282,283],[238,658],{"className":659,"style":660},[429],"margin-right:0.2222em;",[238,662,608],{"className":663},[664],"mbin",[238,666],{"className":667,"style":660},[429],[238,669,671,674,677,680,683],{"className":670},[273],[238,672],{"className":673,"style":653},[277],[238,675,300],{"className":676},[282,283],[238,678],{"className":679,"style":660},[429],[238,681,608],{"className":682},[664],[238,684],{"className":685,"style":660},[429],[238,687,689,692,695,698,701],{"className":688},[273],[238,690],{"className":691,"style":278},[277],[238,693,260],{"className":694,"style":284},[282,283],[238,696],{"className":697,"style":430},[429],[238,699,617],{"className":700},[434],[238,702],{"className":703,"style":430},[429],[238,705,707,711],{"className":706},[273],[238,708],{"className":709,"style":710},[277],"height:0.6889em;",[238,712,621],{"className":713},[282,714],"mathbb"," 为奖励函数，",[238,717,719,751],{"className":718,"translate":242},[241],[238,720,722],{"className":721},[246],[248,723,724],{"xmlns":250},[252,725,726,748],{},[255,727,728,731,734,737,741,743,746],{},[258,729,730],{},"γ",[332,732,733],{},"∈",[332,735,736],{"stretchy":334},"[",[738,739,740],"mn",{},"0",[332,742,354],{"separator":269},[738,744,745],{},"1",[332,747,360],{"stretchy":334},[262,749,750],{"encoding":264},"\\gamma\\in[0,1)",[238,752,754,774],{"className":753,"ariaHidden":269},[268],[238,755,757,761,765,768,771],{"className":756},[273],[238,758],{"className":759,"style":760},[277],"height:0.7335em;vertical-align:-0.1944em;",[238,762,730],{"className":763,"style":764},[282,283],"margin-right:0.0556em;",[238,766],{"className":767,"style":430},[429],[238,769,733],{"className":770},[434],[238,772],{"className":773,"style":430},[429],[238,775,777,780,783,786,789,792,795],{"className":776},[273],[238,778],{"className":779,"style":444},[277],[238,781,736],{"className":782},[381],[238,784,740],{"className":785},[282],[238,787,354],{"className":788},[451],[238,790],{"className":791,"style":455},[429],[238,793,745],{"className":794},[282],[238,796,360],{"className":797},[462]," 为折扣因子。记 ",[238,800,802,824],{"className":801,"translate":242},[241],[238,803,805],{"className":804},[246],[248,806,807],{"xmlns":250},[252,808,809,821],{},[255,810,811],{},[812,813,814,818],"msub",{},[258,815,817],{"mathvariant":816},"script","F",[258,819,820],{},"t",[262,822,823],{"encoding":264},"\\mathcal{F}_t",[238,825,827],{"className":826,"ariaHidden":269},[268],[238,828,830,834],{"className":829},[273],[238,831],{"className":832,"style":833},[277],"height:0.8333em;vertical-align:-0.15em;",[238,835,837,842],{"className":836},[282],[238,838,817],{"className":839,"style":841},[282,840],"mathcal","margin-right:0.0993em;",[238,843,845],{"className":844},[391],[238,846,849,873],{"className":847},[395,848],"vlist-t2",[238,850,852,868],{"className":851},[399],[238,853,856],{"className":854,"style":855},[403],"height:0.2806em;",[238,857,859,862],{"style":858},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[238,860],{"className":861,"style":412},[411],[238,863,865],{"className":864},[416,417,418,419],[238,866,820],{"className":867},[282,283,419],[238,869,872],{"className":870},[871],"vlist-s","​",[238,874,876],{"className":875},[399],[238,877,880],{"className":878,"style":879},[403],"height:0.15em;",[238,881],{}," 为截至时刻 ",[238,884,886,899],{"className":885,"translate":242},[241],[238,887,889],{"className":888},[246],[248,890,891],{"xmlns":250},[252,892,893,897],{},[255,894,895],{},[258,896,820],{},[262,898,820],{"encoding":264},[238,900,902],{"className":901,"ariaHidden":269},[268],[238,903,905,909],{"className":904},[273],[238,906],{"className":907,"style":908},[277],"height:0.6151em;",[238,910,820],{"className":911},[282,283]," 的历史信息。若对任意 ",[238,914,916,929],{"className":915,"translate":242},[241],[238,917,919],{"className":918},[246],[248,920,921],{"xmlns":250},[252,922,923,927],{},[255,924,925],{},[258,926,820],{},[262,928,820],{"encoding":264},[238,930,932],{"className":931,"ariaHidden":269},[268],[238,933,935,938],{"className":934},[273],[238,936],{"className":937,"style":908},[277],[238,939,820],{"className":940},[282,283],"，",[238,943,946],{"className":944,"translate":242},[945],"katex-display",[238,947,949,1098],{"className":948,"translate":242},[241],[238,950,952],{"className":951},[246],[248,953,955],{"xmlns":250,"display":954},"block",[252,956,957,1095],{},[958,959,962],"mtable",{"rowspacing":960,"columnalign":961,"columnspacing":345},"0.25em","right 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s_t=s,a_t=a,\\mathcal{F}_t)\n&=P(s_{t+1}=s'\\mid s_t=s,a_t=a),\n\\end{aligned}",[238,1099,1101],{"className":1100,"ariaHidden":269},[268],[238,1102,1104,1108],{"className":1103},[273],[238,1105],{"className":1106,"style":1107},[277],"height:1.2em;vertical-align:-0.35em;",[238,1109,1111],{"className":1110},[282],[238,1112,1114,1419],{"className":1113},[958],[238,1115,1118],{"className":1116},[1117],"col-align-r",[238,1119,1121,1410],{"className":1120},[395,848],[238,1122,1124,1407],{"className":1123},[399],[238,1125,1128],{"className":1126,"style":1127},[403],"height:0.85em;",[238,1129,1131,1135],{"style":1130},"top:-3.01em;",[238,1132],{"className":1133,"style":1134},[411],"height:3em;",[238,1136,1138,1141,1144,1196,1199,1202,1205,1239,1242,1245,1248,1288,1291,1294,1297,1300,1303,1306,1346,1349,1352,1355,1358,1361,1364,1404],{"className":1137},[282],[238,1139,330],{"className":1140,"style":377},[282,283],[238,1142,335],{"className":1143},[381],[238,1145,1147,1150],{"className":1146},[282],[238,1148,341],{"className":1149},[282,283],[238,1151,1153],{"className":1152},[391],[238,1154,1156,1187],{"className":1155},[395,848],[238,1157,1159,1184],{"className":1158},[399],[238,1160,1163],{"className":1161,"style":1162},[403],"height:0.3011em;",[238,1164,1166,1169],{"style":1165},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[238,1167],{"className":1168,"style":412},[411],[238,1170,1172],{"className":1171},[416,417,418,419],[238,1173,1175,1178,1181],{"className":1174},[282,419],[238,1176,820],{"className":1177},[282,283,419],[238,1179,987],{"className":1180},[664,419],[238,1182,745],{"className":1183},[282,419],[238,1185,872],{"className":1186},[871],[238,1188,1190],{"className":1189},[399],[238,1191,1194],{"className":1192,"style":1193},[403],"height:0.2083em;",[238,1195],{},[238,1197],{"className":1198,"style":430},[429],[238,1200,992],{"className":1201},[434],[238,1203],{"className":1204,"style":430},[429],[238,1206,1208,1211],{"className":1207},[282],[238,1209,341],{"className":1210},[282,283],[238,1212,1214],{"className":1213},[391],[238,1215,1217],{"className":1216},[395],[238,1218,1220],{"className":1219},[399],[238,1221,1224],{"className":1222,"style":1223},[403],"height:0.8019em;",[238,1225,1227,1230],{"style":1226},"top:-3.113em;margin-right:0.05em;",[238,1228],{"className":1229,"style":412},[411],[238,1231,1233],{"className":1232},[416,417,418,419],[238,1234,1236],{"className":1235},[282,419],[238,1237,346],{"className":1238},[282,419],[238,1240],{"className":1241,"style":430},[429],[238,1243,349],{"className":1244},[434],[238,1246],{"className":1247,"style":430},[429],[238,1249,1251,1254],{"className":1250},[282],[238,1252,341],{"className":1253},[282,283],[238,1255,1257],{"className":1256},[391],[238,1258,1260,1280],{"className":1259},[395,848],[238,1261,1263,1277],{"className":1262},[399],[238,1264,1266],{"className":1265,"style":855},[403],[238,1267,1268,1271],{"style":1165},[238,1269],{"className":1270,"style":412},[411],[238,1272,1274],{"className":1273},[416,417,418,419],[238,1275,820],{"className":1276},[282,283,419],[238,1278,872],{"className":1279},[871],[238,1281,1283],{"className":1282},[399],[238,1284,1286],{"className":1285,"style":879},[403],[238,1287],{},[238,1289],{"className":1290,"style":430},[429],[238,1292,992],{"className":1293},[434],[238,1295],{"className":1296,"style":430},[429],[238,1298,341],{"className":1299},[282,283],[238,1301,354],{"className":1302},[451],[238,1304],{"className":1305,"style":455},[429],[238,1307,1309,1312],{"className":1308},[282],[238,1310,357],{"className":1311},[282,283],[238,1313,1315],{"className":1314},[391],[238,1316,1318,1338],{"className":1317},[395,848],[238,1319,1321,1335],{"className":1320},[399],[238,1322,1324],{"className":1323,"style":855},[403],[238,1325,1326,1329],{"style":1165},[238,1327],{"className":1328,"style":412},[411],[238,1330,1332],{"className":1331},[416,417,418,419],[238,1333,820],{"className":1334},[282,283,419],[238,1336,872],{"className":1337},[871],[238,1339,1341],{"className":1340},[399],[238,1342,1344],{"className":1343,"style":879},[403],[238,1345],{},[238,1347],{"className":1348,"style":430},[429],[238,1350,992],{"className":1351},[434],[238,1353],{"className":1354,"style":430},[429],[238,1356,357],{"className":1357},[282,283],[238,1359,354],{"className":1360},[451],[238,1362],{"className":1363,"style":455},[429],[238,1365,1367,1370],{"className":1366},[282],[238,1368,817],{"className":1369,"style":841},[282,840],[238,1371,1373],{"className":1372},[391],[238,1374,1376,1396],{"className":1375},[395,848],[238,1377,1379,1393],{"className":1378},[399],[238,1380,1382],{"className":1381,"style":855},[403],[238,1383,1384,1387],{"style":858},[238,1385],{"className":1386,"style":412},[411],[238,1388,1390],{"className":1389},[416,417,418,419],[238,1391,820],{"className":1392},[282,283,419],[238,1394,872],{"className":1395},[871],[238,1397,1399],{"className":1398},[399],[238,1400,1402],{"className":1401,"style":879},[403],[238,1403],{},[238,1405,360],{"className":1406},[462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Markov decision process，折扣 MDP），记作 ",[238,1693,1695,1734],{"className":1694,"translate":242},[241],[238,1696,1698],{"className":1697},[246],[248,1699,1700],{"xmlns":250},[252,1701,1702,1731],{},[255,1703,1704,1707,1709,1711,1713,1715,1717,1719,1721,1723,1725,1727,1729],{},[258,1705,1706],{"mathvariant":816},"M",[332,1708,992],{},[332,1710,335],{"stretchy":334},[258,1712,260],{},[332,1714,354],{"separator":269},[258,1716,300],{},[332,1718,354],{"separator":269},[258,1720,330],{},[332,1722,354],{"separator":269},[258,1724,600],{},[332,1726,354],{"separator":269},[258,1728,730],{},[332,1730,360],{"stretchy":334},[262,1732,1733],{"encoding":264},"\\mathcal{M}=(S,A,P,r,\\gamma)",[238,1735,1737,1755],{"className":1736,"ariaHidden":269},[268],[238,1738,1740,1743,1746,1749,1752],{"className":1739},[273],[238,1741],{"className":1742,"style":278},[277],[238,1744,1706],{"className":1745},[282,840],[238,1747],{"className":1748,"style":430},[429],[238,1750,992],{"className":1751},[434],[238,1753],{"className":1754,"style":430},[429],[238,1756,1758,1761,1764,1767,1770,1773,1776,1779,1782,1785,1788,1791,1794,1797,1800,1803],{"className":1757},[273],[238,1759],{"className":1760,"style":444},[277],[238,1762,335],{"className":1763},[381],[238,1765,260],{"className":1766,"style":284},[282,283],[238,1768,354],{"className":1769},[451],[238,1771],{"className":1772,"style":455},[429],[238,1774,300],{"className":1775},[282,283],[238,1777,354],{"className":1778},[451],[238,1780],{"className":1781,"style":455},[429],[238,1783,330],{"className":1784,"style":377},[282,283],[238,1786,354],{"className":1787},[451],[238,1789],{"className":1790,"style":455},[429],[238,1792,600],{"className":1793,"style":637},[282,283],[238,1795,354],{"className":1796},[451],[238,1798],{"className":1799,"style":455},[429],[238,1801,730],{"className":1802,"style":764},[282,283],[238,1804,360],{"className":1805},[462],"。",[11,1808,1809,1810,1838,1839,1867,1868,1984,1985,2176,2177,2275],{},"本文默认 ",[238,1811,1813,1826],{"className":1812,"translate":242},[241],[238,1814,1816],{"className":1815},[246],[248,1817,1818],{"xmlns":250},[252,1819,1820,1824],{},[255,1821,1822],{},[258,1823,260],{},[262,1825,260],{"encoding":264},[238,1827,1829],{"className":1828,"ariaHidden":269},[268],[238,1830,1832,1835],{"className":1831},[273],[238,1833],{"className":1834,"style":278},[277],[238,1836,260],{"className":1837,"style":284},[282,283]," 与 ",[238,1840,1842,1855],{"className":1841,"translate":242},[241],[238,1843,1845],{"className":1844},[246],[248,1846,1847],{"xmlns":250},[252,1848,1849,1853],{},[255,1850,1851],{},[258,1852,300],{},[262,1854,300],{"encoding":264},[238,1856,1858],{"className":1857,"ariaHidden":269},[268],[238,1859,1861,1864],{"className":1860},[273],[238,1862],{"className":1863,"style":278},[277],[238,1865,300],{"className":1866},[282,283]," 均为有限集，且存在常数 ",[238,1869,1871,1899],{"className":1870,"translate":242},[241],[238,1872,1874],{"className":1873},[246],[248,1875,1876],{"xmlns":250},[252,1877,1878,1896],{},[255,1879,1880,1890,1893],{},[812,1881,1882,1884,1887],{},[258,1883,621],{},[258,1885,1886],{},"max",[332,1888,1889],{},"⁡",[332,1891,1892],{},"\u003C",[258,1894,1895],{"mathvariant":344},"∞",[262,1897,1898],{"encoding":264},"R_{\\max}\u003C\\infty",[238,1900,1902,1975],{"className":1901,"ariaHidden":269},[268],[238,1903,1905,1908,1966,1969,1972],{"className":1904},[273],[238,1906],{"className":1907,"style":833},[277],[238,1909,1911,1915],{"className":1910},[282],[238,1912,621],{"className":1913,"style":1914},[282,283],"margin-right:0.0077em;",[238,1916,1918],{"className":1917},[391],[238,1919,1921,1958],{"className":1920},[395,848],[238,1922,1924,1955],{"className":1923},[399],[238,1925,1928],{"className":1926,"style":1927},[403],"height:0.1514em;",[238,1929,1931,1934],{"style":1930},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[238,1932],{"className":1933,"style":412},[411],[238,1935,1937],{"className":1936},[416,417,418,419],[238,1938,1940],{"className":1939},[282,419],[238,1941,1944,1948,1951],{"className":1942},[1943,419],"mop",[238,1945,1947],{"className":1946},[419],"m",[238,1949,357],{"className":1950},[419],[238,1952,1954],{"className":1953},[419],"x",[238,1956,872],{"className":1957},[871],[238,1959,1961],{"className":1960},[399],[238,1962,1964],{"className":1963,"style":879},[403],[238,1965],{},[238,1967],{"className":1968,"style":430},[429],[238,1970,1892],{"className":1971},[434],[238,1973],{"className":1974,"style":430},[429],[238,1976,1978,1981],{"className":1977},[273],[238,1979],{"className":1980,"style":489},[277],[238,1982,1895],{"className":1983},[282]," 使得 ",[238,1986,1988,2035],{"className":1987,"translate":242},[241],[238,1989,1991],{"className":1990},[246],[248,1992,1993],{"xmlns":250},[252,1994,1995,2032],{},[255,1996,1997,1999,2001,2003,2005,2007,2009,2011,2017,2019,2021,2024],{},[258,1998,349],{"mathvariant":344},[258,2000,600],{},[332,2002,335],{"stretchy":334},[258,2004,341],{},[332,2006,354],{"separator":269},[258,2008,357],{},[332,2010,354],{"separator":269},[337,2012,2013,2015],{},[258,2014,341],{},[332,2016,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,2018,360],{"stretchy":334},[258,2020,349],{"mathvariant":344},[332,2022,2023],{},"≤",[812,2025,2026,2028,2030],{},[258,2027,621],{},[258,2029,1886],{},[332,2031,1889],{},[262,2033,2034],{"encoding":264},"|r(s,a,s')|\\le R_{\\max}",[238,2036,2038,2118],{"className":2037,"ariaHidden":269},[268],[238,2039,2041,2044,2047,2050,2053,2056,2059,2062,2065,2068,2071,2103,2106,2109,2112,2115],{"className":2040},[273],[238,2042],{"className":2043,"style":373},[277],[238,2045,349],{"className":2046},[282],[238,2048,600],{"className":2049,"style":637},[282,283],[238,2051,335],{"className":2052},[381],[238,2054,341],{"className":2055},[282,283],[238,2057,354],{"className":2058},[451],[238,2060],{"className":2061,"style":455},[429],[238,2063,357],{"className":2064},[282,283],[238,2066,354],{"className":2067},[451],[238,2069],{"className":2070,"style":455},[429],[238,2072,2074,2077],{"className":2073},[282],[238,2075,341],{"className":2076},[282,283],[238,2078,2080],{"className":2079},[391],[238,2081,2083],{"className":2082},[395],[238,2084,2086],{"className":2085},[399],[238,2087,2089],{"className":2088,"style":404},[403],[238,2090,2091,2094],{"style":407},[238,2092],{"className":2093,"style":412},[411],[238,2095,2097],{"className":2096},[416,417,418,419],[238,2098,2100],{"className":2099},[282,419],[238,2101,346],{"className":2102},[282,419],[238,2104,360],{"className":2105},[462],[238,2107,349],{"className":2108},[282],[238,2110],{"className":2111,"style":430},[429],[238,2113,2023],{"className":2114},[434],[238,2116],{"className":2117,"style":430},[429],[238,2119,2121,2124],{"className":2120},[273],[238,2122],{"className":2123,"style":833},[277],[238,2125,2127,2130],{"className":2126},[282],[238,2128,621],{"className":2129,"style":1914},[282,283],[238,2131,2133],{"className":2132},[391],[238,2134,2136,2168],{"className":2135},[395,848],[238,2137,2139,2165],{"className":2138},[399],[238,2140,2142],{"className":2141,"style":1927},[403],[238,2143,2144,2147],{"style":1930},[238,2145],{"className":2146,"style":412},[411],[238,2148,2150],{"className":2149},[416,417,418,419],[238,2151,2153],{"className":2152},[282,419],[238,2154,2156,2159,2162],{"className":2155},[1943,419],[238,2157,1947],{"className":2158},[419],[238,2160,357],{"className":2161},[419],[238,2163,1954],{"className":2164},[419],[238,2166,872],{"className":2167},[871],[238,2169,2171],{"className":2170},[399],[238,2172,2174],{"className":2173,"style":879},[403],[238,2175],{}," 对一切 ",[238,2178,2180,2210],{"className":2179,"translate":242},[241],[238,2181,2183],{"className":2182},[246],[248,2184,2185],{"xmlns":250},[252,2186,2187,2207],{},[255,2188,2189,2191,2193,2195,2197,2199,2205],{},[332,2190,335],{"stretchy":334},[258,2192,341],{},[332,2194,354],{"separator":269},[258,2196,357],{},[332,2198,354],{"separator":269},[337,2200,2201,2203],{},[258,2202,341],{},[332,2204,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,2206,360],{"stretchy":334},[262,2208,2209],{"encoding":264},"(s,a,s')",[238,2211,2213],{"className":2212,"ariaHidden":269},[268],[238,2214,2216,2219,2222,2225,2228,2231,2234,2237,2240,2272],{"className":2215},[273],[238,2217],{"className":2218,"style":373},[277],[238,2220,335],{"className":2221},[381],[238,2223,341],{"className":2224},[282,283],[238,2226,354],{"className":2227},[451],[238,2229],{"className":2230,"style":455},[429],[238,2232,357],{"className":2233},[282,283],[238,2235,354],{"className":2236},[451],[238,2238],{"className":2239,"style":455},[429],[238,2241,2243,2246],{"className":2242},[282],[238,2244,341],{"className":2245},[282,283],[238,2247,2249],{"className":2248},[391],[238,2250,2252],{"className":2251},[395],[238,2253,2255],{"className":2254},[399],[238,2256,2258],{"className":2257,"style":404},[403],[238,2259,2260,2263],{"style":407},[238,2261],{"className":2262,"style":412},[411],[238,2264,2266],{"className":2265},[416,417,418,419],[238,2267,2269],{"className":2268},[282,419],[238,2270,346],{"className":2271},[282,419],[238,2273,360],{"className":2274},[462]," 成立。有限性假设保证最大值可以取到，后文会在第 7 节说明连续状态空间中的相应修改。",[11,2277,2278,2279,2282,2283,2314,2315,2382,2383,2427,2428,2494,2495,2523,2524,2552,2553,941,2581,2646,2647,2676,2677,2705,2706,2709,2710,1806],{},"一个",[1688,2280,2281],{},"策略"," ",[238,2284,2286,2301],{"className":2285,"translate":242},[241],[238,2287,2289],{"className":2288},[246],[248,2290,2291],{"xmlns":250},[252,2292,2293,2298],{},[255,2294,2295],{},[258,2296,2297],{},"π",[262,2299,2300],{"encoding":264},"\\pi",[238,2302,2304],{"className":2303,"ariaHidden":269},[268],[238,2305,2307,2310],{"className":2306},[273],[238,2308],{"className":2309,"style":489},[277],[238,2311,2297],{"className":2312,"style":2313},[282,283],"margin-right:0.0359em;"," 是映射 ",[238,2316,2318,2343],{"className":2317,"translate":242},[241],[238,2319,2321],{"className":2320},[246],[248,2322,2323],{"xmlns":250},[252,2324,2325,2340],{},[255,2326,2327,2329,2331,2334,2336,2338],{},[258,2328,260],{},[332,2330,617],{},[258,2332,2333],{"mathvariant":344},"Δ",[332,2335,335],{"stretchy":334},[258,2337,300],{},[332,2339,360],{"stretchy":334},[262,2341,2342],{"encoding":264},"S\\to\\Delta(A)",[238,2344,2346,2364],{"className":2345,"ariaHidden":269},[268],[238,2347,2349,2352,2355,2358,2361],{"className":2348},[273],[238,2350],{"className":2351,"style":278},[277],[238,2353,260],{"className":2354,"style":284},[282,283],[238,2356],{"className":2357,"style":430},[429],[238,2359,617],{"className":2360},[434],[238,2362],{"className":2363,"style":430},[429],[238,2365,2367,2370,2373,2376,2379],{"className":2366},[273],[238,2368],{"className":2369,"style":444},[277],[238,2371,2333],{"className":2372},[282],[238,2374,335],{"className":2375},[381],[238,2377,300],{"className":2378},[282,283],[238,2380,360],{"className":2381},[462],"，其中 ",[238,2384,2386,2406],{"className":2385,"translate":242},[241],[238,2387,2389],{"className":2388},[246],[248,2390,2391],{"xmlns":250},[252,2392,2393,2403],{},[255,2394,2395,2397,2399,2401],{},[258,2396,2333],{"mathvariant":344},[332,2398,335],{"stretchy":334},[258,2400,300],{},[332,2402,360],{"stretchy":334},[262,2404,2405],{"encoding":264},"\\Delta(A)",[238,2407,2409],{"className":2408,"ariaHidden":269},[268],[238,2410,2412,2415,2418,2421,2424],{"className":2411},[273],[238,2413],{"className":2414,"style":444},[277],[238,2416,2333],{"className":2417},[282],[238,2419,335],{"className":2420},[381],[238,2422,300],{"className":2423},[282,283],[238,2425,360],{"className":2426},[462]," 是动作集上的概率单纯形。",[238,2429,2431,2455],{"className":2430,"translate":242},[241],[238,2432,2434],{"className":2433},[246],[248,2435,2436],{"xmlns":250},[252,2437,2438,2452],{},[255,2439,2440,2442,2444,2446,2448,2450],{},[258,2441,2297],{},[332,2443,335],{"stretchy":334},[258,2445,357],{},[332,2447,349],{},[258,2449,341],{},[332,2451,360],{"stretchy":334},[262,2453,2454],{"encoding":264},"\\pi(a\\mid s)",[238,2456,2458,2482],{"className":2457,"ariaHidden":269},[268],[238,2459,2461,2464,2467,2470,2473,2476,2479],{"className":2460},[273],[238,2462],{"className":2463,"style":444},[277],[238,2465,2297],{"className":2466,"style":2313},[282,283],[238,2468,335],{"className":2469},[381],[238,2471,357],{"className":2472},[282,283],[238,2474],{"className":2475,"style":430},[429],[238,2477,349],{"className":2478},[434],[238,2480],{"className":2481,"style":430},[429],[238,2483,2485,2488,2491],{"className":2484},[273],[238,2486],{"className":2487,"style":444},[277],[238,2489,341],{"className":2490},[282,283],[238,2492,360],{"className":2493},[462]," 表示在状态 ",[238,2496,2498,2511],{"className":2497,"translate":242},[241],[238,2499,2501],{"className":2500},[246],[248,2502,2503],{"xmlns":250},[252,2504,2505,2509],{},[255,2506,2507],{},[258,2508,341],{},[262,2510,341],{"encoding":264},[238,2512,2514],{"className":2513,"ariaHidden":269},[268],[238,2515,2517,2520],{"className":2516},[273],[238,2518],{"className":2519,"style":489},[277],[238,2521,341],{"className":2522},[282,283]," 选择动作 ",[238,2525,2527,2540],{"className":2526,"translate":242},[241],[238,2528,2530],{"className":2529},[246],[248,2531,2532],{"xmlns":250},[252,2533,2534,2538],{},[255,2535,2536],{},[258,2537,357],{},[262,2539,357],{"encoding":264},[238,2541,2543],{"className":2542,"ariaHidden":269},[268],[238,2544,2546,2549],{"className":2545},[273],[238,2547],{"className":2548,"style":489},[277],[238,2550,357],{"className":2551},[282,283]," 的概率。若对每个 ",[238,2554,2556,2569],{"className":2555,"translate":242},[241],[238,2557,2559],{"className":2558},[246],[248,2560,2561],{"xmlns":250},[252,2562,2563,2567],{},[255,2564,2565],{},[258,2566,341],{},[262,2568,341],{"encoding":264},[238,2570,2572],{"className":2571,"ariaHidden":269},[268],[238,2573,2575,2578],{"className":2574},[273],[238,2576],{"className":2577,"style":489},[277],[238,2579,341],{"className":2580},[282,283],[238,2582,2584,2607],{"className":2583,"translate":242},[241],[238,2585,2587],{"className":2586},[246],[248,2588,2589],{"xmlns":250},[252,2590,2591,2605],{},[255,2592,2593,2595,2597,2599,2601,2603],{},[258,2594,2297],{},[332,2596,335],{"stretchy":334},[258,2598,357],{},[332,2600,349],{},[258,2602,341],{},[332,2604,360],{"stretchy":334},[262,2606,2454],{"encoding":264},[238,2608,2610,2634],{"className":2609,"ariaHidden":269},[268],[238,2611,2613,2616,2619,2622,2625,2628,2631],{"className":2612},[273],[238,2614],{"className":2615,"style":444},[277],[238,2617,2297],{"className":2618,"style":2313},[282,283],[238,2620,335],{"className":2621},[381],[238,2623,357],{"className":2624},[282,283],[238,2626],{"className":2627,"style":430},[429],[238,2629,349],{"className":2630},[434],[238,2632],{"className":2633,"style":430},[429],[238,2635,2637,2640,2643],{"className":2636},[273],[238,2638],{"className":2639,"style":444},[277],[238,2641,341],{"className":2642},[282,283],[238,2644,360],{"className":2645},[462]," 只在某一个动作处取 ",[238,2648,2650,2663],{"className":2649,"translate":242},[241],[238,2651,2653],{"className":2652},[246],[248,2654,2655],{"xmlns":250},[252,2656,2657,2661],{},[255,2658,2659],{},[738,2660,745],{},[262,2662,745],{"encoding":264},[238,2664,2666],{"className":2665,"ariaHidden":269},[268],[238,2667,2669,2673],{"className":2668},[273],[238,2670],{"className":2671,"style":2672},[277],"height:0.6444em;",[238,2674,745],{"className":2675},[282],"，则称 ",[238,2678,2680,2693],{"className":2679,"translate":242},[241],[238,2681,2683],{"className":2682},[246],[248,2684,2685],{"xmlns":250},[252,2686,2687,2691],{},[255,2688,2689],{},[258,2690,2297],{},[262,2692,2300],{"encoding":264},[238,2694,2696],{"className":2695,"ariaHidden":269},[268],[238,2697,2699,2702],{"className":2698},[273],[238,2700],{"className":2701,"style":489},[277],[238,2703,2297],{"className":2704,"style":2313},[282,283]," 为",[1688,2707,2708],{},"确定性策略","，并可将其视为映射 ",[238,2711,2713,2735],{"className":2712,"translate":242},[241],[238,2714,2716],{"className":2715},[246],[248,2717,2718],{"xmlns":250},[252,2719,2720,2732],{},[255,2721,2722,2724,2726,2728,2730],{},[258,2723,2297],{},[332,2725,603],{},[258,2727,260],{},[332,2729,617],{},[258,2731,300],{},[262,2733,2734],{"encoding":264},"\\pi:S\\to A",[238,2736,2738,2756,2774],{"className":2737,"ariaHidden":269},[268],[238,2739,2741,2744,2747,2750,2753],{"className":2740},[273],[238,2742],{"className":2743,"style":489},[277],[238,2745,2297],{"className":2746,"style":2313},[282,283],[238,2748],{"className":2749,"style":430},[429],[238,2751,603],{"className":2752},[434],[238,2754],{"className":2755,"style":430},[429],[238,2757,2759,2762,2765,2768,2771],{"className":2758},[273],[238,2760],{"className":2761,"style":278},[277],[238,2763,260],{"className":2764,"style":284},[282,283],[238,2766],{"className":2767,"style":430},[429],[238,2769,617],{"className":2770},[434],[238,2772],{"className":2773,"style":430},[429],[238,2775,2777,2780],{"className":2776},[273],[238,2778],{"className":2779,"style":278},[277],[238,2781,300],{"className":2782},[282,283],[225,2784,2786],{"id":2785},"_12-回报与值函数","1.2 回报与值函数",[230,2788,2791,3043,3325,3361],{"id":2789,"title":2790},"definition-return","定义 1.2（折扣回报）",[11,2792,2793,2794,3042],{},"设轨迹为 ",[238,2795,2797,2844],{"className":2796,"translate":242},[241],[238,2798,2800],{"className":2799},[246],[248,2801,2802],{"xmlns":250},[252,2803,2804,2841],{},[255,2805,2806,2812,2814,2820,2822,2828,2830,2836,2838],{},[812,2807,2808,2810],{},[258,2809,341],{},[738,2811,740],{},[332,2813,354],{"separator":269},[812,2815,2816,2818],{},[258,2817,357],{},[738,2819,740],{},[332,2821,354],{"separator":269},[812,2823,2824,2826],{},[258,2825,341],{},[738,2827,745],{},[332,2829,354],{"separator":269},[812,2831,2832,2834],{},[258,2833,357],{},[738,2835,745],{},[332,2837,354],{"separator":269},[332,2839,2840],{},"…",[262,2842,2843],{"encoding":264},"s_0,a_0,s_1,a_1,\\dots",[238,2845,2847],{"className":2846,"ariaHidden":269},[268],[238,2848,2850,2854,2894,2897,2900,2940,2943,2946,2986,2989,2992,3032,3035,3038],{"className":2849},[273],[238,2851],{"className":2852,"style":2853},[277],"height:0.625em;vertical-align:-0.1944em;",[238,2855,2857,2860],{"className":2856},[282],[238,2858,341],{"className":2859},[282,283],[238,2861,2863],{"className":2862},[391],[238,2864,2866,2886],{"className":2865},[395,848],[238,2867,2869,2883],{"className":2868},[399],[238,2870,2872],{"className":2871,"style":1162},[403],[238,2873,2874,2877],{"style":1165},[238,2875],{"className":2876,"style":412},[411],[238,2878,2880],{"className":2879},[416,417,418,419],[238,2881,740],{"className":2882},[282,419],[238,2884,872],{"className":2885},[871],[238,2887,2889],{"className":2888},[399],[238,2890,2892],{"className":2891,"style":879},[403],[238,2893],{},[238,2895,354],{"className":2896},[451],[238,2898],{"className":2899,"style":455},[429],[238,2901,2903,2906],{"className":2902},[282],[238,2904,357],{"className":2905},[282,283],[238,2907,2909],{"className":2908},[391],[238,2910,2912,2932],{"className":2911},[395,848],[238,2913,2915,2929],{"className":2914},[399],[238,2916,2918],{"className":2917,"style":1162},[403],[238,2919,2920,2923],{"style":1165},[238,2921],{"className":2922,"style":412},[411],[238,2924,2926],{"className":2925},[416,417,418,419],[238,2927,740],{"className":2928},[282,419],[238,2930,872],{"className":2931},[871],[238,2933,2935],{"className":2934},[399],[238,2936,2938],{"className":2937,"style":879},[403],[238,2939],{},[238,2941,354],{"className":2942},[451],[238,2944],{"className":2945,"style":455},[429],[238,2947,2949,2952],{"className":2948},[282],[238,2950,341],{"className":2951},[282,283],[238,2953,2955],{"className":2954},[391],[238,2956,2958,2978],{"className":2957},[395,848],[238,2959,2961,2975],{"className":2960},[399],[238,2962,2964],{"className":2963,"style":1162},[403],[238,2965,2966,2969],{"style":1165},[238,2967],{"className":2968,"style":412},[411],[238,2970,2972],{"className":2971},[416,417,418,419],[238,2973,745],{"className":2974},[282,419],[238,2976,872],{"className":2977},[871],[238,2979,2981],{"className":2980},[399],[238,2982,2984],{"className":2983,"style":879},[403],[238,2985],{},[238,2987,354],{"className":2988},[451],[238,2990],{"className":2991,"style":455},[429],[238,2993,2995,2998],{"className":2994},[282],[238,2996,357],{"className":2997},[282,283],[238,2999,3001],{"className":3000},[391],[238,3002,3004,3024],{"className":3003},[395,848],[238,3005,3007,3021],{"className":3006},[399],[238,3008,3010],{"className":3009,"style":1162},[403],[238,3011,3012,3015],{"style":1165},[238,3013],{"className":3014,"style":412},[411],[238,3016,3018],{"className":3017},[416,417,418,419],[238,3019,745],{"className":3020},[282,419],[238,3022,872],{"className":3023},[871],[238,3025,3027],{"className":3026},[399],[238,3028,3030],{"className":3029,"style":879},[403],[238,3031],{},[238,3033,354],{"className":3034},[451],[238,3036],{"className":3037,"style":455},[429],[238,3039,2840],{"className":3040},[3041],"minner","，记",[238,3044,3046],{"className":3045,"translate":242},[945],[238,3047,3049,3106],{"className":3048,"translate":242},[241],[238,3050,3052],{"className":3051},[246],[248,3053,3054],{"xmlns":250,"display":954},[252,3055,3056,3103],{},[255,3057,3058,3064,3066,3068,3070,3076,3078,3084,3086,3098,3100],{},[812,3059,3060,3062],{},[258,3061,600],{},[258,3063,820],{},[332,3065,992],{},[258,3067,600],{},[332,3069,335],{"stretchy":334},[812,3071,3072,3074],{},[258,3073,341],{},[258,3075,820],{},[332,3077,354],{"separator":269},[812,3079,3080,3082],{},[258,3081,357],{},[258,3083,820],{},[332,3085,354],{"separator":269},[812,3087,3088,3090],{},[258,3089,341],{},[255,3091,3092,3094,3096],{},[258,3093,820],{},[332,3095,987],{},[738,3097,745],{},[332,3099,360],{"stretchy":334},[258,3101,3102],{"mathvariant":344},".",[262,3104,3105],{"encoding":264},"r_t=r(s_t,a_t,s_{t+1}).",[238,3107,3109,3166],{"className":3108,"ariaHidden":269},[268],[238,3110,3112,3116,3157,3160,3163],{"className":3111},[273],[238,3113],{"className":3114,"style":3115},[277],"height:0.5806em;vertical-align:-0.15em;",[238,3117,3119,3122],{"className":3118},[282],[238,3120,600],{"className":3121,"style":637},[282,283],[238,3123,3125],{"className":3124},[391],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",[238,3329,3331,3344],{"className":3330,"translate":242},[241],[238,3332,3334],{"className":3333},[246],[248,3335,3336],{"xmlns":250},[252,3337,3338,3342],{},[255,3339,3340],{},[258,3341,820],{},[262,3343,820],{"encoding":264},[238,3345,3347],{"className":3346,"ariaHidden":269},[268],[238,3348,3350,3353],{"className":3349},[273],[238,3351],{"className":3352,"style":908},[277],[238,3354,820],{"className":3355},[282,283]," 开始的",[1688,3358,3359],{},"折扣回报","定义为",[238,3362,3364],{"className":3363,"translate":242},[945],[238,3365,3367,3425],{"className":3366,"translate":242},[241],[238,3368,3370],{"className":3369},[246],[248,3371,3372],{"xmlns":250,"display":954},[252,3373,3374,3422],{},[255,3375,3376,3383,3385,3402,3408,3420],{},[812,3377,3378,3381],{},[258,3379,3380],{},"G",[258,3382,820],{},[332,3384,992],{},[3386,3387,3388,3391,3400],"munderover",{},[332,3389,3390],{},"∑",[255,3392,3393,3396,3398],{},[258,3394,3395],{},"k",[332,3397,992],{},[738,3399,740],{},[258,3401,1895],{"mathvariant":344},[337,3403,3404,3406],{},[258,3405,730],{},[258,3407,3395],{},[812,3409,3410,3412],{},[258,3411,600],{},[255,3413,3414,3416,3418],{},[258,3415,820],{},[332,3417,987],{},[258,3419,3395],{},[258,3421,3102],{"mathvariant":344},[262,3423,3424],{"encoding":264},"G_t=\\sum_{k=0}^{\\infty}\\gamma^k r_{t+k}.",[238,3426,3428,3483],{"className":3427,"ariaHidden":269},[268],[238,3429,3431,3434,3474,3477,3480],{"className":3430},[273],[238,3432],{"className":3433,"style":833},[277],[238,3435,3437,3440],{"className":3436},[282],[238,3438,3380],{"className":3439},[282,283],[238,3441,3443],{"className":3442},[391],[238,3444,3446,3466],{"className":3445},[395,848],[238,3447,3449,3463],{"className":3448},[399],[238,3450,3452],{"className":3451,"style":855},[403],[238,3453,3454,3457],{"style":1165},[238,3455],{"className":3456,"style":412},[411],[238,3458,3460],{"className":3459},[416,417,418,419],[238,3461,820],{"className":3462},[282,283,419],[238,3464,872],{"className":3465},[871],[238,3467,3469],{"className":3468},[399],[238,3470,3472],{"className":3471,"style":879},[403],[238,3473],{},[238,3475],{"className":3476,"style":430},[429],[238,3478,992],{"className":3479},[434],[238,3481],{"className":3482,"style":430},[429],[238,3484,3486,3490,3567,3570,3600,3650],{"className":3485},[273],[238,3487],{"className":3488,"style":3489},[277],"height:2.9535em;vertical-align:-1.3021em;",[238,3491,3494],{"className":3492},[1943,3493],"op-limits",[238,3495,3497,3558],{"className":3496},[395,848],[238,3498,3500,3555],{"className":3499},[399],[238,3501,3504,3527,3540],{"className":3502,"style":3503},[403],"height:1.6514em;",[238,3505,3507,3511],{"style":3506},"top:-1.8479em;margin-left:0em;",[238,3508],{"className":3509,"style":3510},[411],"height:3.05em;",[238,3512,3514],{"className":3513},[416,417,418,419],[238,3515,3517,3521,3524],{"className":3516},[282,419],[238,3518,3395],{"className":3519,"style":3520},[282,283,419],"margin-right:0.0315em;",[238,3522,992],{"className":3523},[434,419],[238,3525,740],{"className":3526},[282,419],[238,3528,3530,3533],{"style":3529},"top:-3.05em;",[238,3531],{"className":3532,"style":3510},[411],[238,3534,3535],{},[238,3536,3390],{"className":3537},[1943,3538,3539],"op-symbol","large-op",[238,3541,3543,3546],{"style":3542},"top:-4.3em;margin-left:0em;",[238,3544],{"className":3545,"style":3510},[411],[238,3547,3549],{"className":3548},[416,417,418,419],[238,3550,3552],{"className":3551},[282,419],[238,3553,1895],{"className":3554},[282,419],[238,3556,872],{"className":3557},[871],[238,3559,3561],{"className":3560},[399],[238,3562,3565],{"className":3563,"style":3564},[403],"height:1.3021em;",[238,3566],{},[238,3568],{"className":3569,"style":455},[429],[238,3571,3573,3576],{"className":3572},[282],[238,3574,730],{"className":3575,"style":764},[282,283],[238,3577,3579],{"className":3578},[391],[238,3580,3582],{"className":3581},[395],[238,3583,3585],{"className":3584},[399],[238,3586,3589],{"className":3587,"style":3588},[403],"height:0.8991em;",[238,3590,3591,3594],{"style":1226},[238,3592],{"className":3593,"style":412},[411],[238,3595,3597],{"className":3596},[416,417,418,419],[238,3598,3395],{"className":3599,"style":3520},[282,283,419],[238,3601,3603,3606],{"className":3602},[282],[238,3604,600],{"className":3605,"style":637},[282,283],[238,3607,3609],{"className":3608},[391],[238,3610,3612,3642],{"className":3611},[395,848],[238,3613,3615,3639],{"className":3614},[399],[238,3616,3619],{"className":3617,"style":3618},[403],"height:0.3361em;",[238,3620,3621,3624],{"style":3136},[238,3622],{"className":3623,"style":412},[411],[238,3625,3627],{"className":3626},[416,417,418,419],[238,3628,3630,3633,3636],{"className":3629},[282,419],[238,3631,820],{"className":3632},[282,283,419],[238,3634,987],{"className":3635},[664,419],[238,3637,3395],{"className":3638,"style":3520},[282,283,419],[238,3640,872],{"className":3641},[871],[238,3643,3645],{"className":3644},[399],[238,3646,3648],{"className":3647,"style":1193},[403],[238,3649],{},[238,3651,3102],{"className":3652},[282],[11,3654,3655,3656,3726,3727,3797,3798,3884,3885,4019,4020,4357],{},"这里采用「转移奖励」约定，即 ",[238,3657,3659,3677],{"className":3658,"translate":242},[241],[238,3660,3662],{"className":3661},[246],[248,3663,3664],{"xmlns":250},[252,3665,3666,3674],{},[255,3667,3668],{},[812,3669,3670,3672],{},[258,3671,600],{},[258,3673,820],{},[262,3675,3676],{"encoding":264},"r_t",[238,3678,3680],{"className":3679,"ariaHidden":269},[268],[238,3681,3683,3686],{"className":3682},[273],[238,3684],{"className":3685,"style":3115},[277],[238,3687,3689,3692],{"className":3688},[282],[238,3690,600],{"className":3691,"style":637},[282,283],[238,3693,3695],{"className":3694},[391],[238,3696,3698,3718],{"className":3697},[395,848],[238,3699,3701,3715],{"className":3700},[399],[238,3702,3704],{"className":3703,"style":855},[403],[238,3705,3706,3709],{"style":3136},[238,3707],{"className":3708,"style":412},[411],[238,3710,3712],{"className":3711},[416,417,418,419],[238,3713,820],{"className":3714},[282,283,419],[238,3716,872],{"className":3717},[871],[238,3719,3721],{"className":3720},[399],[238,3722,3724],{"className":3723,"style":879},[403],[238,3725],{}," 是在从 ",[238,3728,3730,3748],{"className":3729,"translate":242},[241],[238,3731,3733],{"className":3732},[246],[248,3734,3735],{"xmlns":250},[252,3736,3737,3745],{},[255,3738,3739],{},[812,3740,3741,3743],{},[258,3742,341],{},[258,3744,820],{},[262,3746,3747],{"encoding":264},"s_t",[238,3749,3751],{"className":3750,"ariaHidden":269},[268],[238,3752,3754,3757],{"className":3753},[273],[238,3755],{"className":3756,"style":3115},[277],[238,3758,3760,3763],{"className":3759},[282],[238,3761,341],{"className":3762},[282,283],[238,3764,3766],{"className":3765},[391],[238,3767,3769,3789],{"className":3768},[395,848],[238,3770,3772,3786],{"className":3771},[399],[238,3773,3775],{"className":3774,"style":855},[403],[238,3776,3777,3780],{"style":1165},[238,3778],{"className":3779,"style":412},[411],[238,3781,3783],{"className":3782},[416,417,418,419],[238,3784,820],{"className":3785},[282,283,419],[238,3787,872],{"className":3788},[871],[238,3790,3792],{"className":3791},[399],[238,3793,3795],{"className":3794,"style":879},[403],[238,3796],{}," 转移到 ",[238,3799,3801,3825],{"className":3800,"translate":242},[241],[238,3802,3804],{"className":3803},[246],[248,3805,3806],{"xmlns":250},[252,3807,3808,3822],{},[255,3809,3810],{},[812,3811,3812,3814],{},[258,3813,341],{},[255,3815,3816,3818,3820],{},[258,3817,820],{},[332,3819,987],{},[738,3821,745],{},[262,3823,3824],{"encoding":264},"s_{t+1}",[238,3826,3828],{"className":3827,"ariaHidden":269},[268],[238,3829,3831,3835],{"className":3830},[273],[238,3832],{"className":3833,"style":3834},[277],"height:0.6389em;vertical-align:-0.2083em;",[238,3836,3838,3841],{"className":3837},[282],[238,3839,341],{"className":3840},[282,283],[238,3842,3844],{"className":3843},[391],[238,3845,3847,3876],{"className":3846},[395,848],[238,3848,3850,3873],{"className":3849},[399],[238,3851,3853],{"className":3852,"style":1162},[403],[238,3854,3855,3858],{"style":1165},[238,3856],{"className":3857,"style":412},[411],[238,3859,3861],{"className":3860},[416,417,418,419],[238,3862,3864,3867,3870],{"className":3863},[282,419],[238,3865,820],{"className":3866},[282,283,419],[238,3868,987],{"className":3869},[664,419],[238,3871,745],{"className":3872},[282,419],[238,3874,872],{"className":3875},[871],[238,3877,3879],{"className":3878},[399],[238,3880,3882],{"className":3881,"style":1193},[403],[238,3883],{}," 时获得的奖励。若奖励只依赖于 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成立，则对任意轨迹与任意 ",[238,5506,5508,5521],{"className":5507,"translate":242},[241],[238,5509,5511],{"className":5510},[246],[248,5512,5513],{"xmlns":250},[252,5514,5515,5519],{},[255,5516,5517],{},[258,5518,820],{},[262,5520,820],{"encoding":264},[238,5522,5524],{"className":5523,"ariaHidden":269},[268],[238,5525,5527,5530],{"className":5526},[273],[238,5528],{"className":5529,"style":908},[277],[238,5531,820],{"className":5532},[282,283],[238,5534,5536],{"className":5535,"translate":242},[945],[238,5537,5539,5585],{"className":5538,"translate":242},[241],[238,5540,5542],{"className":5541},[246],[248,5543,5544],{"xmlns":250,"display":954},[252,5545,5546,5582],{},[255,5547,5548,5550,5556,5558,5560,5580],{},[258,5549,349],{"mathvariant":344},[812,5551,5552,5554],{},[258,5553,3380],{},[258,5555,820],{},[258,5557,349],{"mathvariant":344},[332,5559,2023],{},[5561,5562,5563,5571],"mfrac",{},[812,5564,5565,5567,5569],{},[258,5566,621],{},[258,5568,1886],{},[332,5570,1889],{},[255,5572,5573,5575,5578],{},[738,5574,745],{},[332,5576,5577],{},"−",[258,5579,730],{},[258,5581,3102],{"mathvariant":344},[262,5583,5584],{"encoding":264},"|G_t|\\le\\frac{R_{\\max}}{1-\\gamma}.",[238,5586,5588,5649],{"className":5587,"ariaHidden":269},[268],[238,5589,5591,5594,5597,5637,5640,5643,5646],{"className":5590},[273],[238,5592],{"className":5593,"style":444},[277],[238,5595,349],{"className":5596},[282],[238,5598,5600,5603],{"className":5599},[282],[238,5601,3380],{"className":5602},[282,283],[238,5604,5606],{"className":5605},[391],[238,5607,5609,5629],{"className":5608},[395,848],[238,5610,5612,5626],{"className":5611},[399],[238,5613,5615],{"className":5614,"style":855},[403],[238,5616,5617,5620],{"style":1165},[238,5618],{"className":5619,"style":412},[411],[238,5621,5623],{"className":5622},[416,417,418,419],[238,5624,820],{"className":5625},[282,283,419],[238,5627,872],{"className":5628},[871],[238,5630,5632],{"className":5631},[399],[238,5633,5635],{"className":5634,"style":879},[403],[238,5636],{},[238,5638,349],{"className":5639},[282],[238,5641],{"className":5642,"style":430},[429],[238,5644,2023],{"className":5645},[434],[238,5647],{"className":5648,"style":430},[429],[238,5650,5652,5656,5787],{"className":5651},[273],[238,5653],{"className":5654,"style":5655},[277],"height:2.2408em;vertical-align:-0.8804em;",[238,5657,5659,5663,5784],{"className":5658},[282],[238,5660],{"className":5661},[381,5662],"nulldelimiter",[238,5664,5666],{"className":5665},[5561],[238,5667,5669,5775],{"className":5668},[395,848],[238,5670,5672,5772],{"className":5671},[399],[238,5673,5676,5700,5711],{"className":5674,"style":5675},[403],"height:1.3603em;",[238,5677,5679,5682],{"style":5678},"top:-2.314em;",[238,5680],{"className":5681,"style":1134},[411],[238,5683,5685,5688,5691,5694,5697],{"className":5684},[282],[238,5686,745],{"className":5687},[282],[238,5689],{"className":5690,"style":660},[429],[238,5692,5577],{"className":5693},[664],[238,5695],{"className":5696,"style":660},[429],[238,5698,730],{"className":5699,"style":764},[282,283],[238,5701,5703,5706],{"style":5702},"top:-3.23em;",[238,5704],{"className":5705,"style":1134},[411],[238,5707],{"className":5708,"style":5710},[5709],"frac-line","border-bottom-width:0.04em;",[238,5712,5714,5717],{"style":5713},"top:-3.677em;",[238,5715],{"className":5716,"style":1134},[411],[238,5718,5720],{"className":5719},[282],[238,5721,5723,5726],{"className":5722},[282],[238,5724,621],{"className":5725,"style":1914},[282,283],[238,5727,5729],{"className":5728},[391],[238,5730,5732,5764],{"className":5731},[395,848],[238,5733,5735,5761],{"className":5734},[399],[238,5736,5738],{"className":5737,"style":1927},[403],[238,5739,5740,5743],{"style":1930},[238,5741],{"className":5742,"style":412},[411],[238,5744,5746],{"className":5745},[416,417,418,419],[238,5747,5749],{"className":5748},[282,419],[238,5750,5752,5755,5758],{"className":5751},[1943,419],[238,5753,1947],{"className":5754},[419],[238,5756,357],{"className":5757},[419],[238,5759,1954],{"className":5760},[419],[238,5762,872],{"className":5763},[871],[238,5765,5767],{"className":5766},[399],[238,5768,5770],{"className":5769,"style":879},[403],[238,5771],{},[238,5773,872],{"className":5774},[871],[238,5776,5778],{"className":5777},[399],[238,5779,5782],{"className":5780,"style":5781},[403],"height:0.8804em;",[238,5783],{},[238,5785],{"className":5786},[462,5662],[238,5788,3102],{"className":5789},[282],[11,5791,5792,5795],{},[1688,5793,5794],{},"证明"," 由三角不等式与几何级数求和，",[238,5797,5799],{"className":5798,"translate":242},[945],[238,5800,5802,5946],{"className":5801,"translate":242},[241],[238,5803,5805],{"className":5804},[246],[248,5806,5807],{"xmlns":250,"display":954},[252,5808,5809,5943],{},[958,5810,5811,5875],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,5812,5813,5829],{},[966,5814,5815],{},[969,5816,5817],{"scriptlevel":740,"displaystyle":269},[255,5818,5819,5821,5827],{},[258,5820,349],{"mathvariant":344},[812,5822,5823,5825],{},[258,5824,3380],{},[258,5826,820],{},[258,5828,349],{"mathvariant":344},[966,5830,5831],{},[969,5832,5833],{"scriptlevel":740,"displaystyle":269},[255,5834,5835,5837,5839,5853,5859,5861,5873],{},[255,5836],{},[332,5838,2023],{},[3386,5840,5841,5843,5851],{},[332,5842,3390],{},[255,5844,5845,5847,5849],{},[258,5846,3395],{},[332,5848,992],{},[738,5850,740],{},[258,5852,1895],{"mathvariant":344},[337,5854,5855,5857],{},[258,5856,730],{},[258,5858,3395],{},[258,5860,349],{"mathvariant":344},[812,5862,5863,5865],{},[258,5864,600],{},[255,5866,5867,5869,5871],{},[258,5868,820],{},[332,5870,987],{},[258,5872,3395],{},[258,5874,349],{"mathvariant":344},[963,5876,5877,5883],{},[966,5878,5879],{},[969,5880,5881],{"scriptlevel":740,"displaystyle":269},[255,5882],{},[966,5884,5885],{},[969,5886,5887],{"scriptlevel":740,"displaystyle":269},[255,5888,5889,5891,5893,5901,5915,5921,5923,5941],{},[255,5890],{},[332,5892,2023],{},[812,5894,5895,5897,5899],{},[258,5896,621],{},[258,5898,1886],{},[332,5900,1889],{},[3386,5902,5903,5905,5913],{},[332,5904,3390],{},[255,5906,5907,5909,5911],{},[258,5908,3395],{},[332,5910,992],{},[738,5912,740],{},[258,5914,1895],{"mathvariant":344},[337,5916,5917,5919],{},[258,5918,730],{},[258,5920,3395],{},[332,5922,992],{},[5561,5924,5925,5933],{},[812,5926,5927,5929,5931],{},[258,5928,621],{},[258,5930,1886],{},[332,5932,1889],{},[255,5934,5935,5937,5939],{},[738,5936,745],{},[332,5938,5577],{},[258,5940,730],{},[258,5942,3102],{"mathvariant":344},[262,5944,5945],{"encoding":264},"\\begin{aligned}\n|G_t|\n&\\le\\sum_{k=0}^{\\infty}\\gamma^k|r_{t+k}|\\\\\n&\\le R_{\\max}\\sum_{k=0}^{\\infty}\\gamma^k\n=\\frac{R_{\\max}}{1-\\gamma}.\n\\end{aligned}",[238,5947,5949],{"className":5948,"ariaHidden":269},[268],[238,5950,5952,5956],{"className":5951},[273],[238,5953],{"className":5954,"style":5955},[277],"height:6.207em;vertical-align:-2.8535em;",[238,5957,5959],{"className":5958},[282],[238,5960,5962,6052],{"className":5961},[958],[238,5963,5965],{"className":5964},[1117],[238,5966,5968,6043],{"className":5967},[395,848],[238,5969,5971,6040],{"className":5970},[399],[238,5972,5975,6031],{"className":5973,"style":5974},[403],"height:3.3535em;",[238,5976,5978,5982],{"style":5977},"top:-5.3535em;",[238,5979],{"className":5980,"style":5981},[411],"height:3.6514em;",[238,5983,5985,5988,6028],{"className":5984},[282],[238,5986,349],{"className":5987},[282],[238,5989,5991,5994],{"className":5990},[282],[238,5992,3380],{"className":5993},[282,283],[238,5995,5997],{"className":5996},[391],[238,5998,6000,6020],{"className":5999},[395,848],[238,6001,6003,6017],{"className":6002},[399],[238,6004,6006],{"className":6005,"style":855},[403],[238,6007,6008,6011],{"style":1165},[238,6009],{"className":6010,"style":412},[411],[238,6012,6014],{"className":6013},[416,417,418,419],[238,6015,820],{"className":6016},[282,283,419],[238,6018,872],{"className":6019},[871],[238,6021,6023],{"className":6022},[399],[238,6024,6026],{"className":6025,"style":879},[403],[238,6027],{},[238,6029,349],{"className":6030},[282],[238,6032,6034,6037],{"style":6033},"top:-2.1em;",[238,6035],{"className":6036,"style":5981},[411],[238,6038],{"className":6039},[282],[238,6041,872],{"className":6042},[871],[238,6044,6046],{"className":6045},[399],[238,6047,6050],{"className":6048,"style":6049},[403],"height:2.8535em;",[238,6051],{},[238,6053,6055],{"className":6054},[1422],[238,6056,6058,6550],{"className":6057},[395,848],[238,6059,6061,6547],{"className":6060},[399],[238,6062,6064,6238],{"className":6063,"style":5974},[403],[238,6065,6066,6069],{"style":5977},[238,6067],{"className":6068,"style":5981},[411],[238,6070,6072,6075,6078,6081,6084,6151,6154,6183,6186,6235],{"className":6071},[282],[238,6073],{"className":6074},[282],[238,6076],{"className":6077,"style":430},[429],[238,6079,2023],{"className":6080},[434],[238,6082],{"className":6083,"style":430},[429],[238,6085,6087],{"className":6086},[1943,3493],[238,6088,6090,6143],{"className":6089},[395,848],[238,6091,6093,6140],{"className":6092},[399],[238,6094,6096,6116,6126],{"className":6095,"style":3503},[403],[238,6097,6098,6101],{"style":3506},[238,6099],{"className":6100,"style":3510},[411],[238,6102,6104],{"className":6103},[416,417,418,419],[238,6105,6107,6110,6113],{"className":6106},[282,419],[238,6108,3395],{"className":6109,"style":3520},[282,283,419],[238,6111,992],{"className":6112},[434,419],[238,6114,740],{"className":6115},[282,419],[238,6117,6118,6121],{"style":3529},[238,6119],{"className":6120,"style":3510},[411],[238,6122,6123],{},[238,6124,3390],{"className":6125},[1943,3538,3539],[238,6127,6128,6131],{"style":3542},[238,6129],{"className":6130,"style":3510},[411],[238,6132,6134],{"className":6133},[416,417,418,419],[238,6135,6137],{"className":6136},[282,419],[238,6138,1895],{"className":6139},[282,419],[238,6141,872],{"className":6142},[871],[238,6144,6146],{"className":6145},[399],[238,6147,6149],{"className":6148,"style":3564},[403],[238,6150],{},[238,6152],{"className":6153,"style":455},[429],[238,6155,6157,6160],{"className":6156},[282],[238,6158,730],{"className":6159,"style":764},[282,283],[238,6161,6163],{"className":6162},[391],[238,6164,6166],{"className":6165},[395],[238,6167,6169],{"className":6168},[399],[238,6170,6172],{"className":6171,"style":3588},[403],[238,6173,6174,6177],{"style":1226},[238,6175],{"className":6176,"style":412},[411],[238,6178,6180],{"className":6179},[416,417,418,419],[238,6181,3395],{"className":6182,"style":3520},[282,283,419],[238,6184,349],{"className":6185},[282],[238,6187,6189,6192],{"className":6188},[282],[238,6190,600],{"className":6191,"style":637},[282,283],[238,6193,6195],{"className":6194},[391],[238,6196,6198,6227],{"className":6197},[395,848],[238,6199,6201,6224],{"className":6200},[399],[238,6202,6204],{"className":6203,"style":3618},[403],[238,6205,6206,6209],{"style":3136},[238,6207],{"className":6208,"style":412},[411],[238,6210,6212],{"className":6211},[416,417,418,419],[238,6213,6215,6218,6221],{"className":6214},[282,419],[238,6216,820],{"className":6217},[282,283,419],[238,6219,987],{"className":6220},[664,419],[238,6222,3395],{"className":6223,"style":3520},[282,283,419],[238,6225,872],{"className":6226},[871],[238,6228,6230],{"className":6229},[399],[238,6231,6233],{"className":6232,"style":1193},[403],[238,6234],{},[238,6236,349],{"className":6237},[282],[238,6239,6240,6243],{"style":6033},[238,6241],{"className":6242,"style":5981},[411],[238,6244,6246,6249,6252,6255,6258,6310,6313,6380,6383,6412,6415,6418,6421,6544],{"className":6245},[282],[238,6247],{"className":6248},[282],[238,6250],{"className":6251,"style":430},[429],[238,6253,2023],{"className":6254},[434],[238,6256],{"className":6257,"style":430},[429],[238,6259,6261,6264],{"className":6260},[282],[238,6262,621],{"className":6263,"style":1914},[282,283],[238,6265,6267],{"className":6266},[391],[238,6268,6270,6302],{"className":6269},[395,848],[238,6271,6273,6299],{"className":6272},[399],[238,6274,6276],{"className":6275,"style":1927},[403],[238,6277,6278,6281],{"style":1930},[238,6279],{"className":6280,"style":412},[411],[238,6282,6284],{"className":6283},[416,417,418,419],[238,6285,6287],{"className":6286},[282,419],[238,6288,6290,6293,6296],{"className":6289},[1943,419],[238,6291,1947],{"className":6292},[419],[238,6294,357],{"className":6295},[419],[238,6297,1954],{"className":6298},[419],[238,6300,872],{"className":6301},[871],[238,6303,6305],{"className":6304},[399],[238,6306,6308],{"className":6307,"style":879},[403],[238,6309],{},[238,6311],{"className":6312,"style":455},[429],[238,6314,6316],{"className":6315},[1943,3493],[238,6317,6319,6372],{"className":6318},[395,848],[238,6320,6322,6369],{"className":6321},[399],[238,6323,6325,6345,6355],{"className":6324,"style":3503},[403],[238,6326,6327,6330],{"style":3506},[238,6328],{"className":6329,"style":3510},[411],[238,6331,6333],{"className":6332},[416,417,418,419],[238,6334,6336,6339,6342],{"className":6335},[282,419],[238,6337,3395],{"className":6338,"style":3520},[282,283,419],[238,6340,992],{"className":6341},[434,419],[238,6343,740],{"className":6344},[282,419],[238,6346,6347,6350],{"style":3529},[238,6348],{"className":6349,"style":3510},[411],[238,6351,6352],{},[238,6353,3390],{"className":6354},[1943,3538,3539],[238,6356,6357,6360],{"style":3542},[238,6358],{"className":6359,"style":3510},[411],[238,6361,6363],{"className":6362},[416,417,418,419],[238,6364,6366],{"className":6365},[282,419],[238,6367,1895],{"className":6368},[282,419],[238,6370,872],{"className":6371},[871],[238,6373,6375],{"className":6374},[399],[238,6376,6378],{"className":6377,"style":3564},[403],[238,6379],{},[238,6381],{"className":6382,"style":455},[429],[238,6384,6386,6389],{"className":6385},[282],[238,6387,730],{"className":6388,"style":764},[282,283],[238,6390,6392],{"className":6391},[391],[238,6393,6395],{"className":6394},[395],[238,6396,6398],{"className":6397},[399],[238,6399,6401],{"className":6400,"style":3588},[403],[238,6402,6403,6406],{"style":1226},[238,6404],{"className":6405,"style":412},[411],[238,6407,6409],{"className":6408},[416,417,418,419],[238,6410,3395],{"className":6411,"style":3520},[282,283,419],[238,6413],{"className":6414,"style":430},[429],[238,6416,992],{"className":6417},[434],[238,6419],{"className":6420,"style":430},[429],[238,6422,6424,6427,6541],{"className":6423},[282],[238,6425],{"className":6426},[381,5662],[238,6428,6430],{"className":6429},[5561],[238,6431,6433,6533],{"className":6432},[395,848],[238,6434,6436,6530],{"className":6435},[399],[238,6437,6439,6462,6470],{"className":6438,"style":5675},[403],[238,6440,6441,6444],{"style":5678},[238,6442],{"className":6443,"style":1134},[411],[238,6445,6447,6450,6453,6456,6459],{"className":6446},[282],[238,6448,745],{"className":6449},[282],[238,6451],{"className":6452,"style":660},[429],[238,6454,5577],{"className":6455},[664],[238,6457],{"className":6458,"style":660},[429],[238,6460,730],{"className":6461,"style":764},[282,283],[238,6463,6464,6467],{"style":5702},[238,6465],{"className":6466,"style":1134},[411],[238,6468],{"className":6469,"style":5710},[5709],[238,6471,6472,6475],{"style":5713},[238,6473],{"className":6474,"style":1134},[411],[238,6476,6478],{"className":6477},[282],[238,6479,6481,6484],{"className":6480},[282],[238,6482,621],{"className":6483,"style":1914},[282,283],[238,6485,6487],{"className":6486},[391],[238,6488,6490,6522],{"className":6489},[395,848],[238,6491,6493,6519],{"className":6492},[399],[238,6494,6496],{"className":6495,"style":1927},[403],[238,6497,6498,6501],{"style":1930},[238,6499],{"className":6500,"style":412},[411],[238,6502,6504],{"className":6503},[416,417,418,419],[238,6505,6507],{"className":6506},[282,419],[238,6508,6510,6513,6516],{"className":6509},[1943,419],[238,6511,1947],{"className":6512},[419],[238,6514,357],{"className":6515},[419],[238,6517,1954],{"className":6518},[419],[238,6520,872],{"className":6521},[871],[238,6523,6525],{"className":6524},[399],[238,6526,6528],{"className":6527,"style":879},[403],[238,6529],{},[238,6531,872],{"className":6532},[871],[238,6534,6536],{"className":6535},[399],[238,6537,6539],{"className":6538,"style":5781},[403],[238,6540],{},[238,6542],{"className":6543},[462,5662],[238,6545,3102],{"className":6546},[282],[238,6548,872],{"className":6549},[871],[238,6551,6553],{"className":6552},[399],[238,6554,6556],{"className":6555,"style":6049},[403],[238,6557],{},[11,6559,6560,6561,1838,6621,6681],{},"因此 ",[238,6562,6564,6582],{"className":6563,"translate":242},[241],[238,6565,6567],{"className":6566},[246],[248,6568,6569],{"xmlns":250},[252,6570,6571,6579],{},[255,6572,6573],{},[337,6574,6575,6577],{},[258,6576,4429],{},[258,6578,2297],{},[262,6580,6581],{"encoding":264},"V^\\pi",[238,6583,6585],{"className":6584,"ariaHidden":269},[268],[238,6586,6588,6591],{"className":6587},[273],[238,6589],{"className":6590,"style":278},[277],[238,6592,6594,6597],{"className":6593},[282],[238,6595,4429],{"className":6596,"style":660},[282,283],[238,6598,6600],{"className":6599},[391],[238,6601,6603],{"className":6602},[395],[238,6604,6606],{"className":6605},[399],[238,6607,6610],{"className":6608,"style":6609},[403],"height:0.6644em;",[238,6611,6612,6615],{"style":407},[238,6613],{"className":6614,"style":412},[411],[238,6616,6618],{"className":6617},[416,417,418,419],[238,6619,2297],{"className":6620,"style":2313},[282,283,419],[238,6622,6624,6642],{"className":6623,"translate":242},[241],[238,6625,6627],{"className":6626},[246],[248,6628,6629],{"xmlns":250},[252,6630,6631,6639],{},[255,6632,6633],{},[337,6634,6635,6637],{},[258,6636,4494],{},[258,6638,2297],{},[262,6640,6641],{"encoding":264},"Q^\\pi",[238,6643,6645],{"className":6644,"ariaHidden":269},[268],[238,6646,6648,6652],{"className":6647},[273],[238,6649],{"className":6650,"style":6651},[277],"height:0.8778em;vertical-align:-0.1944em;",[238,6653,6655,6658],{"className":6654},[282],[238,6656,4494],{"className":6657},[282,283],[238,6659,6661],{"className":6660},[391],[238,6662,6664],{"className":6663},[395],[238,6665,6667],{"className":6666},[399],[238,6668,6670],{"className":6669,"style":6609},[403],[238,6671,6672,6675],{"style":407},[238,6673],{"className":6674,"style":412},[411],[238,6676,6678],{"className":6677},[416,417,418,419],[238,6679,2297],{"className":6680,"style":2313},[282,283,419]," 均为有界函数。证毕。",[11,6683,6684],{},"有界性并不是一个无关紧要的技术条件。它保证值函数属于一个有界函数空间，而该空间在 sup 范数下是完备的。后文的压缩映射论证正是在这个空间上进行的。",[225,6686,6688],{"id":6687},"_13-两个基本恒等式","1.3 两个基本恒等式",[230,6690,6693,6725,6990,6993],{"id":6691,"title":6692},"proposition-basic-identities","命题 1.5（值函数的基本恒等式）",[11,6694,6695,6696,6724],{},"对任意策略 ",[238,6697,6699,6712],{"className":6698,"translate":242},[241],[238,6700,6702],{"className":6701},[246],[248,6703,6704],{"xmlns":250},[252,6705,6706,6710],{},[255,6707,6708],{},[258,6709,2297],{},[262,6711,2300],{"encoding":264},[238,6713,6715],{"className":6714,"ariaHidden":269},[268],[238,6716,6718,6721],{"className":6717},[273],[238,6719],{"className":6720,"style":489},[277],[238,6722,2297],{"className":6723,"style":2313},[282,283],"，有",[238,6726,6728],{"className":6727,"translate":242},[945],[238,6729,6731,6794],{"className":6730,"translate":242},[241],[238,6732,6734],{"className":6733},[246],[248,6735,6736],{"xmlns":250,"display":954},[252,6737,6738,6791],{},[255,6739,6740,6746,6748,6750,6752,6754,6761,6763,6765,6767,6769,6771,6773,6779,6781,6783,6785,6787,6789],{},[337,6741,6742,6744],{},[258,6743,4429],{},[258,6745,2297],{},[332,6747,335],{"stretchy":334},[258,6749,341],{},[332,6751,360],{"stretchy":334},[332,6753,992],{},[6755,6756,6757,6759],"munder",{},[332,6758,3390],{},[258,6760,357],{},[258,6762,2297],{},[332,6764,335],{"stretchy":334},[258,6766,357],{},[332,6768,349],{},[258,6770,341],{},[332,6772,360],{"stretchy":334},[337,6774,6775,6777],{},[258,6776,4494],{},[258,6778,2297],{},[332,6780,335],{"stretchy":334},[258,6782,341],{},[332,6784,354],{"separator":269},[258,6786,357],{},[332,6788,360],{"stretchy":334},[332,6790,354],{"separator":269},[262,6792,6793],{"encoding":264},"V^\\pi(s)=\\sum_{a}\\pi(a\\mid s)Q^\\pi(s,a),",[238,6795,6797,6850,6928],{"className":6796,"ariaHidden":269},[268],[238,6798,6800,6803,6832,6835,6838,6841,6844,6847],{"className":6799},[273],[238,6801],{"className":6802,"style":444},[277],[238,6804,6806,6809],{"className":6805},[282],[238,6807,4429],{"className":6808,"style":660},[282,283],[238,6810,6812],{"className":6811},[391],[238,6813,6815],{"className":6814},[395],[238,6816,6818],{"className":6817},[399],[238,6819,6821],{"className":6820,"style":4620},[403],[238,6822,6823,6826],{"style":1226},[238,6824],{"className":6825,"style":412},[411],[238,6827,6829],{"className":6828},[416,417,418,419],[238,6830,2297],{"className":6831,"style":2313},[282,283,419],[238,6833,335],{"className":6834},[381],[238,6836,341],{"className":6837},[282,283],[238,6839,360],{"className":6840},[462],[238,6842],{"className":6843,"style":430},[429],[238,6845,992],{"className":6846},[434],[238,6848],{"className":6849,"style":430},[429],[238,6851,6853,6857,6907,6910,6913,6916,6919,6922,6925],{"className":6852},[273],[238,6854],{"className":6855,"style":6856},[277],"height:2.3em;vertical-align:-1.25em;",[238,6858,6860],{"className":6859},[1943,3493],[238,6861,6863,6898],{"className":6862},[395,848],[238,6864,6866,6895],{"className":6865},[399],[238,6867,6870,6885],{"className":6868,"style":6869},[403],"height:1.05em;",[238,6871,6873,6876],{"style":6872},"top:-1.9em;margin-left:0em;",[238,6874],{"className":6875,"style":3510},[411],[238,6877,6879],{"className":6878},[416,417,418,419],[238,6880,6882],{"className":6881},[282,419],[238,6883,357],{"className":6884},[282,283,419],[238,6886,6887,6890],{"style":3529},[238,6888],{"className":6889,"style":3510},[411],[238,6891,6892],{},[238,6893,3390],{"className":6894},[1943,3538,3539],[238,6896,872],{"className":6897},[871],[238,6899,6901],{"className":6900},[399],[238,6902,6905],{"className":6903,"style":6904},[403],"height:1.25em;",[238,6906],{},[238,6908],{"className":6909,"style":455},[429],[238,6911,2297],{"className":6912,"style":2313},[282,283],[238,6914,335],{"className":6915},[381],[238,6917,357],{"className":6918},[282,283],[238,6920],{"className":6921,"style":430},[429],[238,6923,349],{"className":6924},[434],[238,6926],{"className":6927,"style":430},[429],[238,6929,6931,6934,6937,6940,6969,6972,6975,6978,6981,6984,6987],{"className":6930},[273],[238,6932],{"className":6933,"style":444},[277],[238,6935,341],{"className":6936},[282,283],[238,6938,360],{"className":6939},[462],[238,6941,6943,6946],{"className":6942},[282],[238,6944,4494],{"className":6945},[282,283],[238,6947,6949],{"className":6948},[391],[238,6950,6952],{"className":6951},[395],[238,6953,6955],{"className":6954},[399],[238,6956,6958],{"className":6957,"style":4620},[403],[238,6959,6960,6963],{"style":1226},[238,6961],{"className":6962,"style":412},[411],[238,6964,6966],{"className":6965},[416,417,418,419],[238,6967,2297],{"className":6968,"style":2313},[282,283,419],[238,6970,335],{"className":6971},[381],[238,6973,341],{"className":6974},[282,283],[238,6976,354],{"className":6977},[451],[238,6979],{"className":6980,"style":455},[429],[238,6982,357],{"className":6983},[282,283],[238,6985,360],{"className":6986},[462],[238,6988,354],{"className":6989},[451],[11,6991,6992],{},"以及",[238,6994,6996],{"className":6995,"translate":242},[945],[238,6997,6999,7144],{"className":6998,"translate":242},[241],[238,7000,7002],{"className":7001},[246],[248,7003,7004],{"xmlns":250,"display":954},[252,7005,7006,7141],{},[958,7007,7008,7072],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,7009,7010,7032],{},[966,7011,7012],{},[969,7013,7014],{"scriptlevel":740,"displaystyle":269},[255,7015,7016,7022,7024,7026,7028,7030],{},[337,7017,7018,7020],{},[258,7019,4494],{},[258,7021,2297],{},[332,7023,335],{"stretchy":334},[258,7025,341],{},[332,7027,354],{"separator":269},[258,7029,357],{},[332,7031,360],{"stretchy":334},[966,7033,7034],{},[969,7035,7036],{"scriptlevel":740,"displaystyle":269},[255,7037,7038,7040,7042,7052,7054,7056,7062,7064,7066,7068,7070],{},[255,7039],{},[332,7041,992],{},[6755,7043,7044,7046],{},[332,7045,3390],{},[337,7047,7048,7050],{},[258,7049,341],{},[332,7051,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,7053,330],{},[332,7055,335],{"stretchy":334},[337,7057,7058,7060],{},[258,7059,341],{},[332,7061,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,7063,349],{},[258,7065,341],{},[332,7067,354],{"separator":269},[258,7069,357],{},[332,7071,360],{"stretchy":334},[963,7073,7074,7080],{},[966,7075,7076],{},[969,7077,7078],{"scriptlevel":740,"displaystyle":269},[255,7079],{},[966,7081,7082],{},[969,7083,7084],{"scriptlevel":740,"displaystyle":269},[255,7085,7086,7088,7091,7093,7139],{},[255,7087],{},[429,7089],{"width":7090},"1em",[332,7092,608],{},[255,7094,7095,7097,7099,7101,7103,7105,7107,7109,7115,7117,7119,7121,7127,7129,7135,7137],{},[332,7096,736],{"fence":269},[258,7098,600],{},[332,7100,335],{"stretchy":334},[258,7102,341],{},[332,7104,354],{"separator":269},[258,7106,357],{},[332,7108,354],{"separator":269},[337,7110,7111,7113],{},[258,7112,341],{},[332,7114,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,7116,360],{"stretchy":334},[332,7118,987],{},[258,7120,730],{},[337,7122,7123,7125],{},[258,7124,4429],{},[258,7126,2297],{},[332,7128,335],{"stretchy":334},[337,7130,7131,7133],{},[258,7132,341],{},[332,7134,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,7136,360],{"stretchy":334},[332,7138,4479],{"fence":269},[258,7140,3102],{"mathvariant":344},[262,7142,7143],{"encoding":264},"\\begin{aligned}\nQ^\\pi(s,a)\n&=\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right].\n\\end{aligned}",[238,7145,7147],{"className":7146,"ariaHidden":269},[268],[238,7148,7150,7154],{"className":7149},[273],[238,7151],{"className":7152,"style":7153},[277],"height:3.844em;vertical-align:-1.672em;",[238,7155,7157],{"className":7156},[282],[238,7158,7160,7250],{"className":7159},[958],[238,7161,7163],{"className":7162},[1117],[238,7164,7166,7241],{"className":7165},[395,848],[238,7167,7169,7238],{"className":7168},[399],[238,7170,7173,7229],{"className":7171,"style":7172},[403],"height:2.172em;",[238,7174,7176,7179],{"style":7175},"top:-4.172em;",[238,7177],{"className":7178,"style":3510},[411],[238,7180,7182,7211,7214,7217,7220,7223,7226],{"className":7181},[282],[238,7183,7185,7188],{"className":7184},[282],[238,7186,4494],{"className":7187},[282,283],[238,7189,7191],{"className":7190},[391],[238,7192,7194],{"className":7193},[395],[238,7195,7197],{"className":7196},[399],[238,7198,7200],{"className":7199,"style":4620},[403],[238,7201,7202,7205],{"style":1226},[238,7203],{"className":7204,"style":412},[411],[238,7206,7208],{"className":7207},[416,417,418,419],[238,7209,2297],{"className":7210,"style":2313},[282,283,419],[238,7212,335],{"className":7213},[381],[238,7215,341],{"className":7216},[282,283],[238,7218,354],{"className":7219},[451],[238,7221],{"className":7222,"style":455},[429],[238,7224,357],{"className":7225},[282,283],[238,7227,360],{"className":7228},[462],[238,7230,7232,7235],{"style":7231},"top:-1.738em;",[238,7233],{"className":7234,"style":3510},[411],[238,7236],{"className":7237},[282],[238,7239,872],{"className":7240},[871],[238,7242,7244],{"className":7243},[399],[238,7245,7248],{"className":7246,"style":7247},[403],"height:1.672em;",[238,7249],{},[238,7251,7253],{"className":7252},[1422],[238,7254,7256,7605],{"className":7255},[395,848],[238,7257,7259,7602],{"className":7258},[399],[238,7260,7262,7425],{"className":7261,"style":7172},[403],[238,7263,7264,7267],{"style":7175},[238,7265],{"className":7266,"style":3510},[411],[238,7268,7270,7273,7276,7279,7282,7360,7363,7366,7369,7401,7404,7407,7410,7413,7416,7419,7422],{"className":7269},[282],[238,7271],{"className":7272},[282],[238,7274],{"className":7275,"style":430},[429],[238,7277,992],{"className":7278},[434],[238,7280],{"className":7281,"style":430},[429],[238,7283,7285],{"className":7284},[1943,3493],[238,7286,7288,7351],{"className":7287},[395,848],[238,7289,7291,7348],{"className":7290},[399],[238,7292,7294,7338],{"className":7293,"style":6869},[403],[238,7295,7297,7300],{"style":7296},"top:-1.856em;margin-left:0em;",[238,7298],{"className":7299,"style":3510},[411],[238,7301,7303],{"className":7302},[416,417,418,419],[238,7304,7306],{"className":7305},[282,419],[238,7307,7309,7312],{"className":7308},[282,419],[238,7310,341],{"className":7311},[282,283,419],[238,7313,7315],{"className":7314},[391],[238,7316,7318],{"className":7317},[395],[238,7319,7321],{"className":7320},[399],[238,7322,7324],{"className":7323,"style":4196},[403],[238,7325,7326,7329],{"style":4199},[238,7327],{"className":7328,"style":4203},[411],[238,7330,7332],{"className":7331},[416,4207,4208,419],[238,7333,7335],{"className":7334},[282,419],[238,7336,346],{"className":7337},[282,419],[238,7339,7340,7343],{"style":3529},[238,7341],{"className":7342,"style":3510},[411],[238,7344,7345],{},[238,7346,3390],{"className":7347},[1943,3538,3539],[238,7349,872],{"className":7350},[871],[238,7352,7354],{"className":7353},[399],[238,7355,7358],{"className":7356,"style":7357},[403],"height:1.294em;",[238,7359],{},[238,7361],{"className":7362,"style":455},[429],[238,7364,330],{"className":7365,"style":377},[282,283],[238,7367,335],{"className":7368},[381],[238,7370,7372,7375],{"className":7371},[282],[238,7373,341],{"className":7374},[282,283],[238,7376,7378],{"className":7377},[391],[238,7379,7381],{"className":7380},[395],[238,7382,7384],{"className":7383},[399],[238,7385,7387],{"className":7386,"style":1223},[403],[238,7388,7389,7392],{"style":1226},[238,7390],{"className":7391,"style":412},[411],[238,7393,7395],{"className":7394},[416,417,418,419],[238,7396,7398],{"className":7397},[282,419],[238,7399,346],{"className":7400},[282,419],[238,7402],{"className":7403,"style":430},[429],[238,7405,349],{"className":7406},[434],[238,7408],{"className":7409,"style":430},[429],[238,7411,341],{"className":7412},[282,283],[238,7414,354],{"className":7415},[451],[238,7417],{"className":7418,"style":455},[429],[238,7420,357],{"className":7421},[282,283],[238,7423,360],{"className":7424},[462],[238,7426,7427,7430],{"style":7231},[238,7428],{"className":7429,"style":3510},[411],[238,7431,7433,7436,7440,7443,7446,7449,7596,7599],{"className":7432},[282],[238,7434],{"className":7435},[282],[238,7437],{"className":7438,"style":7439},[429],"margin-right:1em;",[238,7441],{"className":7442,"style":660},[429],[238,7444,608],{"className":7445},[664],[238,7447],{"className":7448,"style":660},[429],[238,7450,7452,7455,7458,7461,7464,7467,7470,7473,7476,74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第一式由全期望公式直接得到。对于第二式，利用 ",[238,7619,7621,7663],{"className":7620,"translate":242},[241],[238,7622,7624],{"className":7623},[246],[248,7625,7626],{"xmlns":250},[252,7627,7628,7660],{},[255,7629,7630,7636,7638,7644,7646,7648],{},[812,7631,7632,7634],{},[258,7633,3380],{},[258,7635,820],{},[332,7637,992],{},[812,7639,7640,7642],{},[258,7641,600],{},[258,7643,820],{},[332,7645,987],{},[258,7647,730],{},[812,7649,7650,7652],{},[258,7651,3380],{},[255,7653,7654,7656,7658],{},[258,7655,820],{},[332,7657,987],{},[738,7659,745],{},[262,7661,7662],{"encoding":264},"G_t=r_t+\\gamma 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",[238,8594,8596,8658],{"className":8595,"translate":242},[241],[238,8597,8599],{"className":8598},[246],[248,8600,8601],{"xmlns":250},[252,8602,8603,8655],{},[255,8604,8605,8615,8617,8619,8625,8627,8629,8631,8633,8635,8637,8639,8641,8643,8645,8647,8653],{},[812,8606,8607,8609],{},[332,8608,3390],{},[337,8610,8611,8613],{},[258,8612,341],{},[332,8614,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,8616,330],{},[332,8618,335],{"stretchy":334},[337,8620,8621,8623],{},[258,8622,341],{},[332,8624,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,8626,349],{},[258,8628,341],{},[332,8630,354],{"separator":269},[258,8632,357],{},[332,8634,360],{"stretchy":334},[258,8636,600],{},[332,8638,335],{"stretchy":334},[258,8640,341],{},[332,8642,354],{"separator":269},[258,8644,357],{},[332,8646,354],{"separator":269},[337,8648,8649,8651],{},[258,8650,341],{},[332,8652,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,8654,360],{"stretchy":334},[262,8656,8657],{"encoding":264},"\\sum_{s'}P(s'\\mid s,a)r(s,a,s')",[238,8659,8661,8789],{"className":8660,"ariaHidden":269},[268],[238,8662,8664,8667,8739,8742,8745,8748,8780,8783,8786],{"className":8663},[273],[238,8665],{"className":8666,"style":4144},[277],[238,8668,8670,8673],{"className":8669},[1943],[238,8671,3390],{"className":8672,"style":4152},[1943,3538,4151],[238,8674,8676],{"className":8675},[391],[238,8677,8679,8731],{"className":8678},[395,848],[238,8680,8682,8728],{"className":8681},[399],[238,8683,8685],{"className":8684,"style":4165},[403],[238,8686,8687,8690],{"style":4168},[238,8688],{"className":8689,"style":412},[411],[238,8691,8693],{"className":8692},[416,417,418,419],[238,8694,8696],{"className":8695},[282,419],[238,8697,8699,8702],{"className":8698},[282,419],[238,8700,341],{"className":8701},[282,283,419],[238,8703,8705],{"className":8704},[391],[238,8706,8708],{"className":8707},[395],[238,8709,8711],{"className":8710},[399],[238,8712,8714],{"className":8713,"style":4196},[403],[238,8715,8716,8719],{"style":4199},[238,8717],{"className":8718,"style":4203},[411],[238,8720,8722],{"className":8721},[416,4207,4208,419],[238,8723,8725],{"className":8724},[282,419],[238,8726,346],{"className":8727},[282,419],[238,8729,872],{"className":8730},[871],[238,8732,8734],{"className":8733},[399],[238,8735,8737],{"className":8736,"style":4224},[403],[238,8738],{},[238,8740],{"className":8741,"style":455},[429],[238,8743,330],{"className":8744,"style":377},[282,283],[238,8746,335],{"className":8747},[381],[238,8749,8751,8754],{"className":8750},[282],[238,8752,341],{"className":8753},[282,283],[238,8755,8757],{"className":8756},[391],[238,8758,8760],{"className":8759},[395],[238,8761,8763],{"className":8762},[399],[238,8764,8766],{"className":8765,"style":404},[403],[238,8767,8768,8771],{"style":407},[238,8769],{"className":8770,"style":412},[411],[238,8772,8774],{"className":8773},[416,417,418,419],[238,8775,8777],{"className":8776},[282,419],[238,8778,346],{"className":8779},[282,419],[238,8781],{"className":8782,"style":430},[429],[238,8784,349],{"className":8785},[434],[238,8787],{"className":8788,"style":430},[429],[238,8790,8792,8795,8798,8801,8804,8807,8810,8813,8816,8819,8822,8825,8828,8831,8834,8866],{"className":8791},[273],[238,8793],{"className":8794,"style":373},[277],[238,8796,341],{"className":8797},[282,283],[238,8799,354],{"className":8800},[451],[238,8802],{"className":8803,"style":455},[429],[238,8805,357],{"className":8806},[282,283],[238,8808,360],{"className":8809},[462],[238,8811,600],{"className":8812,"style":637},[282,283],[238,8814,335],{"className":8815},[381],[238,8817,341],{"className":8818},[282,283],[238,8820,354],{"className":8821},[451],[238,8823],{"className":8824,"style":455},[429],[238,8826,357],{"className":8827},[282,283],[238,8829,354],{"className":8830},[451],[238,8832],{"className":8833,"style":455},[429],[238,8835,8837,8840],{"className":8836},[282],[238,8838,341],{"className":8839},[282,283],[238,8841,8843],{"className":8842},[391],[238,8844,8846],{"className":8845},[395],[238,8847,8849],{"className":8848},[399],[238,8850,8852],{"className":8851,"style":404},[403],[238,8853,8854,8857],{"style":407},[238,8855],{"className":8856,"style":412},[411],[238,8858,8860],{"className":8859},[416,417,418,419],[238,8861,8863],{"className":8862},[282,419],[238,8864,346],{"className":8865},[282,419],[238,8867,360],{"className":8868},[462],"。第二项先用全期望公式对 ",[238,8871,8873,8896],{"className":8872,"translate":242},[241],[238,8874,8876],{"className":8875},[246],[248,8877,8878],{"xmlns":250},[252,8879,8880,8894],{},[255,8881,8882],{},[812,8883,8884,8886],{},[258,8885,341],{},[255,8887,8888,8890,8892],{},[258,8889,820],{},[332,8891,987],{},[738,8893,745],{},[262,8895,3824],{"encoding":264},[238,8897,8899],{"className":8898,"ariaHidden":269},[268],[238,8900,8902,8905],{"className":8901},[273],[238,8903],{"className":8904,"style":3834},[277],[238,8906,8908,8911],{"className":8907},[282],[238,8909,341],{"className":8910},[282,283],[238,8912,8914],{"className":8913},[391],[238,8915,8917,8946],{"className":8916},[395,848],[238,8918,8920,8943],{"className":8919},[399],[238,8921,8923],{"className":8922,"style":1162},[403],[238,8924,8925,8928],{"style":1165},[238,8926],{"className":8927,"style":412},[411],[238,8929,8931],{"className":8930},[416,417,418,419],[238,8932,8934,8937,8940],{"className":8933},[282,419],[238,8935,820],{"className":8936},[282,283,419],[238,8938,987],{"className":8939},[664,419],[238,8941,745],{"className":8942},[282,419],[238,8944,872],{"className":8945},[871],[238,8947,8949],{"className":8948},[399],[238,8950,8952],{"className":8951,"style":1193},[403],[238,8953],{}," 分解，再利用马尔可夫性与策略的平稳性：",[238,8956,8958],{"className":8957,"translate":242},[945],[238,8959,8961,9198],{"className":8960,"translate":242},[241],[238,8962,8964],{"className":8963},[246],[248,8965,8966],{"xmlns":250,"display":954},[252,8967,8968,9195],{},[958,8969,8970,9064,9129],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,8971,8972,9024],{},[966,8973,8974],{},[969,8975,8976],{"scriptlevel":740,"displaystyle":269},[255,8977,8978,8984,8986,8998,9000,9006,9008,9010,9012,9018,9020,9022],{},[812,8979,8980,8982],{},[258,8981,4452],{"mathvariant":620},[258,8983,2297],{},[332,8985,736],{"stretchy":334},[812,8987,8988,8990],{},[258,8989,3380],{},[255,8991,8992,8994,8996],{},[258,8993,820],{},[332,8995,987],{},[738,8997,745],{},[332,8999,349],{},[812,9001,9002,9004],{},[258,9003,341],{},[258,9005,820],{},[332,9007,992],{},[258,9009,341],{},[332,9011,354],{"separator":269},[812,9013,9014,9016],{},[258,9015,357],{},[258,9017,820],{},[332,9019,992],{},[258,9021,357],{},[332,9023,4479],{"stretchy":334},[966,9025,9026],{},[969,9027,9028],{"scriptlevel":740,"displaystyle":269},[255,9029,9030,9032,9034,9044,9046,9048,9054,9056,9058,9060,9062],{},[255,9031],{},[332,9033,992],{},[6755,9035,9036,9038],{},[332,9037,3390],{},[337,9039,9040,9042],{},[258,9041,341],{},[332,9043,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,9045,330],{},[332,9047,335],{"stretchy":334},[337,9049,9050,9052],{},[258,9051,341],{},[332,9053,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,9055,349],{},[258,9057,341],{},[332,9059,354],{"separator":269},[258,9061,357],{},[332,9063,360],{"stretchy":334},[963,9065,9066,9072],{},[966,9067,9068],{},[969,9069,9070],{"scriptlevel":740,"displaystyle":269},[255,9071],{},[966,9073,9074],{},[969,9075,9076],{"scriptlevel":740,"displaystyle":269},[255,9077,9078,9080,9082,9085,9091,9093,9105,9107,9119,9121,9127],{},[255,9079],{},[429,9081],{"width":7090},[332,9083,9084],{},"⋅",[812,9086,9087,9089],{},[258,9088,4452],{"mathvariant":620},[258,9090,2297],{},[332,9092,736],{"stretchy":334},[812,9094,9095,9097],{},[258,9096,3380],{},[255,9098,9099,9101,9103],{},[258,9100,820],{},[332,9102,987],{},[738,9104,745],{},[332,9106,349],{},[812,9108,9109,9111],{},[258,9110,341],{},[255,9112,9113,9115,9117],{},[258,9114,820],{},[332,9116,987],{},[738,9118,745],{},[332,9120,992],{},[337,9122,9123,9125],{},[258,9124,341],{},[332,9126,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,9128,4479],{"stretchy":334},[963,9130,9131,9137],{},[966,9132,9133],{},[969,9134,9135],{"scriptlevel":740,"displaystyle":269},[255,9136],{},[966,9138,9139],{},[969,9140,9141],{"scriptlevel":740,"displaystyle":269},[255,9142,9143,9145,9147,9157,9159,9161,9167,9169,9171,9173,9175,9177,9183,9185,9191,9193],{},[255,9144],{},[332,9146,992],{},[6755,9148,9149,9151],{},[332,9150,3390],{},[337,9152,9153,9155],{},[258,9154,341],{},[332,9156,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,9158,330],{},[332,9160,335],{"stretchy":334},[337,9162,9163,9165],{},[258,9164,341],{},[332,9166,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,9168,349],{},[258,9170,341],{},[332,9172,354],{"separator":269},[258,9174,357],{},[332,9176,360],{"stretchy":334},[337,9178,9179,9181],{},[258,9180,4429],{},[258,9182,2297],{},[332,9184,335],{"stretchy":334},[337,9186,9187,9189],{},[258,9188,341],{},[332,9190,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,9192,360],{"stretchy":334},[258,9194,3102],{"mathvariant":344},[262,9196,9197],{"encoding":264},"\\begin{aligned}\n\\mathbb{E}_\\pi[G_{t+1}\\mid s_t=s,a_t=a]\n&=\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\cdot\\mathbb{E}_\\pi[G_{t+1}\\mid s_{t+1}=s']\\\\\n&=\\sum_{s'}P(s'\\mid s,a)V^\\pi(s').\n\\end{aligned}",[238,9199,9201],{"className":9200,"ariaHidden":269},[268],[238,9202,9204,9208],{"className":9203},[273],[238,9205],{"className":9206,"style":9207},[277],"height:6.488em;vertical-align:-2.994em;",[238,9209,9211],{"className":9210},[282],[238,9212,9214,9480],{"className":9213},[958],[238,9215,9217],{"className":9216},[1117],[238,9218,9220,9471],{"className":9219},[395,848],[238,9221,9223,9468],{"className":9222},[399],[238,9224,9227,9450,9459],{"className":9225,"style":9226},[403],"height:3.494em;",[238,9228,9230,9233],{"style":9229},"top:-5.494em;",[238,9231],{"className":9232,"style":3510},[411],[238,9234,9236,9276,9279,9328,9331,9334,9337,9377,9380,9383,9386,9389,9392,9395,9435,9438,9441,9444,9447],{"className":9235},[282],[238,9237,9239,9242],{"className":9238},[282],[238,9240,4452],{"className":9241},[282,714],[238,9243,9245],{"className":9244},[391],[238,9246,9248,9268],{"className":9247},[395,848],[238,9249,9251,9265],{"className":9250},[399],[238,9252,9254],{"className":9253,"style":1927},[403],[238,9255,9256,9259],{"style":1165},[238,9257],{"className":9258,"style":412},[411],[238,9260,9262],{"className":9261},[416,417,418,419],[238,9263,2297],{"className":9264,"style":2313},[282,283,419],[238,9266,872],{"className":9267},[871],[238,9269,9271],{"className":9270},[399],[238,9272,9274],{"className":9273,"style":879},[403],[238,9275],{},[238,9277,736],{"className":9278},[381],[238,9280,9282,9285],{"className":9281},[282],[238,9283,3380],{"className":9284},[282,283],[238,9286,9288],{"className":9287},[391],[238,9289,9291,9320],{"className":9290},[395,848],[238,9292,9294,9317],{"className":9293},[399],[238,9295,9297],{"className":9296,"style":1162},[403],[238,9298,9299,9302],{"style":1165},[238,9300],{"className":9301,"style":412},[411],[238,9303,9305],{"className":9304},[416,417,418,419],[238,9306,9308,9311,9314],{"className":9307},[282,419],[238,9309,820],{"className":9310},[282,283,419],[238,9312,987],{"className":9313},[664,419],[238,9315,745],{"className":9316},[282,419],[238,9318,872],{"className":9319},[871],[238,9321,9323],{"className":9322},[399],[238,9324,9326],{"className":9325,"style":1193},[403],[238,9327],{},[238,9329],{"className":9330,"style":430},[429],[238,9332,349],{"className":9333},[434],[238,9335],{"className":9336,"style":430},[429],[238,9338,9340,9343],{"className":9339},[282],[238,9341,341],{"className":9342},[282,283],[238,9344,9346],{"className":9345},[391],[238,9347,9349,9369],{"className":9348},[395,848],[238,9350,9352,9366],{"className":9351},[399],[238,9353,9355],{"className":9354,"style":855},[403],[238,9356,9357,9360],{"style":1165},[238,9358],{"className":9359,"style":412},[411],[238,9361,9363],{"className":9362},[416,417,418,419],[238,9364,820],{"className":9365},[282,283,419],[238,9367,872],{"className":9368},[871],[238,9370,9372],{"className":9371},[399],[238,9373,9375],{"className":9374,"style":879},[403],[238,9376],{},[238,9378],{"className":9379,"style":430},[429],[238,9381,992],{"className":9382},[434],[238,9384],{"className":9385,"style":430},[429],[238,9387,341],{"className":9388},[282,283],[238,9390,354],{"className":9391},[451],[238,9393],{"className":9394,"style":455},[429],[238,9396,9398,9401],{"className":9397},[282],[238,9399,357],{"className":9400},[282,283],[238,9402,9404],{"className":9403},[391],[238,9405,9407,9427],{"className":9406},[395,848],[238,9408,9410,9424],{"className":9409},[399],[238,9411,9413],{"className":9412,"style":855},[403],[238,9414,9415,9418],{"style":1165},[238,9416],{"className":9417,"style":412},[411],[238,9419,9421],{"className":9420},[416,417,418,419],[238,9422,820],{"className":9423},[282,283,419],[238,9425,872],{"className":9426},[871],[238,9428,9430],{"className":9429},[399],[238,9431,9433],{"className":9432,"style":879},[403],[238,9434],{},[238,9436],{"className":9437,"style":430},[429],[238,9439,992],{"className":9440},[434],[238,9442],{"className":9443,"style":430},[429],[238,9445,357],{"className":9446},[282,283],[238,9448,4479],{"className":9449},[462],[238,9451,9453,9456],{"style":9452},"top:-3.06em;",[238,9454],{"className":9455,"style":3510},[411],[238,9457],{"className":9458},[282],[238,9460,9462,9465],{"style":9461},"top:-1.35em;",[238,9463],{"className":9464,"style":3510},[411],[238,9466],{"className":9467},[282],[238,9469,872],{"className":9470},[871],[238,9472,9474],{"className":9473},[399],[238,9475,9478],{"className":9476,"style":9477},[403],"height:2.994em;",[238,9479],{},[238,9481,9483],{"className":9482},[1422],[238,9484,9486,10104],{"className":9485},[395,848],[238,9487,9489,10101],{"className":9488},[399],[238,9490,9492,9653,9870],{"className":9491,"style":9226},[403],[238,9493,9494,9497],{"style":9229},[238,9495],{"className":9496,"style":3510},[411],[238,9498,9500,9503,9506,9509,9512,9588,9591,9594,9597,9629,9632,9635,9638,9641,9644,9647,9650],{"className":9499},[282],[238,9501],{"className":9502},[282],[238,9504],{"className":9505,"style":430},[429],[238,9507,992],{"className":9508},[434],[238,9510],{"className":9511,"style":430},[429],[238,9513,9515],{"className":9514},[1943,3493],[238,9516,9518,9580],{"className":9517},[395,848],[238,9519,9521,9577],{"className":9520},[399],[238,9522,9524,9567],{"className":9523,"style":6869},[403],[238,9525,9526,9529],{"style":7296},[238,9527],{"className":9528,"style":3510},[411],[238,9530,9532],{"className":9531},[416,417,418,419],[238,9533,9535],{"className":9534},[282,419],[238,9536,9538,9541],{"className":9537},[282,419],[238,9539,341],{"className":9540},[282,283,419],[238,9542,9544],{"className":9543},[391],[238,9545,9547],{"className":9546},[395],[238,9548,9550],{"className":9549},[399],[238,9551,9553],{"className":9552,"style":4196},[403],[238,9554,9555,9558],{"style":4199},[238,9556],{"className":9557,"style":4203},[411],[238,9559,9561],{"className":9560},[416,4207,4208,419],[238,9562,9564],{"className":9563},[282,419],[238,9565,346],{"className":9566},[282,419],[238,9568,9569,9572],{"style":3529},[238,9570],{"className":9571,"style":3510},[411],[238,9573,9574],{},[238,9575,3390],{"className":9576},[1943,3538,3539],[238,9578,872],{"className":9579},[871],[238,9581,9583],{"className":9582},[399],[238,9584,9586],{"className":9585,"style":7357},[403],[238,9587],{},[238,9589],{"className":9590,"style":455},[429],[238,9592,330],{"className":9593,"style":377},[282,283],[238,9595,335],{"className":9596},[381],[238,9598,9600,9603],{"className":9599},[282],[238,9601,341],{"className":9602},[282,283],[238,9604,9606],{"className":9605},[391],[238,9607,9609],{"className":9608},[395],[238,9610,9612],{"className":9611},[399],[238,9613,9615],{"className":9614,"style":1223},[403],[238,9616,9617,9620],{"style":1226},[238,9618],{"className":9619,"style":412},[411],[238,9621,9623],{"className":9622},[416,417,418,419],[238,9624,9626],{"className":9625},[282,419],[238,9627,346],{"className":9628},[282,419],[238,9630],{"className":9631,"style":430},[429],[238,9633,349],{"className":9634},[434],[238,9636],{"className":9637,"style":430},[429],[238,9639,341],{"className":9640},[282,283],[238,9642,354],{"className":9643},[451],[238,9645],{"className":9646,"style":455},[429],[238,9648,357],{"className":9649},[282,283],[238,9651,360],{"className":9652},[462],[238,9654,9655,9658],{"style":9452},[238,9656],{"className":9657,"style":3510},[411],[238,9659,9661,9664,9667,9670,9673,9676,9716,9719,9768,9771,9774,9777,9826,9829,9832,9835,9867],{"className":9660},[282],[238,9662],{"className":9663},[282],[238,9665],{"className":9666,"style":7439},[429],[238,9668],{"className":9669,"style":660},[429],[238,9671,9084],{"className":9672},[664],[238,9674],{"className":9675,"style":660},[429],[238,9677,9679,9682],{"className":9678},[282],[238,9680,4452],{"className":9681},[282,714],[238,9683,9685],{"className":9684},[391],[238,9686,9688,9708],{"className":9687},[395,848],[238,9689,9691,9705],{"className":9690},[399],[238,9692,9694],{"className":9693,"style":1927},[403],[238,9695,9696,9699],{"style":1165},[238,9697],{"className":9698,"style":412},[411],[238,9700,9702],{"className":9701},[416,417,418,419],[238,9703,2297],{"className":9704,"style":2313},[282,283,419],[238,9706,872],{"className":9707},[871],[238,9709,9711],{"className":9710},[399],[238,9712,9714],{"className":9713,"style":879},[403],[238,9715],{},[238,9717,736],{"className":9718},[381],[238,9720,9722,9725],{"className":9721},[282],[238,9723,3380],{"className":9724},[282,283],[238,9726,9728],{"className":9727},[391],[238,9729,9731,9760],{"className":9730},[395,848],[238,9732,9734,9757],{"className":9733},[399],[238,9735,9737],{"className":9736,"style":1162},[403],[238,9738,9739,9742],{"style":1165},[238,9740],{"className":9741,"style":412},[411],[238,9743,9745],{"className":9744},[416,417,418,419],[238,9746,9748,9751,9754],{"className":9747},[282,419],[238,9749,820],{"className":9750},[282,283,419],[238,9752,987],{"className":9753},[664,419],[238,9755,745],{"className":9756},[282,419],[238,9758,872],{"className":9759},[871],[238,9761,9763],{"className":9762},[399],[238,9764,9766],{"className":9765,"style":1193},[403],[238,9767],{},[238,9769],{"className":9770,"style":430},[429],[238,9772,349],{"className":9773},[434],[238,9775],{"className":9776,"style":430},[429],[238,9778,9780,9783],{"className":9779},[282],[238,9781,341],{"className":9782},[282,283],[238,9784,9786],{"className":9785},[391],[238,9787,9789,9818],{"className":9788},[395,848],[238,9790,9792,9815],{"className":9791},[399],[238,9793,9795],{"className":9794,"style":1162},[403],[238,9796,9797,9800],{"style":1165},[238,9798],{"className":9799,"style":412},[411],[238,9801,9803],{"className":9802},[416,417,418,419],[238,9804,9806,9809,9812],{"className":9805},[282,419],[238,9807,820],{"className":9808},[282,283,419],[238,9810,987],{"className":9811},[664,419],[238,9813,745],{"className":9814},[282,419],[238,9816,872],{"className":9817},[871],[238,9819,9821],{"className":9820},[399],[238,9822,9824],{"className":9823,"style":1193},[403],[238,9825],{},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",[238,10119,10121,10142],{"className":10120,"translate":242},[241],[238,10122,10124],{"className":10123},[246],[248,10125,10126],{"xmlns":250},[252,10127,10128,10139],{},[255,10129,10130],{},[10131,10132,10133,10135,10137],"msubsup",{},[258,10134,4429],{},[258,10136,820],{},[258,10138,2297],{},[262,10140,10141],{"encoding":264},"V_t^\\pi",[238,10143,10145],{"className":10144,"ariaHidden":269},[268],[238,10146,10148,10152],{"className":10147},[273],[238,10149],{"className":10150,"style":10151},[277],"height:0.9303em;vertical-align:-0.247em;",[238,10153,10155,10158],{"className":10154},[282],[238,10156,4429],{"className":10157,"style":660},[282,283],[238,10159,10161],{"className":10160},[391],[238,10162,10164,10196],{"className":10163},[395,848],[238,10165,10167,10193],{"className":10166},[399],[238,10168,10170,10182],{"className":10169,"style":6609},[403],[238,10171,10173,10176],{"style":10172},"top:-2.453em;margin-left:-0.2222em;margin-right:0.05em;",[238,10174],{"className":10175,"style":412},[411],[238,10177,10179],{"className":10178},[416,417,418,419],[238,10180,820],{"className":10181},[282,283,419],[238,10183,10184,10187],{"style":407},[238,10185],{"className":10186,"style":412},[411],[238,10188,10190],{"className":10189},[416,417,418,419],[238,10191,2297],{"className":10192,"style":2313},[282,283,419],[238,10194,872],{"className":10195},[871],[238,10197,10199],{"className":10198},[399],[238,10200,10203],{"className":10201,"style":10202},[403],"height:0.247em;",[238,10204],{},"，递归形式仍然成立，但函数会随 ",[238,10207,10209,10222],{"className":10208,"translate":242},[241],[238,10210,10212],{"className":10211},[246],[248,10213,10214],{"xmlns":250},[252,10215,10216,10220],{},[255,10217,10218],{},[258,10219,820],{},[262,10221,820],{"encoding":264},[238,10223,10225],{"className":10224,"ariaHidden":269},[268],[238,10226,10228,10231],{"className":10227},[273],[238,10229],{"className":10230,"style":908},[277],[238,10232,820],{"className":10233},[282,283]," 变化。本文只讨论平稳策略，因此可以省略时间下标。",[220,10236,10238],{"id":10237},"二贝尔曼期望方程","二、贝尔曼期望方程",[225,10240,10242],{"id":10241},"_21-方程的推导","2.1 方程的推导",[11,10244,10245,10246,10250],{},"将",[357,10247,10249],{"href":10248},"#proposition-basic-identities","命题 1.5"," 的第二式代入第一式，立即得到策略评估的核心方程。",[230,10252,10255,10287],{"id":10253,"title":10254},"theorem-bellman-expectation","定理 2.1（贝尔曼期望方程）",[11,10256,10257,10258,10286],{},"对任意平稳策略 ",[238,10259,10261,10274],{"className":10260,"translate":242},[241],[238,10262,10264],{"className":10263},[246],[248,10265,10266],{"xmlns":250},[252,10267,10268,10272],{},[255,10269,10270],{},[258,10271,2297],{},[262,10273,2300],{"encoding":264},[238,10275,10277],{"className":10276,"ariaHidden":269},[268],[238,10278,10280,10283],{"className":10279},[273],[238,10281],{"className":10282,"style":489},[277],[238,10284,2297],{"className":10285,"style":2313},[282,283],"，状态值函数满足",[238,10288,10290],{"className":10289,"translate":242},[945],[238,10291,10293,10465],{"className":10292,"translate":242},[241],[238,10294,10296],{"className":10295},[246],[248,10297,10298],{"xmlns":250,"display":954},[252,10299,10300,10462],{},[958,10301,10302,10380],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,10303,10304,10322],{},[966,10305,10306],{},[969,10307,10308],{"scriptlevel":740,"displaystyle":269},[255,10309,10310,10316,10318,10320],{},[337,10311,10312,10314],{},[258,10313,4429],{},[258,10315,2297],{},[332,10317,335],{"stretchy":334},[258,10319,341],{},[332,10321,360],{"stretchy":334},[966,10323,10324],{},[969,10325,10326],{"scriptlevel":740,"displaystyle":269},[255,10327,10328,10330,10332,10338,10340,10342,10344,10346,10348,10350,10360,10362,10364,10370,10372,10374,10376,10378],{},[255,10329],{},[332,10331,992],{},[6755,10333,10334,10336],{},[332,10335,3390],{},[258,10337,357],{},[258,10339,2297],{},[332,10341,335],{"stretchy":334},[258,10343,357],{},[332,10345,349],{},[258,10347,341],{},[332,10349,360],{"stretchy":334},[6755,10351,10352,10354],{},[332,10353,3390],{},[337,10355,10356,10358],{},[258,10357,341],{},[332,10359,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,10361,330],{},[332,10363,335],{"stretchy":334},[337,10365,10366,10368],{},[258,10367,341],{},[332,10369,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,10371,349],{},[258,10373,341],{},[332,10375,354],{"separator":269},[258,10377,357],{},[332,10379,360],{"stretchy":334},[963,10381,10382,10388],{},[966,10383,10384],{},[969,10385,10386],{"scriptlevel":740,"displaystyle":269},[255,10387],{},[966,10389,10390],{},[969,10391,10392],{"scriptlevel":740,"displaystyle":269},[255,10393,10394,10396,10398,10400,10446,10448,10451,10454,10456,10458,10460],{},[255,10395],{},[429,10397],{"width":7090},[332,10399,608],{},[255,10401,10402,10404,10406,10408,10410,10412,10414,10416,10422,10424,10426,10428,10434,10436,10442,10444],{},[332,10403,736],{"fence":269},[258,10405,600],{},[332,10407,335],{"stretchy":334},[258,10409,341],{},[332,10411,354],{"separator":269},[258,10413,357],{},[332,10415,354],{"separator":269},[337,10417,10418,10420],{},[258,10419,341],{},[332,10421,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,10423,360],{"stretchy":334},[332,10425,987],{},[258,10427,730],{},[337,10429,10430,10432],{},[258,10431,4429],{},[258,10433,2297],{},[332,10435,335],{"stretchy":334},[337,10437,10438,10440],{},[258,10439,341],{},[332,10441,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,10443,360],{"stretchy":334},[332,10445,4479],{"fence":269},[332,10447,354],{"separator":269},[429,10449],{"width":10450},"2em",[258,10452,10453],{"mathvariant":344},"∀",[258,10455,341],{},[332,10457,733],{},[258,10459,260],{},[258,10461,3102],{"mathvariant":344},[262,10463,10464],{"encoding":264},"\\begin{aligned}\nV^\\pi(s)\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right],\n\\qquad \\forall s\\in S.\n\\end{aligned}",[238,10466,10468],{"className":10467,"ariaHidden":269},[268],[238,10469,10471,10474],{"className":10470},[273],[238,10472],{"className":10473,"style":7153},[277],[238,10475,10477],{"className":10476},[282],[238,10478,10480,10557],{"className":10479},[958],[238,10481,10483],{"className":10482},[1117],[238,10484,10486,10549],{"className":10485},[395,848],[238,10487,10489,10546],{"className":10488},[399],[238,10490,10492,10538],{"className":10491,"style":7172},[403],[238,10493,10494,10497],{"style":7175},[238,10495],{"className":10496,"style":3510},[411],[238,10498,10500,10529,10532,10535],{"className":10499},[282],[238,10501,10503,10506],{"className":10502},[282],[238,10504,4429],{"className":10505,"style":660},[282,283],[238,10507,10509],{"className":10508},[391],[238,10510,10512],{"className":10511},[395],[238,10513,10515],{"className":10514},[399],[238,10516,10518],{"className":10517,"style":4620},[403],[238,10519,10520,10523],{"style":1226},[238,10521],{"className":10522,"style":412},[411],[238,10524,10526],{"className":10525},[416,417,418,419],[238,10527,2297],{"className":10528,"style":2313},[282,283,419],[238,10530,335],{"className":10531},[381],[238,10533,341],{"className":10534},[282,283],[238,10536,360],{"className":10537},[462],[238,10539,10540,10543],{"style":7231},[238,10541],{"className":10542,"style":3510},[411],[238,10544],{"className":10545},[282],[238,10547,872],{"className":10548},[871],[238,10550,10552],{"className":10551},[399],[238,10553,10555],{"className":10554,"style":7247},[403],[238,10556],{},[238,10558,10560],{"className":10559},[1422],[238,10561,10563,11014],{"className":10562},[395,848],[238,10564,10566,11011],{"className":10565},[399],[238,10567,10569,10807],{"className":10568,"style":7172},[403],[238,10570,10571,10574],{"style":7175},[238,10572],{"className":10573,"style":3510},[411],[238,10575,10577,10580,10583,10586,10589,10636,10639,10642,10645,10648,10651,10654,10657,10660,10663,10666,10742,10745,10748,10751,10783,10786,10789,10792,10795,10798,10801,10804],{"className":10576},[282],[238,10578],{"className":10579},[282],[238,10581],{"className":10582,"style":430},[429],[238,10584,992],{"className":10585},[434],[238,10587],{"className":10588,"style":430},[429],[238,10590,10592],{"className":10591},[1943,3493],[238,10593,10595,10628],{"className":10594},[395,848],[238,10596,10598,10625],{"className":10597},[399],[238,10599,10601,10615],{"className":10600,"style":6869},[403],[238,10602,10603,10606],{"style":6872},[238,10604],{"className":10605,"style":3510},[411],[238,10607,10609],{"className":10608},[416,417,418,419],[238,10610,10612],{"className":10611},[282,419],[238,10613,357],{"className":10614},[282,283,419],[238,10616,10617,10620],{"style":3529},[238,10618],{"className":10619,"style":3510},[411],[238,10621,10622],{},[238,10623,3390],{"className":10624},[1943,3538,3539],[238,10626,872],{"className":10627},[871],[238,10629,10631],{"className":10630},[399],[238,10632,10634],{"className":10633,"style":6904},[403],[238,10635],{},[238,10637],{"className":10638,"style":455},[429],[238,10640,2297],{"className":10641,"style":2313},[282,283],[238,10643,335],{"className":10644},[381],[238,10646,357],{"className":10647},[282,283],[238,10649],{"className":10650,"style":430},[429],[238,10652,349],{"className":10653},[434],[238,10655],{"className":10656,"style":430},[429],[238,10658,341],{"className":10659},[282,283],[238,10661,360],{"className":10662},[462],[238,10664],{"className":10665,"style":455},[429],[238,10667,10669],{"className":10668},[1943,3493],[238,10670,10672,10734],{"className":10671},[395,848],[238,10673,10675,10731],{"className":10674},[399],[238,10676,10678,10721],{"className":10677,"style":6869},[403],[238,10679,10680,10683],{"style":7296},[238,10681],{"className":10682,"style":3510},[411],[238,10684,10686],{"className":10685},[416,417,418,419],[238,10687,10689],{"className":10688},[282,419],[238,10690,10692,10695],{"className":10691},[282,419],[238,10693,341],{"className":10694},[282,283,419],[238,10696,10698],{"className":10697},[391],[238,10699,10701],{"className":10700},[395],[238,10702,10704],{"className":10703},[399],[238,10705,10707],{"className":10706,"style":4196},[403],[238,10708,10709,10712],{"style":4199},[238,10710],{"className":10711,"style":4203},[411],[238,10713,10715],{"className":10714},[416,4207,4208,419],[238,10716,10718],{"className":10717},[282,419],[238,10719,346],{"className":10720},[282,419],[238,10722,10723,10726],{"style":3529},[238,10724],{"className":10725,"style":3510},[411],[238,10727,10728],{},[238,10729,3390],{"className":10730},[1943,3538,3539],[238,10732,872],{"className":10733},[871],[238,10735,10737],{"className":10736},[399],[238,10738,10740],{"className":10739,"style":7357},[403],[238,10741],{},[238,10743],{"className":10744,"style":455},[429],[238,10746,330],{"className":10747,"style":377},[282,283],[238,10749,335],{"className":10750},[381],[238,10752,10754,10757],{"className":10753},[282],[238,10755,341],{"className":10756},[282,283],[238,10758,10760],{"className":10759},[391],[238,10761,10763],{"className":10762},[395],[238,10764,10766],{"className":10765},[399],[238,10767,10769],{"className":10768,"style":1223},[403],[238,10770,10771,10774],{"style":1226},[238,10772],{"className":10773,"style":412},[411],[238,10775,10777],{"className":10776},[416,417,418,419],[238,10778,10780],{"className":10779},[282,419],[238,10781,346],{"className":10782},[282,419],[238,10784],{"className":10785,"style":430},[429],[238,10787,349],{"className":10788},[434],[238,10790],{"className":10791,"style":430},[429],[238,10793,341],{"className":10794},[282,283],[238,10796,354],{"className":10797},[451],[238,10799],{"className":10800,"style":455},[429],[238,10802,357],{"className":10803},[282,283],[238,10805,360],{"className":10806},[462],[238,10808,10809,10812],{"style":7231},[238,10810],{"className":10811,"style":3510},[411],[238,10813,10815,10818,10821,10824,10827,10830,10977,10980,10983,10987,10990,10993,10996,10999,11002,11005,11008],{"className":10814},[282],[238,10816],{"className":10817},[282],[238,10819],{"className":10820,"style":7439},[429],[238,10822],{"className":10823,"style":660},[429],[238,10825,608],{"className":10826},[664],[238,10828],{"className":10829,"style":660},[429],[238,10831,10833,10836,10839,10842,10845,10848,10851,10854,10857,10860,10892,10895,10898,10901,10904,10907,10936,10939,10971,10974],{"className":10832},[3041],[238,10834,736],{"className":10835,"style":4791},[381,4790],[238,10837,600],{"className":10838,"style":637},[282,283],[238,10840,335],{"className":10841},[381],[238,10843,341],{"className":10844},[282,283],[238,10846,354],{"className":10847},[451],[238,10849],{"className":10850,"style":455},[429],[238,10852,357],{"className":10853},[282,283],[238,10855,354],{"className":10856},[451],[238,10858],{"className":10859,"style":455},[429],[238,10861,10863,10866],{"className":10862},[282],[238,10864,341],{"className":10865},[282,283],[238,10867,10869],{"className":10868},[391],[238,10870,10872],{"className":10871},[395],[238,10873,10875],{"className":10874},[399],[238,10876,10878],{"className":10877,"style":1223},[403],[238,10879,10880,10883],{"style":1226},[238,10881],{"className":10882,"style":412},[411],[238,10884,10886],{"className":10885},[416,417,418,419],[238,10887,10889],{"className":10888},[282,419],[238,10890,346],{"className":10891},[282,419],[238,10893,360],{"className":10894},[462],[238,10896],{"className":10897,"style":660},[429],[238,10899,987],{"className":10900},[664],[238,10902],{"className":10903,"style":660},[429],[238,10905,730],{"className":10906,"style":764},[282,283],[238,10908,10910,10913],{"className":10909},[282],[238,10911,4429],{"className":10912,"style":660},[282,283],[238,10914,10916],{"className":10915},[391],[238,10917,10919],{"className":10918},[395],[238,10920,10922],{"className":10921},[399],[238,10923,10925],{"className":10924,"style":4620},[403],[238,10926,10927,10930],{"style":1226},[238,10928],{"className":10929,"style":412},[411],[238,10931,10933],{"className":10932},[416,417,418,419],[238,10934,2297],{"className":10935,"style":2313},[282,283,419],[238,10937,335],{"className":10938},[381],[238,10940,10942,10945],{"className":10941},[282],[238,10943,341],{"className":10944},[282,283],[238,10946,10948],{"className":10947},[391],[238,10949,10951],{"className":10950},[395],[238,10952,10954],{"className":10953},[399],[238,10955,10957],{"className":10956,"style":1223},[403],[238,10958,10959,10962],{"style":1226},[238,10960],{"className":10961,"style":412},[411],[238,10963,10965],{"className":10964},[416,417,418,419],[238,10966,10968],{"className":10967},[282,419],[238,10969,346],{"className":10970},[282,419],[238,10972,360],{"className":10973},[462],[238,10975,4479],{"className":10976,"style":4791},[462,4790],[238,10978],{"className":10979,"style":455},[429],[238,10981,354],{"className":10982},[451],[238,10984],{"className":10985,"style":10986},[429],"margin-right:2em;",[238,10988],{"className":10989,"style":455},[429],[238,10991,10453],{"className":10992},[282],[238,10994,341],{"className":10995},[282,283],[238,10997],{"className":10998,"style":430},[429],[238,11000,733],{"className":11001},[434],[238,11003],{"className":11004,"style":430},[429],[238,11006,260],{"className":11007,"style":284},[282,283],[238,11009,3102],{"className":11010},[282],[238,11012,872],{"className":11013},[871],[238,11015,11017],{"className":11016},[399],[238,11018,11020],{"className":11019,"style":7247},[403],[238,11021],{},[11,11023,11024,11026,11027,941],{},[1688,11025,5794],{}," 由",[357,11028,10249],{"href":10248},[238,11030,11032],{"className":11031,"translate":242},[945],[238,11033,11035,11225],{"className":11034,"translate":242},[241],[238,11036,11038],{"className":11037},[246],[248,11039,11040],{"xmlns":250,"display":954},[252,11041,11042,11222],{},[958,11043,11044,11108],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,11045,11046,11064],{},[966,11047,11048],{},[969,11049,11050],{"scriptlevel":740,"displaystyle":269},[255,11051,11052,11058,11060,11062],{},[337,11053,11054,11056],{},[258,11055,4429],{},[258,11057,2297],{},[332,11059,335],{"stretchy":334},[258,11061,341],{},[332,11063,360],{"stretchy":334},[966,11065,11066],{},[969,11067,11068],{"scriptlevel":740,"displaystyle":269},[255,11069,11070,11072,11074,11080,11082,11084,11086,11088,11090,11092,11098,11100,11102,11104,11106],{},[255,11071],{},[332,11073,992],{},[6755,11075,11076,11078],{},[332,11077,3390],{},[258,11079,357],{},[258,11081,2297],{},[332,11083,335],{"stretchy":334},[258,11085,357],{},[332,11087,349],{},[258,11089,341],{},[332,11091,360],{"stretchy":334},[337,11093,11094,11096],{},[258,11095,4494],{},[258,11097,2297],{},[332,11099,335],{"stretchy":334},[258,11101,341],{},[332,11103,354],{"separator":269},[258,11105,357],{},[332,11107,360],{"stretchy":334},[963,11109,11110,11116],{},[966,11111,11112],{},[969,11113,11114],{"scriptlevel":740,"displaystyle":269},[255,11115],{},[966,11117,11118],{},[969,11119,11120],{"scriptlevel":740,"displaystyle":269},[255,11121,11122,11124,11126,11132,11134,11136,11138,11140,11142,11144,11154,11156,11158,11164,11166,11168,11170,11172,11174,11220],{},[255,11123],{},[332,11125,992],{},[6755,11127,11128,11130],{},[332,11129,3390],{},[258,11131,357],{},[258,11133,2297],{},[332,11135,335],{"stretchy":334},[258,11137,357],{},[332,11139,349],{},[258,11141,341],{},[332,11143,360],{"stretchy":334},[6755,11145,11146,11148],{},[332,11147,3390],{},[337,11149,11150,11152],{},[258,11151,341],{},[332,11153,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,11155,330],{},[332,11157,335],{"stretchy":334},[337,11159,11160,11162],{},[258,11161,341],{},[332,11163,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,11165,349],{},[258,11167,341],{},[332,11169,354],{"separator":269},[258,11171,357],{},[332,11173,360],{"stretchy":334},[255,11175,11176,11178,11180,11182,11184,11186,11188,11190,11196,11198,11200,11202,11208,11210,11216,11218],{},[332,11177,736],{"fence":269},[258,11179,600],{},[332,11181,335],{"stretchy":334},[258,11183,341],{},[332,11185,354],{"separator":269},[258,11187,357],{},[332,11189,354],{"separator":269},[337,11191,11192,11194],{},[258,11193,341],{},[332,11195,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,11197,360],{"stretchy":334},[332,11199,987],{},[258,11201,730],{},[337,11203,11204,11206],{},[258,11205,4429],{},[258,11207,2297],{},[332,11209,335],{"stretchy":334},[337,11211,11212,11214],{},[258,11213,341],{},[332,11215,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,11217,360],{"stretchy":334},[332,11219,4479],{"fence":269},[258,11221,3102],{"mathvariant":344},[262,11223,11224],{"encoding":264},"\\begin{aligned}\nV^\\pi(s)\n&=\\sum_{a}\\pi(a\\mid s)Q^\\pi(s,a)\\\\\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right].\n\\end{aligned}",[238,11226,11228],{"className":11227,"ariaHidden":269},[268],[238,11229,11231,11235],{"className":11230},[273],[238,11232],{"className":11233,"style":11234},[277],"height:4.944em;vertical-align:-2.222em;",[238,11236,11238],{"className":11237},[282],[238,11239,11241,11322],{"className":11240},[958],[238,11242,11244],{"className":11243},[1117],[238,11245,11247,11313],{"className":11246},[395,848],[238,11248,11250,11310],{"className":11249},[399],[238,11251,11254,11301],{"className":11252,"style":11253},[403],"height:2.722em;",[238,11255,11257,11260],{"style":11256},"top:-4.722em;",[238,11258],{"className":11259,"style":3510},[411],[238,11261,11263,11292,11295,11298],{"className":11262},[282],[238,11264,11266,11269],{"className":11265},[282],[238,11267,4429],{"className":11268,"style":660},[282,283],[238,11270,11272],{"className":11271},[391],[238,11273,11275],{"className":11274},[395],[238,11276,11278],{"className":11277},[399],[238,11279,11281],{"className":11280,"style":4620},[403],[238,11282,11283,11286],{"style":1226},[238,11284],{"className":11285,"style":412},[411],[238,11287,11289],{"className":11288},[416,417,418,419],[238,11290,2297],{"className":11291,"style":2313},[282,283,419],[238,11293,335],{"className":11294},[381],[238,11296,341],{"className":11297},[282,283],[238,11299,360],{"className":11300},[462],[238,11302,11304,11307],{"style":11303},"top:-2.122em;",[238,11305],{"className":11306,"style":3510},[411],[238,11308],{"className":11309},[282],[238,11311,872],{"className":11312},[871],[238,11314,11316],{"className":11315},[399],[238,11317,11320],{"className":11318,"style":11319},[403],"height:2.222em;",[238,11321],{},[238,11323,11325],{"className":11324},[1422],[238,11326,11328,11872],{"className":11327},[395,848],[238,11329,11331,11869],{"className":11330},[399],[238,11332,11334,11475],{"className":11333,"style":11253},[403],[238,11335,11336,11339],{"style":11256},[238,11337],{"className":11338,"style":3510},[411],[238,11340,11342,11345,11348,11351,11354,11401,11404,11407,11410,11413,11416,11419,11422,11425,11428,11457,11460,11463,11466,11469,11472],{"className":11341},[282],[238,11343],{"className":11344},[282],[238,11346],{"className":11347,"style":430},[429],[238,11349,992],{"className":11350},[434],[238,11352],{"className":11353,"style":430},[429],[238,11355,11357],{"className":11356},[1943,3493],[238,11358,11360,11393],{"className":11359},[395,848],[238,11361,11363,11390],{"className":11362},[399],[238,11364,11366,11380],{"className":11365,"style":6869},[403],[238,11367,11368,11371],{"style":6872},[238,11369],{"className":11370,"style":3510},[411],[238,11372,11374],{"className":11373},[416,417,418,419],[238,11375,11377],{"className":11376},[282,419],[238,11378,357],{"className":11379},[282,283,419],[238,11381,11382,11385],{"style":3529},[238,11383],{"className":11384,"style":3510},[411],[238,11386,11387],{},[238,11388,3390],{"className":11389},[1943,3538,3539],[238,11391,872],{"className":11392},[871],[238,11394,11396],{"className":11395},[399],[238,11397,11399],{"className":11398,"style":6904},[403],[238,11400],{},[238,11402],{"className":11403,"style":455},[429],[238,11405,2297],{"className":11406,"style":2313},[282,283],[238,11408,335],{"className":11409},[381],[238,11411,357],{"className":11412},[282,283],[238,11414],{"className":11415,"style":430},[429],[238,11417,349],{"className":11418},[434],[238,11420],{"className":11421,"style":430},[429],[238,11423,341],{"className":11424},[282,283],[238,11426,360],{"className":11427},[462],[238,11429,11431,11434],{"className":11430},[282],[238,11432,4494],{"className":11433},[282,283],[238,11435,11437],{"className":11436},[391],[238,11438,11440],{"className":11439},[395],[238,11441,11443],{"className":11442},[399],[238,11444,11446],{"className":11445,"style":4620},[403],[238,11447,11448,11451],{"style":1226},[238,11449],{"className":11450,"style":412},[411],[238,11452,11454],{"className":11453},[416,417,418,419],[238,11455,2297],{"className":11456,"style":2313},[282,283,419],[238,11458,335],{"className":11459},[381],[238,11461,341],{"className":11462},[282,283],[238,11464,354],{"className":11465},[451],[238,11467],{"className":11468,"style":455},[429],[238,11470,357],{"className":11471},[282,283],[238,11473,360],{"className":11474},[462],[238,11476,11477,11480],{"style":11303},[238,11478],{"className":11479,"style":3510},[411],[238,11481,11483,11486,11489,11492,11495,11542,11545,11548,11551,11554,11557,11560,11563,11566,11569,11572,11648,11651,11654,11657,11689,11692,11695,11698,11701,11704,11707,11710,11713,11716,11863,11866],{"className":11482},[282],[238,11484],{"className":11485},[282],[238,11487],{"className":11488,"style":430},[429],[238,11490,992],{"className":11491},[434],[238,11493],{"className":11494,"style":430},[429],[238,11496,11498],{"className":11497},[1943,3493],[238,11499,11501,11534],{"className":11500},[395,848],[238,11502,11504,11531],{"className":11503},[399],[238,11505,11507,11521],{"className":11506,"style":6869},[403],[238,11508,11509,11512],{"style":6872},[238,11510],{"className":11511,"style":3510},[411],[238,11513,11515],{"className":11514},[416,417,418,419],[238,11516,11518],{"className":11517},[282,419],[238,11519,357],{"className":11520},[282,283,419],[238,11522,11523,11526],{"style":3529},[238,11524],{"className":11525,"style":3510},[411],[238,11527,11528],{},[238,11529,3390],{"className":11530},[1943,3538,3539],[238,11532,872],{"className":11533},[871],[238,11535,11537],{"className":11536},[399],[238,11538,11540],{"className":11539,"style":6904},[403],[238,11541],{},[238,11543],{"className":11544,"style":455},[429],[238,11546,2297],{"className":11547,"style":2313},[282,283],[238,11549,335],{"className":11550},[381],[238,11552,357],{"className":11553},[282,283],[238,11555],{"className":11556,"style":430},[429],[238,11558,349],{"className":11559},[434],[238,11561],{"className":11562,"style":430},[429],[238,11564,341],{"className":11565},[282,283],[238,11567,360],{"className":11568},[462],[238,11570],{"className":11571,"style":455},[429],[238,11573,11575],{"className":11574},[1943,3493],[238,11576,11578,11640],{"className":11577},[395,848],[238,11579,11581,11637],{"className":11580},[399],[238,11582,11584,11627],{"className":11583,"style":6869},[403],[238,11585,11586,11589],{"style":7296},[238,11587],{"className":11588,"style":3510},[411],[238,11590,11592],{"className":11591},[416,417,418,419],[238,11593,11595],{"className":11594},[282,419],[238,11596,11598,11601],{"className":11597},[282,419],[238,11599,341],{"className":11600},[282,283,419],[238,11602,11604],{"className":11603},[391],[238,11605,11607],{"className":11606},[395],[238,11608,11610],{"className":11609},[399],[238,11611,11613],{"className":11612,"style":4196},[403],[238,11614,11615,11618],{"style":4199},[238,11616],{"className":11617,"style":4203},[411],[238,11619,11621],{"className":11620},[416,4207,4208,419],[238,11622,11624],{"className":11623},[282,419],[238,11625,346],{"className":11626},[282,419],[238,11628,11629,11632],{"style":3529},[238,11630],{"className":11631,"style":3510},[411],[238,11633,11634],{},[238,11635,3390],{"className":11636},[1943,3538,3539],[238,11638,872],{"className":11639},[871],[238,11641,11643],{"className":11642},[399],[238,11644,11646],{"className":11645,"style":7357},[403],[238,11647],{},[238,11649],{"className":11650,"style":455},[429],[238,11652,330],{"className":11653,"style":377},[282,283],[238,11655,335],{"className":11656},[381],[238,11658,11660,11663],{"className":11659},[282],[238,11661,341],{"className":11662},[282,283],[238,11664,11666],{"className":11665},[391],[238,11667,11669],{"className":11668},[395],[238,11670,11672],{"className":11671},[399],[238,11673,11675],{"className":11674,"style":1223},[403],[238,11676,11677,11680],{"style":1226},[238,11678],{"className":11679,"style":412},[411],[238,11681,11683],{"className":11682},[416,417,418,419],[238,11684,11686],{"className":11685},[282,419],[238,11687,346],{"className":11688},[282,419],[238,11690],{"className":11691,"style":430},[429],[238,11693,349],{"className":11694},[434],[238,11696],{"className":11697,"style":430},[429],[238,11699,341],{"className":11700},[282,283],[238,11702,354],{"className":11703},[451],[238,11705],{"className":11706,"style":455},[429],[238,11708,357],{"className":11709},[282,283],[238,11711,360],{"className":11712},[462],[238,11714],{"className":11715,"style":455},[429],[238,11717,11719,11722,11725,11728,11731,11734,11737,11740,11743,11746,11778,11781,11784,11787,11790,11793,11822,11825,11857,11860],{"className":11718},[3041],[238,11720,736],{"className":11721,"style":4791},[381,4790],[238,11723,600],{"className":11724,"style":637},[282,283],[238,11726,335],{"className":11727},[381],[238,11729,341],{"className":11730},[282,283],[238,11732,354],{"className":11733},[451],[238,11735],{"className":11736,"style":455},[429],[238,11738,357],{"className":11739},[282,283],[238,11741,354],{"className":11742},[451],[238,11744],{"className":11745,"style":455},[429],[238,11747,11749,11752],{"className":11748},[282],[238,11750,341],{"className":11751},[282,283],[238,11753,11755],{"className":11754},[391],[238,11756,11758],{"className":11757},[395],[238,11759,11761],{"className":11760},[399],[238,11762,11764],{"className":11763,"style":1223},[403],[238,11765,11766,11769],{"style":1226},[238,11767],{"className":11768,"style":412},[411],[238,11770,11772],{"className":11771},[416,417,418,419],[238,11773,11775],{"className":11774},[282,419],[238,11776,346],{"className":11777},[282,419],[238,11779,360],{"className":11780},[462],[238,11782],{"className":11783,"style":660},[429],[238,11785,987],{"className":11786},[664],[238,11788],{"className":11789,"style":660},[429],[238,11791,730],{"className":11792,"style":764},[282,283],[238,11794,11796,11799],{"className":11795},[282],[238,11797,4429],{"className":11798,"style":660},[282,283],[238,11800,11802],{"className":11801},[391],[238,11803,11805],{"className":11804},[395],[238,11806,11808],{"className":11807},[399],[238,11809,11811],{"className":11810,"style":4620},[403],[238,11812,11813,11816],{"style":1226},[238,11814],{"className":11815,"style":412},[411],[238,11817,11819],{"className":11818},[416,417,418,419],[238,11820,2297],{"className":11821,"style":2313},[282,283,419],[238,11823,335],{"className":11824},[381],[238,11826,11828,11831],{"className":11827},[282],[238,11829,341],{"className":11830},[282,283],[238,11832,11834],{"className":11833},[391],[238,11835,11837],{"className":11836},[395],[238,11838,11840],{"className":11839},[399],[238,11841,11843],{"className":11842,"style":1223},[403],[238,11844,11845,11848],{"style":1226},[238,11846],{"className":11847,"style":412},[411],[238,11849,11851],{"className":11850},[416,417,418,419],[238,11852,11854],{"className":11853},[282,419],[238,11855,346],{"className":11856},[282,419],[238,11858,360],{"className":11859},[462],[238,11861,4479],{"className":11862,"style":4791},[462,4790],[238,11864],{"className":11865,"style":455},[429],[238,11867,3102],{"className":11868},[282],[238,11870,872],{"className":11871},[871],[238,11873,11875],{"className":11874},[399],[238,11876,11878],{"className":11877,"style":11319},[403],[238,11879],{},[11,11881,11882],{},"证毕。",[11,11884,11885,11886,11944],{},"这个方程的意义在于，它把一个涉及无限期回报的期望，转化为一个只涉及当前奖励与下一状态值函数的方程。方程右端仍然含有未知函数 ",[238,11887,11889,11906],{"className":11888,"translate":242},[241],[238,11890,11892],{"className":11891},[246],[248,11893,11894],{"xmlns":250},[252,11895,11896,11904],{},[255,11897,11898],{},[337,11899,11900,11902],{},[258,11901,4429],{},[258,11903,2297],{},[262,11905,6581],{"encoding":264},[238,11907,11909],{"className":11908,"ariaHidden":269},[268],[238,11910,11912,11915],{"className":11911},[273],[238,11913],{"className":11914,"style":278},[277],[238,11916,11918,11921],{"className":11917},[282],[238,11919,4429],{"className":11920,"style":660},[282,283],[238,11922,11924],{"className":11923},[391],[238,11925,11927],{"className":11926},[395],[238,11928,11930],{"className":11929},[399],[238,11931,11933],{"className":11932,"style":6609},[403],[238,11934,11935,11938],{"style":407},[238,11936],{"className":11937,"style":412},[411],[238,11939,11941],{"className":11940},[416,417,418,419],[238,11942,2297],{"className":11943,"style":2313},[282,283,419],"，因此它不是普通的递推式，而是一个泛函方程。",[225,11946,11948],{"id":11947},"_22-算子形式","2.2 算子形式",[11,11950,11951,11952,2282,11955,2705],{},"为了把上述方程写成更紧凑的形式，定义",[1688,11953,11954],{},"贝尔曼期望算子",[238,11956,11958,11977],{"className":11957,"translate":242},[241],[238,11959,11961],{"className":11960},[246],[248,11962,11963],{"xmlns":250},[252,11964,11965,11974],{},[255,11966,11967],{},[337,11968,11969,11972],{},[258,11970,11971],{"mathvariant":816},"T",[258,11973,2297],{},[262,11975,11976],{"encoding":264},"\\mathcal{T}^\\pi",[238,11978,11980],{"className":11979,"ariaHidden":269},[268],[238,11981,11983,11986],{"className":11982},[273],[238,11984],{"className":11985,"style":278},[277],[238,11987,11989,11993],{"className":11988},[282],[238,11990,11971],{"className":11991,"style":11992},[282,840],"margin-right:0.2542em;",[238,11994,11996],{"className":11995},[391],[238,11997,11999],{"className":11998},[395],[238,12000,12002],{"className":12001},[399],[238,12003,12005],{"className":12004,"style":6609},[403],[238,12006,12007,12010],{"style":407},[238,12008],{"className":12009,"style":412},[411],[238,12011,12013],{"className":12012},[416,417,418,419],[238,12014,2297],{"className":12015,"style":2313},[282,283,419],[238,12017,12019],{"className":12018,"translate":242},[945],[238,12020,12022,12182],{"className":12021,"translate":242},[241],[238,12023,12025],{"className":12024},[246],[248,12026,12027],{"xmlns":250,"display":954},[252,12028,12029,12179],{},[958,12030,12031,12115],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,12032,12033,12057],{},[966,12034,12035],{},[969,12036,12037],{"scriptlevel":740,"displaystyle":269},[255,12038,12039,12041,12047,12049,12051,12053,12055],{},[332,12040,335],{"stretchy":334},[337,12042,12043,12045],{},[258,12044,11971],{"mathvariant":816},[258,12046,2297],{},[258,12048,4429],{},[332,12050,360],{"stretchy":334},[332,12052,335],{"stretchy":334},[258,12054,341],{},[332,12056,360],{"stretchy":334},[966,12058,12059],{},[969,12060,12061],{"scriptlevel":740,"displaystyle":269},[255,12062,12063,12065,12067,12073,12075,12077,12079,12081,12083,12085,12095,12097,12099,12105,12107,12109,12111,12113],{},[255,12064],{},[332,12066,992],{},[6755,12068,12069,12071],{},[332,12070,3390],{},[258,12072,357],{},[258,12074,2297],{},[332,12076,335],{"stretchy":334},[258,12078,357],{},[332,12080,349],{},[258,12082,341],{},[332,12084,360],{"stretchy":334},[6755,12086,12087,12089],{},[332,12088,3390],{},[337,12090,12091,12093],{},[258,12092,341],{},[332,12094,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,12096,330],{},[332,12098,335],{"stretchy":334},[337,12100,12101,12103],{},[258,12102,341],{},[332,12104,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,12106,349],{},[258,12108,341],{},[332,12110,354],{"separator":269},[258,12112,357],{},[332,12114,360],{"stretchy":334},[963,12116,12117,12123],{},[966,12118,12119],{},[969,12120,12121],{"scriptlevel":740,"displaystyle":269},[255,12122],{},[966,12124,12125],{},[969,12126,12127],{"scriptlevel":740,"displaystyle":269},[255,12128,12129,12131,12133,12135,12177],{},[255,12130],{},[429,12132],{"width":7090},[332,12134,608],{},[255,12136,12137,12139,12141,12143,12145,12147,12149,12151,12157,12159,12161,12163,12165,12167,12173,12175],{},[332,12138,736],{"fence":269},[258,12140,600],{},[332,12142,335],{"stretchy":334},[258,12144,341],{},[332,12146,354],{"separator":269},[258,12148,357],{},[332,12150,354],{"separator":269},[337,12152,12153,12155],{},[258,12154,341],{},[332,12156,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,12158,360],{"stretchy":334},[332,12160,987],{},[258,12162,730],{},[258,12164,4429],{},[332,12166,335],{"stretchy":334},[337,12168,12169,12171],{},[258,12170,341],{},[332,12172,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,12174,360],{"stretchy":334},[332,12176,4479],{"fence":269},[258,12178,3102],{"mathvariant":344},[262,12180,12181],{"encoding":264},"\\begin{aligned}\n(\\mathcal{T}^\\pi V)(s)\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V(s')\\right].\n\\end{aligned}",[238,12183,12185],{"className":12184,"ariaHidden":269},[268],[238,12186,12188,12191],{"className":12187},[273],[238,12189],{"className":12190,"style":7153},[277],[238,12192,12194],{"className":12193},[282],[238,12195,12197,12283],{"className":12196},[958],[238,12198,12200],{"className":12199},[1117],[238,12201,12203,12275],{"className":12202},[395,848],[238,12204,12206,12272],{"className":12205},[399],[238,12207,12209,12264],{"className":12208,"style":7172},[403],[238,12210,12211,12214],{"style":7175},[238,12212],{"className":12213,"style":3510},[411],[238,12215,12217,12220,12249,12252,12255,12258,12261],{"className":12216},[282],[238,12218,335],{"className":12219},[381],[238,12221,12223,12226],{"className":12222},[282],[238,12224,11971],{"className":12225,"style":11992},[282,840],[238,12227,12229],{"className":12228},[391],[238,12230,12232],{"className":12231},[395],[238,12233,12235],{"className":12234},[399],[238,12236,12238],{"className":12237,"style":4620},[403],[238,12239,12240,12243],{"style":1226},[238,12241],{"className":12242,"style":412},[411],[238,12244,12246],{"className":12245},[416,417,418,419],[238,12247,2297],{"className":12248,"style":2313},[282,283,419],[238,12250,4429],{"className":12251,"style":660},[282,283],[238,12253,360],{"className":12254},[462],[238,12256,335],{"className":12257},[381],[238,12259,341],{"className":12260},[282,283],[238,12262,360],{"className":12263},[462],[238,12265,12266,12269],{"style":7231},[238,12267],{"className":12268,"style":3510},[411],[238,12270],{"className":12271},[282],[238,12273,872],{"className":12274},[871],[238,12276,12278],{"className":12277},[399],[238,12279,12281],{"className":12280,"style":7247},[403],[238,12282],{},[238,12284,12286],{"className":12285},[1422],[238,12287,12289,12686],{"className":12288},[395,848],[238,12290,12292,12683],{"className":12291},[399],[238,12293,12295,12533],{"className":12294,"style":7172},[403],[238,12296,12297,12300],{"style":7175},[238,12298],{"className":12299,"style":3510},[411],[238,12301,12303,12306,12309,12312,12315,12362,12365,12368,12371,12374,12377,12380,12383,12386,12389,12392,12468,12471,12474,12477,12509,12512,12515,12518,12521,12524,12527,12530],{"className":12302},[282],[238,12304],{"className":12305},[282],[238,12307],{"className":12308,"style":430},[429],[238,12310,992],{"className":12311},[434],[238,12313],{"className":12314,"style":430},[429],[238,12316,12318],{"className":12317},[1943,3493],[238,12319,12321,12354],{"className":12320},[395,848],[238,12322,12324,12351],{"className":12323},[399],[238,12325,12327,12341],{"className":12326,"style":6869},[403],[238,12328,12329,12332],{"style":6872},[238,12330],{"className":12331,"style":3510},[411],[238,12333,12335],{"className":12334},[416,417,418,419],[238,12336,12338],{"className":12337},[282,419],[238,12339,357],{"className":12340},[282,283,419],[238,12342,12343,12346],{"style":3529},[238,12344],{"className":12345,"style":3510},[411],[238,12347,12348],{},[238,12349,3390],{"className":12350},[1943,3538,3539],[238,12352,872],{"className":12353},[871],[238,12355,12357],{"className":12356},[399],[238,12358,12360],{"className":12359,"style":6904},[403],[238,12361],{},[238,12363],{"className":12364,"style":455},[429],[238,12366,2297],{"className":12367,"style":2313},[282,283],[238,12369,335],{"className":12370},[381],[238,12372,357],{"className":12373},[282,283],[238,12375],{"className":12376,"style":430},[429],[238,12378,349],{"className":12379},[434],[238,12381],{"className":12382,"style":430},[429],[238,12384,341],{"className":12385},[282,283],[238,12387,360],{"className":12388},[462],[238,12390],{"className":12391,"style":455},[429],[238,12393,12395],{"className":12394},[1943,3493],[238,12396,12398,12460],{"className":12397},[395,848],[238,12399,12401,12457],{"className":12400},[399],[238,12402,12404,12447],{"className":12403,"style":6869},[403],[238,12405,12406,12409],{"style":7296},[238,12407],{"className":12408,"style":3510},[411],[238,12410,12412],{"className":12411},[416,417,418,419],[238,12413,12415],{"className":12414},[282,419],[238,12416,12418,12421],{"className":12417},[282,419],[238,12419,341],{"className":12420},[282,283,419],[238,12422,12424],{"className":12423},[391],[238,12425,12427],{"className":12426},[395],[238,12428,12430],{"className":12429},[399],[238,12431,12433],{"className":12432,"style":4196},[403],[238,12434,12435,12438],{"style":4199},[238,12436],{"className":12437,"style":4203},[411],[238,12439,12441],{"className":12440},[416,4207,4208,419],[238,12442,12444],{"className":12443},[282,419],[238,12445,346],{"className":12446},[282,419],[238,12448,12449,12452],{"style":3529},[238,12450],{"className":12451,"style":3510},[411],[238,12453,12454],{},[238,12455,3390],{"className":12456},[1943,3538,3539],[238,12458,872],{"className":12459},[871],[238,12461,12463],{"className":12462},[399],[238,12464,12466],{"className":12465,"style":7357},[403],[238,12467],{},[238,12469],{"className":12470,"style":455},[429],[238,12472,330],{"className":12473,"style":377},[282,283],[238,12475,335],{"className":12476},[381],[238,12478,12480,12483],{"className":12479},[282],[238,12481,341],{"className":12482},[282,283],[238,12484,12486],{"className":12485},[391],[238,12487,12489],{"className":12488},[395],[238,12490,12492],{"className":12491},[399],[238,12493,12495],{"className":12494,"style":1223},[403],[238,12496,12497,12500],{"style":1226},[238,12498],{"className":12499,"style":412},[411],[238,12501,12503],{"className":12502},[416,417,418,419],[238,12504,12506],{"className":12505},[282,419],[238,12507,346],{"className":12508},[282,419],[238,12510],{"className":12511,"style":430},[429],[238,12513,349],{"className":12514},[434],[238,12516],{"className":12517,"style":430},[429],[238,12519,341],{"className":12520},[282,283],[238,12522,354],{"className":12523},[451],[238,12525],{"className":12526,"style":455},[429],[238,12528,357],{"className":12529},[282,283],[238,12531,360],{"className":12532},[462],[238,12534,12535,12538],{"style":7231},[238,12536],{"className":12537,"style":3510},[411],[238,12539,12541,12544,12547,12550,12553,12556,12677,12680],{"className":12540},[282],[238,12542],{"className":12543},[282],[238,12545],{"className":12546,"style":7439},[429],[238,12548],{"className":12549,"style":660},[429],[238,12551,608],{"className":12552},[664],[238,12554],{"className":12555,"style":660},[429],[238,12557,12559,12562,12565,12568,12571,12574,12577,12580,12583,12586,12618,12621,12624,12627,12630,12633,12636,12639,12671,12674],{"className":12558},[3041],[238,12560,736],{"className":12561,"style":4791},[381,4790],[238,12563,600],{"className":12564,"style":637},[282,283],[238,12566,335],{"className":12567},[381],[238,12569,341],{"className":12570},[282,283],[238,12572,354],{"className":12573},[451],[238,12575],{"className":12576,"style":455},[429],[238,12578,357],{"className":12579},[282,283],[238,12581,354],{"className":12582},[451],[238,12584],{"className":12585,"style":455},[429],[238,12587,12589,12592],{"className":12588},[282],[238,12590,341],{"className":12591},[282,283],[238,12593,12595],{"className":12594},[391],[238,12596,12598],{"className":12597},[395],[238,12599,12601],{"className":12600},[399],[238,12602,12604],{"className":12603,"style":1223},[403],[238,12605,12606,12609],{"style":1226},[238,12607],{"className":12608,"style":412},[411],[238,12610,12612],{"className":12611},[416,417,418,419],[238,12613,12615],{"className":12614},[282,419],[238,12616,346],{"className":12617},[282,419],[238,12619,360],{"className":12620},[462],[238,12622],{"className":12623,"style":660},[429],[238,12625,987],{"className":12626},[664],[238,12628],{"className":12629,"style":660},[429],[238,12631,730],{"className":12632,"style":764},[282,283],[238,12634,4429],{"className":12635,"style":660},[282,283],[238,12637,335],{"className":12638},[381],[238,12640,12642,12645],{"className":12641},[282],[238,12643,341],{"className":12644},[282,283],[238,12646,12648],{"className":12647},[391],[238,12649,12651],{"className":12650},[395],[238,12652,12654],{"className":12653},[399],[238,12655,12657],{"className":12656,"style":1223},[403],[238,12658,12659,12662],{"style":1226},[238,12660],{"className":12661,"style":412},[411],[238,12663,12665],{"className":12664},[416,417,418,419],[238,12666,12668],{"className":12667},[282,419],[238,12669,346],{"className":12670},[282,419],[238,12672,360],{"className":12673},[462],[238,12675,4479],{"className":12676,"style":4791},[462,4790],[238,12678],{"className":12679,"style":455},[429],[238,12681,3102],{"className":12682},[282],[238,12684,872],{"className":12685},[871],[238,12687,12689],{"className":12688},[399],[238,12690,12692],{"className":12691,"style":7247},[403],[238,12693],{},[11,12695,12696],{},"于是贝尔曼期望方程可以写成不动点形式",[238,12698,12700],{"className":12699,"translate":242},[945],[238,12701,12703,12737],{"className":12702,"translate":242},[241],[238,12704,12706],{"className":12705},[246],[248,12707,12708],{"xmlns":250,"display":954},[252,12709,12710,12734],{},[255,12711,12712,12718,12720,12726,12732],{},[337,12713,12714,12716],{},[258,12715,4429],{},[258,12717,2297],{},[332,12719,992],{},[337,12721,12722,12724],{},[258,12723,11971],{"mathvariant":816},[258,12725,2297],{},[337,12727,12728,12730],{},[258,12729,4429],{},[258,12731,2297],{},[258,12733,3102],{"mathvariant":344},[262,12735,12736],{"encoding":264},"V^\\pi=\\mathcal{T}^\\pi V^\\pi.",[238,12738,12740,12784],{"className":12739,"ariaHidden":269},[268],[238,12741,12743,12746,12775,12778,12781],{"className":12742},[273],[238,12744],{"className":12745,"style":4620},[277],[238,12747,12749,12752],{"className":12748},[282],[238,12750,4429],{"className":12751,"style":660},[282,283],[238,12753,12755],{"className":12754},[391],[238,12756,12758],{"className":12757},[395],[238,12759,12761],{"className":12760},[399],[238,12762,12764],{"className":12763,"style":4620},[403],[238,12765,12766,12769],{"style":1226},[238,12767],{"className":12768,"style":412},[411],[238,12770,12772],{"className":12771},[416,417,418,419],[238,12773,2297],{"className":12774,"style":2313},[282,283,419],[238,12776],{"className":12777,"style":430},[429],[238,12779,992],{"className":12780},[434],[238,12782],{"className":12783,"style":430},[429],[238,12785,12787,12790,12819,12848],{"className":12786},[273],[238,12788],{"className":12789,"style":4620},[277],[238,12791,12793,12796],{"className":12792},[282],[238,12794,11971],{"className":12795,"style":11992},[282,840],[238,12797,12799],{"className":12798},[391],[238,12800,12802],{"className":12801},[395],[238,12803,12805],{"className":12804},[399],[238,12806,12808],{"className":12807,"style":4620},[403],[238,12809,12810,12813],{"style":1226},[238,12811],{"className":12812,"style":412},[411],[238,12814,12816],{"className":12815},[416,417,418,419],[238,12817,2297],{"className":12818,"style":2313},[282,283,419],[238,12820,12822,12825],{"className":12821},[282],[238,12823,4429],{"className":12824,"style":660},[282,283],[238,12826,12828],{"className":12827},[391],[238,12829,12831],{"className":12830},[395],[238,12832,12834],{"className":12833},[399],[238,12835,12837],{"className":12836,"style":4620},[403],[238,12838,12839,12842],{"style":1226},[238,12840],{"className":12841,"style":412},[411],[238,12843,12845],{"className":12844},[416,417,418,419],[238,12846,2297],{"className":12847,"style":2313},[282,283,419],[238,12849,3102],{"className":12850},[282],[11,12852,12853,12854,12912,12913,12971,12972,13000,13001,13065,13066,13124],{},"这里的 ",[238,12855,12857,12874],{"className":12856,"translate":242},[241],[238,12858,12860],{"className":12859},[246],[248,12861,12862],{"xmlns":250},[252,12863,12864,12872],{},[255,12865,12866],{},[337,12867,12868,12870],{},[258,12869,4429],{},[258,12871,2297],{},[262,12873,6581],{"encoding":264},[238,12875,12877],{"className":12876,"ariaHidden":269},[268],[238,12878,12880,12883],{"className":12879},[273],[238,12881],{"className":12882,"style":278},[277],[238,12884,12886,12889],{"className":12885},[282],[238,12887,4429],{"className":12888,"style":660},[282,283],[238,12890,12892],{"className":12891},[391],[238,12893,12895],{"className":12894},[395],[238,12896,12898],{"className":12897},[399],[238,12899,12901],{"className":12900,"style":6609},[403],[238,12902,12903,12906],{"style":407},[238,12904],{"className":12905,"style":412},[411],[238,12907,12909],{"className":12908},[416,417,418,419],[238,12910,2297],{"className":12911,"style":2313},[282,283,419]," 不是某个数，而是函数空间中的一个点。算子 ",[238,12914,12916,12933],{"className":12915,"translate":242},[241],[238,12917,12919],{"className":12918},[246],[248,12920,12921],{"xmlns":250},[252,12922,12923,12931],{},[255,12924,12925],{},[337,12926,12927,12929],{},[258,12928,11971],{"mathvariant":816},[258,12930,2297],{},[262,12932,11976],{"encoding":264},[238,12934,12936],{"className":12935,"ariaHidden":269},[268],[238,12937,12939,12942],{"className":12938},[273],[238,12940],{"className":12941,"style":278},[277],[238,12943,12945,12948],{"className":12944},[282],[238,12946,11971],{"className":12947,"style":11992},[282,840],[238,12949,12951],{"className":12950},[391],[238,12952,12954],{"className":12953},[395],[238,12955,12957],{"className":12956},[399],[238,12958,12960],{"className":12959,"style":6609},[403],[238,12961,12962,12965],{"style":407},[238,12963],{"className":12964,"style":412},[411],[238,12966,12968],{"className":12967},[416,417,418,419],[238,12969,2297],{"className":12970,"style":2313},[282,283,419]," 把一个有界函数 ",[238,12973,12975,12988],{"className":12974,"translate":242},[241],[238,12976,12978],{"className":12977},[246],[248,12979,12980],{"xmlns":250},[252,12981,12982,12986],{},[255,12983,12984],{},[258,12985,4429],{},[262,12987,4429],{"encoding":264},[238,12989,12991],{"className":12990,"ariaHidden":269},[268],[238,12992,12994,12997],{"className":12993},[273],[238,12995],{"className":12996,"style":278},[277],[238,12998,4429],{"className":12999,"style":660},[282,283]," 映射为另一个有界函数 ",[238,13002,13004,13024],{"className":13003,"translate":242},[241],[238,13005,13007],{"className":13006},[246],[248,13008,13009],{"xmlns":250},[252,13010,13011,13021],{},[255,13012,13013,13019],{},[337,13014,13015,13017],{},[258,13016,11971],{"mathvariant":816},[258,13018,2297],{},[258,13020,4429],{},[262,13022,13023],{"encoding":264},"\\mathcal{T}^\\pi V",[238,13025,13027],{"className":13026,"ariaHidden":269},[268],[238,13028,13030,13033,13062],{"className":13029},[273],[238,13031],{"className":13032,"style":278},[277],[238,13034,13036,13039],{"className":13035},[282],[238,13037,11971],{"className":13038,"style":11992},[282,840],[238,13040,13042],{"className":13041},[391],[238,13043,13045],{"className":13044},[395],[238,13046,13048],{"className":13047},[399],[238,13049,13051],{"className":13050,"style":6609},[403],[238,13052,13053,13056],{"style":407},[238,13054],{"className":13055,"style":412},[411],[238,13057,13059],{"className":13058},[416,417,418,419],[238,13060,2297],{"className":13061,"style":2313},[282,283,419],[238,13063,4429],{"className":13064,"style":660},[282,283],"。因此，求解贝尔曼方程等价于寻找算子 ",[238,13067,13069,13086],{"className":13068,"translate":242},[241],[238,13070,13072],{"className":13071},[246],[248,13073,13074],{"xmlns":250},[252,13075,13076,13084],{},[255,13077,13078],{},[337,13079,13080,13082],{},[258,13081,11971],{"mathvariant":816},[258,13083,2297],{},[262,13085,11976],{"encoding":264},[238,13087,13089],{"className":13088,"ariaHidden":269},[268],[238,13090,13092,13095],{"className":13091},[273],[238,13093],{"className":13094,"style":278},[277],[238,13096,13098,13101],{"className":13097},[282],[238,13099,11971],{"className":13100,"style":11992},[282,840],[238,13102,13104],{"className":13103},[391],[238,13105,13107],{"className":13106},[395],[238,13108,13110],{"className":13109},[399],[238,13111,13113],{"className":13112,"style":6609},[403],[238,13114,13115,13118],{"style":407},[238,13116],{"className":13117,"style":412},[411],[238,13119,13121],{"className":13120},[416,417,418,419],[238,13122,2297],{"className":13123,"style":2313},[282,283,419]," 的不动点。",[225,13126,13128],{"id":13127},"_23-有限状态下的矩阵解","2.3 有限状态下的矩阵解",[11,13130,13131,13132,13225,13226,1838,13308,13420,13421,13548],{},"当 ",[238,13133,13135,13168],{"className":13134,"translate":242},[241],[238,13136,13138],{"className":13137},[246],[248,13139,13140],{"xmlns":250},[252,13141,13142,13165],{},[255,13143,13144,13146,13148,13151,13153,13155,13157,13159,13162],{},[258,13145,260],{},[332,13147,992],{},[332,13149,13150],{"stretchy":334},"{",[738,13152,745],{},[332,13154,354],{"separator":269},[332,13156,2840],{},[332,13158,354],{"separator":269},[258,13160,13161],{},"n",[332,13163,13164],{"stretchy":334},"}",[262,13166,13167],{"encoding":264},"S=\\{1,\\dots,n\\}",[238,13169,13171,13189],{"className":13170,"ariaHidden":269},[268],[238,13172,13174,13177,13180,13183,13186],{"className":13173},[273],[238,13175],{"className":13176,"style":278},[277],[238,13178,260],{"className":13179,"style":284},[282,283],[238,13181],{"className":13182,"style":430},[429],[238,13184,992],{"className":13185},[434],[238,13187],{"className":13188,"style":430},[429],[238,13190,13192,13195,13198,13201,13204,13207,13210,13213,13216,13219,13222],{"className":13191},[273],[238,13193],{"className":13194,"style":444},[277],[238,13196,13150],{"className":13197},[381],[238,13199,745],{"className":13200},[282],[238,13202,354],{"className":13203},[451],[238,13205],{"className":13206,"style":455},[429],[238,13208,2840],{"className":13209},[3041],[238,13211],{"className":13212,"style":455},[429],[238,13214,354],{"className":13215},[451],[238,13217],{"className":13218,"style":455},[429],[238,13220,13161],{"className":13221},[282,283],[238,13223,13164],{"className":13224},[462]," 为有限集时，贝尔曼期望方程可以写成线性方程组。定义向量 ",[238,13227,13229,13251],{"className":13228,"translate":242},[241],[238,13230,13232],{"className":13231},[246],[248,13233,13234],{"xmlns":250},[252,13235,13236,13248],{},[255,13237,13238,13240,13242],{},[258,13239,4429],{},[332,13241,733],{},[337,13243,13244,13246],{},[258,13245,621],{"mathvariant":620},[258,13247,13161],{},[262,13249,13250],{"encoding":264},"V\\in\\mathbb{R}^n",[238,13252,13254,13273],{"className":13253,"ariaHidden":269},[268],[238,13255,13257,13261,13264,13267,13270],{"className":13256},[273],[238,13258],{"className":13259,"style":13260},[277],"height:0.7224em;vertical-align:-0.0391em;",[238,13262,4429],{"className":13263,"style":660},[282,283],[238,13265],{"className":13266,"style":430},[429],[238,13268,733],{"className":13269},[434],[238,13271],{"className":13272,"style":430},[429],[238,13274,13276,13279],{"className":13275},[273],[238,13277],{"className":13278,"style":710},[277],[238,13280,13282,13285],{"className":13281},[282],[238,13283,621],{"className":13284},[282,714],[238,13286,13288],{"className":13287},[391],[238,13289,13291],{"className":13290},[395],[238,13292,13294],{"className":13293},[399],[238,13295,13297],{"className":13296,"style":6609},[403],[238,13298,13299,13302],{"style":407},[238,13300],{"className":13301,"style":412},[411],[238,13303,13305],{"className":13304},[416,417,418,419],[238,13306,13161],{"className":13307},[282,283,419],[238,13309,13311,13337],{"className":13310,"translate":242},[241],[238,13312,13314],{"className":13313},[246],[248,13315,13316],{"xmlns":250},[252,13317,13318,13334],{},[255,13319,13320,13326,13328],{},[337,13321,13322,13324],{},[258,13323,600],{},[258,13325,2297],{},[332,13327,733],{},[337,13329,13330,13332],{},[258,13331,621],{"mathvariant":620},[258,13333,13161],{},[262,13335,13336],{"encoding":264},"r^\\pi\\in\\mathbb{R}^n",[238,13338,13340,13385],{"className":13339,"ariaHidden":269},[268],[238,13341,13343,13347,13376,13379,13382],{"className":13342},[273],[238,13344],{"className":13345,"style":13346},[277],"height:0.7035em;vertical-align:-0.0391em;",[238,13348,13350,13353],{"className":13349},[282],[238,13351,600],{"className":13352,"style":637},[282,283],[238,13354,13356],{"className":13355},[391],[238,13357,13359],{"className":13358},[395],[238,13360,13362],{"className":13361},[399],[238,13363,13365],{"className":13364,"style":6609},[403],[238,13366,13367,13370],{"style":407},[238,13368],{"className":13369,"style":412},[411],[238,13371,13373],{"className":13372},[416,417,418,419],[238,13374,2297],{"className":13375,"style":2313},[282,283,419],[238,13377],{"className":13378,"style":430},[429],[238,13380,733],{"className":13381},[434],[238,13383],{"className":13384,"style":430},[429],[238,13386,13388,13391],{"className":13387},[273],[238,13389],{"className":13390,"style":710},[277],[238,13392,13394,13397],{"className":13393},[282],[238,13395,621],{"className":13396},[282,714],[238,13398,13400],{"className":13399},[391],[238,13401,13403],{"className":13402},[395],[238,13404,13406],{"className":13405},[399],[238,13407,13409],{"className":13408,"style":6609},[403],[238,13410,13411,13414],{"style":407},[238,13412],{"className":13413,"style":412},[411],[238,13415,13417],{"className":13416},[416,417,418,419],[238,13418,13161],{"className":13419},[282,283,419],"，以及矩阵 ",[238,13422,13424,13456],{"className":13423,"translate":242},[241],[238,13425,13427],{"className":13426},[246],[248,13428,13429],{"xmlns":250},[252,13430,13431,13453],{},[255,13432,13433,13439,13441],{},[337,13434,13435,13437],{},[258,13436,330],{},[258,13438,2297],{},[332,13440,733],{},[337,13442,13443,13445],{},[258,13444,621],{"mathvariant":620},[255,13446,13447,13449,13451],{},[258,13448,13161],{},[332,13450,608],{},[258,13452,13161],{},[262,13454,13455],{"encoding":264},"P^\\pi\\in\\mathbb{R}^{n\\times 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i)\\sum_{j}P(j\\mid i,a)r(i,a,j),\\\\\nP^\\pi_{ij}&=\\sum_{a}\\pi(a\\mid i)P(j\\mid 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P^\\pi)V^\\pi=r^\\pi.",[238,14605,14607,14629,14708],{"className":14606,"ariaHidden":269},[268],[238,14608,14610,14613,14616,14620,14623,14626],{"className":14609},[273],[238,14611],{"className":14612,"style":444},[277],[238,14614,335],{"className":14615},[381],[238,14617,14572],{"className":14618,"style":14619},[282,283],"margin-right:0.0785em;",[238,14621],{"className":14622,"style":660},[429],[238,14624,5577],{"className":14625},[664],[238,14627],{"className":14628,"style":660},[429],[238,14630,14632,14635,14638,14667,14670,14699,14702,14705],{"className":14631},[273],[238,14633],{"className":14634,"style":444},[277],[238,14636,730],{"className":14637,"style":764},[282,283],[238,14639,14641,14644],{"className":14640},[282],[238,14642,330],{"className":14643,"style":377},[282,283],[238,14645,14647],{"className":14646},[391],[238,14648,14650],{"className":14649},[395],[238,14651,14653],{"className":14652},[399],[238,14654,14656],{"className":14655,"style":4620},[403],[238,14657,14658,14661],{"style":1226},[238,14659],{"className":14660,"style":412},[411],[238,14662,14664],{"className":14663},[416,417,418,419],[238,14665,2297],{"className":14666,"style":2313},[282,283,419],[238,14668,360],{"className":14669},[462],[238,14671,14673,14676],{"className":14672},[282],[238,14674,4429],{"className":14675,"style":660},[282,283],[238,14677,14679],{"className":14678},[391],[238,14680,14682],{"className":14681},[395],[238,14683,14685],{"className":14684},[399],[238,14686,14688],{"className":14687,"style":4620},[403],[238,14689,14690,14693],{"style":1226},[238,14691],{"className":14692,"style":412},[411],[238,14694,14696],{"className":14695},[416,417,418,419],[238,14697,2297],{"className":14698,"style":2313},[282,283,419],[238,14700],{"className":14701,"style":430},[429],[238,14703,992],{"className":14704},[434],[238,14706],{"className":14707,"style":430},[429],[238,14709,14711,14714,14743],{"className":14710},[273],[238,14712],{"className":14713,"style":4620},[277],[238,14715,14717,14720],{"className":14716},[282],[238,14718,600],{"className":14719,"style":637},[282,283],[238,14721,14723],{"className":14722},[391],[238,14724,14726],{"className":14725},[395],[238,14727,14729],{"className":14728},[399],[238,14730,14732],{"className":14731,"style":4620},[403],[238,14733,14734,14737],{"style":1226},[238,14735],{"className":14736,"style":412},[411],[238,14738,14740],{"className":14739},[416,417,418,419],[238,14741,2297],{"className":14742,"style":2313},[282,283,419],[238,14744,3102],{"className":14745},[282],[230,14747,14750,14914],{"id":14748,"title":14749},"proposition-matrix-solution","命题 2.2（矩阵解与 Neumann 级数）",[11,14751,236,14752,14826,14827,14913],{},[238,14753,14755,14777],{"className":14754,"translate":242},[241],[238,14756,14758],{"className":14757},[246],[248,14759,14760],{"xmlns":250},[252,14761,14762,14774],{},[255,14763,14764,14766,14768,14770,14772],{},[738,14765,740],{},[332,14767,2023],{},[258,14769,730],{},[332,14771,1892],{},[738,14773,745],{},[262,14775,14776],{"encoding":264},"0\\le\\gamma\u003C1",[238,14778,14780,14799,14817],{"className":14779,"ariaHidden":269},[268],[238,14781,14783,14787,14790,14793,14796],{"className":14782},[273],[238,14784],{"className":14785,"style":14786},[277],"height:0.7804em;vertical-align:-0.136em;",[238,14788,740],{"className":14789},[282],[238,14791],{"className":14792,"style":430},[429],[238,14794,2023],{"className":14795},[434],[238,14797],{"className":14798,"style":430},[429],[238,14800,14802,14805,14808,14811,14814],{"className":14801},[273],[238,14803],{"className":14804,"style":760},[277],[238,14806,730],{"className":14807,"style":764},[282,283],[238,14809],{"className":14810,"style":430},[429],[238,14812,1892],{"className":14813},[434],[238,14815],{"className":14816,"style":430},[429],[238,14818,14820,14823],{"className":14819},[273],[238,14821],{"className":14822,"style":2672},[277],[238,14824,745],{"className":14825},[282],"。则 ",[238,14828,14830,14854],{"className":14829,"translate":242},[241],[238,14831,14833],{"className":14832},[246],[248,14834,14835],{"xmlns":250},[252,14836,14837,14851],{},[255,14838,14839,14841,14843,14845],{},[258,14840,14572],{},[332,14842,5577],{},[258,14844,730],{},[337,14846,14847,14849],{},[258,14848,330],{},[258,14850,2297],{},[262,14852,14853],{"encoding":264},"I-\\gamma P^\\pi",[238,14855,14857,14875],{"className":14856,"ariaHidden":269},[268],[238,14858,14860,14863,14866,14869,14872],{"className":14859},[273],[238,14861],{"className":14862,"style":653},[277],[238,14864,14572],{"className":14865,"style":14619},[282,283],[238,14867],{"className":14868,"style":660},[429],[238,14870,5577],{"className":14871},[664],[238,14873],{"className":14874,"style":660},[429],[238,14876,14878,14881,14884],{"className":14877},[273],[238,14879],{"className":14880,"style":6651},[277],[238,14882,730],{"className":14883,"style":764},[282,283],[238,14885,14887,14890],{"className":14886},[282],[238,14888,330],{"className":14889,"style":377},[282,283],[238,14891,14893],{"className":14892},[391],[238,14894,14896],{"className":14895},[395],[238,14897,14899],{"className":14898},[399],[238,14900,14902],{"className":14901,"style":6609},[403],[238,14903,14904,14907],{"style":407},[238,14905],{"className":14906,"style":412},[411],[238,14908,14910],{"className":14909},[416,417,418,419],[238,14911,2297],{"className":14912,"style":2313},[282,283,419]," 可逆，且",[238,14915,14917],{"className":14916,"translate":242},[945],[238,14918,14920,15014],{"className":14919,"translate":242},[241],[238,14921,14923],{"className":14922},[246],[248,14924,14925],{"xmlns":250,"display":954},[252,14926,14927,15011],{},[255,14928,14929,14935,14937,14939,14941,14943,14945,14951,14961,14967,14969,14983,14989,14991,14997,15003,15009],{},[337,14930,14931,14933],{},[258,14932,4429],{},[258,14934,2297],{},[332,14936,992],{},[332,14938,335],{"stretchy":334},[258,14940,14572],{},[332,14942,5577],{},[258,14944,730],{},[337,14946,14947,14949],{},[258,14948,330],{},[258,14950,2297],{},[337,14952,14953,14955],{},[332,14954,360],{"stretchy":334},[255,14956,14957,14959],{},[332,14958,5577],{},[738,14960,745],{},[337,14962,14963,14965],{},[258,14964,600],{},[258,14966,2297],{},[332,14968,992],{},[3386,14970,14971,14973,14981],{},[332,14972,3390],{},[255,14974,14975,14977,14979],{},[258,14976,3395],{},[332,14978,992],{},[738,14980,740],{},[258,14982,1895],{"mathvariant":344},[337,14984,14985,14987],{},[258,14986,730],{},[258,14988,3395],{},[332,14990,335],{"stretchy":334},[337,14992,14993,14995],{},[258,14994,330],{},[258,14996,2297],{},[337,14998,14999,15001],{},[332,15000,360],{"stretchy":334},[258,15002,3395],{},[337,15004,15005,15007],{},[258,15006,600],{},[258,15008,2297],{},[258,15010,3102],{"mathvariant":344},[262,15012,15013],{"encoding":264},"V^\\pi=(I-\\gamma P^\\pi)^{-1}r^\\pi\n=\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi.",[238,15015,15017,15061,15082,15195],{"className":15016,"ariaHidden":269},[268],[238,15018,15020,15023,15052,15055,15058],{"className":15019},[273],[238,15021],{"className":15022,"style":4620},[277],[238,15024,15026,15029],{"className":15025},[282],[238,15027,4429],{"className":15028,"style":660},[282,283],[238,15030,15032],{"className":15031},[391],[238,15033,15035],{"className":15034},[395],[238,15036,15038],{"className":15037},[399],[238,15039,15041],{"className":15040,"style":4620},[403],[238,15042,15043,15046],{"style":1226},[238,15044],{"className":15045,"style":412},[411],[238,15047,15049],{"className":15048},[416,417,418,419],[238,15050,2297],{"className":15051,"style":2313},[282,283,419],[238,15053],{"className":15054,"style":430},[429],[238,15056,992],{"className":15057},[434],[238,15059],{"className":15060,"style":430},[429],[238,15062,15064,15067,15070,15073,15076,15079],{"className":15063},[273],[238,15065],{"className":15066,"style":444},[277],[238,15068,335],{"className":15069},[381],[238,15071,14572],{"className":15072,"style":14619},[282,283],[238,15074],{"className":15075,"style":660},[429],[238,15077,5577],{"className":15078},[664],[238,15080],{"className":15081,"style":660},[429],[238,15083,15085,15089,15092,15121,15157,15186,15189,15192],{"className":15084},[273],[238,15086],{"className":15087,"style":15088},[277],"height:1.1141em;vertical-align:-0.25em;",[238,15090,730],{"className":15091,"style":764},[282,283],[238,15093,15095,15098],{"className":15094},[282],[238,15096,330],{"className":15097,"style":377},[282,283],[238,15099,15101],{"className":15100},[391],[238,15102,15104],{"className":15103},[395],[238,15105,15107],{"className":15106},[399],[238,15108,15110],{"className":15109,"style":4620},[403],[238,15111,15112,15115],{"style":1226},[238,15113],{"className":15114,"style":412},[411],[238,15116,15118],{"className":15117},[416,417,418,419],[238,15119,2297],{"className":15120,"style":2313},[282,283,419],[238,15122,15124,15127],{"className":15123},[462],[238,15125,360],{"className":15126},[462],[238,15128,15130],{"className":15129},[391],[238,15131,15133],{"className":15132},[395],[238,15134,15136],{"className":15135},[399],[238,15137,15140],{"className":15138,"style":15139},[403],"height:0.8641em;",[238,15141,15142,15145],{"style":1226},[238,15143],{"className":15144,"style":412},[411],[238,15146,15148],{"className":15147},[416,417,418,419],[238,15149,15151,15154],{"className":15150},[282,419],[238,15152,5577],{"className":15153},[282,419],[238,15155,745],{"className":15156},[282,419],[238,15158,15160,15163],{"className":15159},[282],[238,15161,600],{"className":15162,"style":637},[282,283],[238,15164,15166],{"className":15165},[391],[238,15167,15169],{"className":15168},[395],[238,15170,15172],{"className":15171},[399],[238,15173,15175],{"className":15174,"style":4620},[403],[238,15176,15177,15180],{"style":1226},[238,15178],{"className":15179,"style":412},[411],[238,15181,15183],{"className":15182},[416,417,418,419],[238,15184,2297],{"className":15185,"style":2313},[282,283,419],[238,15187],{"className":15188,"style":430},[429],[238,15190,992],{"className":15191},[434],[238,15193],{"className":15194,"style":430},[429],[238,15196,15198,15201,15268,15271,15300,15303,15332,15361,15390],{"className":15197},[273],[238,15199],{"className":15200,"style":3489},[277],[238,15202,15204],{"className":15203},[1943,3493],[238,15205,15207,15260],{"className":15206},[395,848],[238,15208,15210,15257],{"className":15209},[399],[238,15211,15213,15233,15243],{"className":15212,"style":3503},[403],[238,15214,15215,15218],{"style":3506},[238,15216],{"className":15217,"style":3510},[411],[238,15219,15221],{"className":15220},[416,417,418,419],[238,15222,15224,15227,15230],{"className":15223},[282,419],[238,15225,3395],{"className":15226,"style":3520},[282,283,419],[238,15228,992],{"className":15229},[434,419],[238,15231,740],{"className":15232},[282,419],[238,15234,15235,15238],{"style":3529},[238,15236],{"className":15237,"style":3510},[411],[238,15239,15240],{},[238,15241,3390],{"className":15242},[1943,3538,3539],[238,15244,15245,15248],{"style":3542},[238,15246],{"className":15247,"style":3510},[411],[238,15249,15251],{"className":15250},[416,417,418,419],[238,15252,15254],{"className":15253},[282,419],[238,15255,1895],{"className":15256},[282,419],[238,15258,872],{"className":15259},[871],[238,15261,15263],{"className":15262},[399],[238,15264,15266],{"className":15265,"style":3564},[403],[238,15267],{},[238,15269],{"className":15270,"style":455},[429],[238,15272,15274,15277],{"className":15273},[282],[238,15275,730],{"className":15276,"style":764},[282,283],[238,15278,15280],{"className":15279},[391],[238,15281,15283],{"className":15282},[395],[238,15284,15286],{"className":15285},[399],[238,15287,15289],{"className":15288,"style":3588},[403],[238,15290,15291,15294],{"style":1226},[238,15292],{"className":15293,"style":412},[411],[238,15295,15297],{"className":15296},[416,417,418,419],[238,15298,3395],{"className":15299,"style":3520},[282,283,419],[238,15301,335],{"className":15302},[381],[238,15304,15306,15309],{"className":15305},[282],[238,15307,330],{"className":15308,"style":377},[282,283],[238,15310,15312],{"className":15311},[391],[238,15313,15315],{"className":15314},[395],[238,15316,15318],{"className":15317},[399],[238,15319,15321],{"className":15320,"style":4620},[403],[238,15322,15323,15326],{"style":1226},[238,15324],{"className":15325,"style":412},[411],[238,15327,15329],{"className":15328},[416,417,418,419],[238,15330,2297],{"className":15331,"style":2313},[282,283,419],[238,15333,15335,15338],{"className":15334},[462],[238,15336,360],{"className":15337},[462],[238,15339,15341],{"className":15340},[391],[238,15342,15344],{"className":15343},[395],[238,15345,15347],{"className":15346},[399],[238,15348,15350],{"className":15349,"style":3588},[403],[238,15351,15352,15355],{"style":1226},[238,15353],{"className":15354,"style":412},[411],[238,15356,15358],{"className":15357},[416,417,418,419],[238,15359,3395],{"className":15360,"style":3520},[282,283,419],[238,15362,15364,15367],{"className":15363},[282],[238,15365,600],{"className":15366,"style":637},[282,283],[238,15368,15370],{"className":15369},[391],[238,15371,15373],{"className":15372},[395],[238,15374,15376],{"className":15375},[399],[238,15377,15379],{"className":15378,"style":4620},[403],[238,15380,15381,15384],{"style":1226},[238,15382],{"className":15383,"style":412},[411],[238,15385,15387],{"className":15386},[416,417,418,419],[238,15388,2297],{"className":15389,"style":2313},[282,283,419],[238,15391,3102],{"className":15392},[282],[11,15394,15395,15397,15398,15484,15485,15544,15545,15573,15574,15707,15708,15873,15874,16016],{},[1688,15396,5794],{}," 记 ",[238,15399,15401,15425],{"className":15400,"translate":242},[241],[238,15402,15404],{"className":15403},[246],[248,15405,15406],{"xmlns":250},[252,15407,15408,15422],{},[255,15409,15410,15412,15414,15416],{},[258,15411,4494],{},[332,15413,992],{},[258,15415,730],{},[337,15417,15418,15420],{},[258,15419,330],{},[258,15421,2297],{},[262,15423,15424],{"encoding":264},"Q=\\gamma P^\\pi",[238,15426,15428,15446],{"className":15427,"ariaHidden":269},[268],[238,15429,15431,15434,15437,15440,15443],{"className":15430},[273],[238,15432],{"className":15433,"style":6651},[277],[238,15435,4494],{"className":15436},[282,283],[238,15438],{"className":15439,"style":430},[429],[238,15441,992],{"className":15442},[434],[238,15444],{"className":15445,"style":430},[429],[238,15447,15449,15452,15455],{"className":15448},[273],[238,15450],{"className":15451,"style":6651},[277],[238,15453,730],{"className":15454,"style":764},[282,283],[238,15456,15458,15461],{"className":15457},[282],[238,15459,330],{"className":15460,"style":377},[282,283],[238,15462,15464],{"className":15463},[391],[238,15465,15467],{"className":15466},[395],[238,15468,15470],{"className":15469},[399],[238,15471,15473],{"className":15472,"style":6609},[403],[238,15474,15475,15478],{"style":407},[238,15476],{"className":15477,"style":412},[411],[238,15479,15481],{"className":15480},[416,417,418,419],[238,15482,2297],{"className":15483,"style":2313},[282,283,419],"。由于 ",[238,15486,15488,15506],{"className":15487,"translate":242},[241],[238,15489,15491],{"className":15490},[246],[248,15492,15493],{"xmlns":250},[252,15494,15495,15503],{},[255,15496,15497],{},[337,15498,15499,15501],{},[258,15500,330],{},[258,15502,2297],{},[262,15504,15505],{"encoding":264},"P^\\pi",[238,15507,15509],{"className":15508,"ariaHidden":269},[268],[238,15510,15512,15515],{"className":15511},[273],[238,15513],{"className":15514,"style":278},[277],[238,15516,15518,15521],{"className":15517},[282],[238,15519,330],{"className":15520,"style":377},[282,283],[238,15522,15524],{"className":15523},[391],[238,15525,15527],{"className":15526},[395],[238,15528,15530],{"className":15529},[399],[238,15531,15533],{"className":15532,"style":6609},[403],[238,15534,15535,15538],{"style":407},[238,15536],{"className":15537,"style":412},[411],[238,15539,15541],{"className":15540},[416,417,418,419],[238,15542,2297],{"className":15543,"style":2313},[282,283,419]," 的每一行之和为 ",[238,15546,15548,15561],{"className":15547,"translate":242},[241],[238,15549,15551],{"className":15550},[246],[248,15552,15553],{"xmlns":250},[252,15554,15555,15559],{},[255,15556,15557],{},[738,15558,745],{},[262,15560,745],{"encoding":264},[238,15562,15564],{"className":15563,"ariaHidden":269},[268],[238,15565,15567,15570],{"className":15566},[273],[238,15568],{"className":15569,"style":2672},[277],[238,15571,745],{"className":15572},[282],"，有 ",[238,15575,15577,15608],{"className":15576,"translate":242},[241],[238,15578,15580],{"className":15579},[246],[248,15581,15582],{"xmlns":250},[252,15583,15584,15605],{},[255,15585,15586,15589,15595,15601,15603],{},[258,15587,15588],{"mathvariant":344},"∥",[337,15590,15591,15593],{},[258,15592,330],{},[258,15594,2297],{},[812,15596,15597,15599],{},[258,15598,15588],{"mathvariant":344},[258,15600,1895],{"mathvariant":344},[332,15602,992],{},[738,15604,745],{},[262,15606,15607],{"encoding":264},"\\|P^\\pi\\|_\\infty=1",[238,15609,15611,15698],{"className":15610,"ariaHidden":269},[268],[238,15612,15614,15617,15620,15649,15689,15692,15695],{"className":15613},[273],[238,15615],{"className":15616,"style":444},[277],[238,15618,15588],{"className":15619},[282],[238,15621,15623,15626],{"className":15622},[282],[238,15624,330],{"className":15625,"style":377},[282,283],[238,15627,15629],{"className":15628},[391],[238,15630,15632],{"className":15631},[395],[238,15633,15635],{"className":15634},[399],[238,15636,15638],{"className":15637,"style":6609},[403],[238,15639,15640,15643],{"style":407},[238,15641],{"className":15642,"style":412},[411],[238,15644,15646],{"className":15645},[416,417,418,419],[238,15647,2297],{"className":15648,"style":2313},[282,283,419],[238,15650,15652,15655],{"className":15651},[282],[238,15653,15588],{"className":15654},[282],[238,15656,15658],{"className":15657},[391],[238,15659,15661,15681],{"className":15660},[395,848],[238,15662,15664,15678],{"className":15663},[399],[238,15665,15667],{"className":15666,"style":1927},[403],[238,15668,15669,15672],{"style":1165},[238,15670],{"className":15671,"style":412},[411],[238,15673,15675],{"className":15674},[416,417,418,419],[238,15676,1895],{"className":15677},[282,419],[238,15679,872],{"className":15680},[871],[238,15682,15684],{"className":15683},[399],[238,15685,15687],{"className":15686,"style":879},[403],[238,15688],{},[238,15690],{"className":15691,"style":430},[429],[238,15693,992],{"className":15694},[434],[238,15696],{"className":15697,"style":430},[429],[238,15699,15701,15704],{"className":15700},[273],[238,15702],{"className":15703,"style":2672},[277],[238,15705,745],{"className":15706},[282],"，从而 ",[238,15709,15711,15745],{"className":15710,"translate":242},[241],[238,15712,15714],{"className":15713},[246],[248,15715,15716],{"xmlns":250},[252,15717,15718,15742],{},[255,15719,15720,15722,15728,15734,15736],{},[258,15721,15588],{"mathvariant":344},[337,15723,15724,15726],{},[258,15725,4494],{},[258,15727,3395],{},[812,15729,15730,15732],{},[258,15731,15588],{"mathvariant":344},[258,15733,1895],{"mathvariant":344},[332,15735,2023],{},[337,15737,15738,15740],{},[258,15739,730],{},[258,15741,3395],{},[262,15743,15744],{"encoding":264},"\\|Q^k\\|_\\infty\\le\\gamma^k",[238,15746,15748,15837],{"className":15747,"ariaHidden":269},[268],[238,15749,15751,15755,15758,15788,15828,15831,15834],{"className":15750},[273],[238,15752],{"className":15753,"style":15754},[277],"height:1.0991em;vertical-align:-0.25em;",[238,15756,15588],{"className":15757},[282],[238,15759,15761,15764],{"className":15760},[282],[238,15762,4494],{"className":15763},[282,283],[238,15765,15767],{"className":15766},[391],[238,15768,15770],{"className":15769},[395],[238,15771,15773],{"className":15772},[399],[238,15774,15777],{"className":15775,"style":15776},[403],"height:0.8491em;",[238,15778,15779,15782],{"style":407},[238,15780],{"className":15781,"style":412},[411],[238,15783,15785],{"className":15784},[416,417,418,419],[238,15786,3395],{"className":15787,"style":3520},[282,283,419],[238,15789,15791,15794],{"className":15790},[282],[238,15792,15588],{"className":15793},[282],[238,15795,15797],{"className":15796},[391],[238,15798,15800,15820],{"className":15799},[395,848],[238,15801,15803,15817],{"className":15802},[399],[238,15804,15806],{"className":15805,"style":1927},[403],[238,15807,15808,15811],{"style":1165},[238,15809],{"className":15810,"style":412},[411],[238,15812,15814],{"className":15813},[416,417,418,419],[238,15815,1895],{"className":15816},[282,419],[238,15818,872],{"className":15819},[871],[238,15821,15823],{"className":15822},[399],[238,15824,15826],{"className":15825,"style":879},[403],[238,15827],{},[238,15829],{"className":15830,"style":430},[429],[238,15832,2023],{"className":15833},[434],[238,15835],{"className":15836,"style":430},[429],[238,15838,15840,15844],{"className":15839},[273],[238,15841],{"className":15842,"style":15843},[277],"height:1.0435em;vertical-align:-0.1944em;",[238,15845,15847,15850],{"className":15846},[282],[238,15848,730],{"className":15849,"style":764},[282,283],[238,15851,15853],{"className":15852},[391],[238,15854,15856],{"className":15855},[395],[238,15857,15859],{"className":15858},[399],[238,15860,15862],{"className":15861,"style":15776},[403],[238,15863,15864,15867],{"style":407},[238,15865],{"className":15866,"style":412},[411],[238,15868,15870],{"className":15869},[416,417,418,419],[238,15871,3395],{"className":15872,"style":3520},[282,283,419],"。因此 Neumann 级数 ",[238,15875,15877,15909],{"className":15876,"translate":242},[241],[238,15878,15880],{"className":15879},[246],[248,15881,15882],{"xmlns":250},[252,15883,15884,15906],{},[255,15885,15886,15900],{},[10131,15887,15888,15890,15898],{},[332,15889,3390],{},[255,15891,15892,15894,15896],{},[258,15893,3395],{},[332,15895,992],{},[738,15897,740],{},[258,15899,1895],{"mathvariant":344},[337,15901,15902,15904],{},[258,15903,4494],{},[258,15905,3395],{},[262,15907,15908],{"encoding":264},"\\sum_{k=0}^{\\infty}Q^k",[238,15910,15912],{"className":15911,"ariaHidden":269},[268],[238,15913,15915,15919,15984,15987],{"className":15914},[273],[238,15916],{"className":15917,"style":15918},[277],"height:1.1488em;vertical-align:-0.2997em;",[238,15920,15922,15925],{"className":15921},[1943],[238,15923,3390],{"className":15924,"style":4152},[1943,3538,4151],[238,15926,15928],{"className":15927},[391],[238,15929,15931,15976],{"className":15930},[395,848],[238,15932,15934,15973],{"className":15933},[399],[238,15935,15938,15958],{"className":15936,"style":15937},[403],"height:0.8043em;",[238,15939,15940,15943],{"style":4168},[238,15941],{"className":15942,"style":412},[411],[238,15944,15946],{"className":15945},[416,417,418,419],[238,15947,15949,15952,15955],{"className":15948},[282,419],[238,15950,3395],{"className":15951,"style":3520},[282,283,419],[238,15953,992],{"className":15954},[434,419],[238,15956,740],{"className":15957},[282,419],[238,15959,15961,15964],{"style":15960},"top:-3.2029em;margin-right:0.05em;",[238,15962],{"className":15963,"style":412},[411],[238,15965,15967],{"className":15966},[416,417,418,419],[238,15968,15970],{"className":15969},[282,419],[238,15971,1895],{"className":15972},[282,419],[238,15974,872],{"className":15975},[871],[238,15977,15979],{"className":15978},[399],[238,15980,15982],{"className":15981,"style":4224},[403],[238,15983],{},[238,15985],{"className":15986,"style":455},[429],[238,15988,15990,15993],{"className":15989},[282],[238,15991,4494],{"className":15992},[282,283],[238,15994,15996],{"className":15995},[391],[238,15997,15999],{"className":15998},[395],[238,16000,16002],{"className":16001},[399],[238,16003,16005],{"className":16004,"style":15776},[403],[238,16006,16007,16010],{"style":407},[238,16008],{"className":16009,"style":412},[411],[238,16011,16013],{"className":16012},[416,417,418,419],[238,16014,3395],{"className":16015,"style":3520},[282,283,419]," 在算子范数下绝对收敛。并且",[238,16018,16020],{"className":16019,"translate":242},[945],[238,16021,16023,16090],{"className":16022,"translate":242},[241],[238,16024,16026],{"className":16025},[246],[248,16027,16028],{"xmlns":250,"display":954},[252,16029,16030,16087],{},[255,16031,16032,16034,16036,16038,16040,16042,16056,16062,16064,16066,16068,16080,16083,16085],{},[332,16033,335],{"stretchy":334},[258,16035,14572],{},[332,16037,5577],{},[258,16039,4494],{},[332,16041,360],{"stretchy":334},[3386,16043,16044,16046,16054],{},[332,16045,3390],{},[255,16047,16048,16050,16052],{},[258,16049,3395],{},[332,16051,992],{},[738,16053,740],{},[258,16055,13161],{},[337,16057,16058,16060],{},[258,16059,4494],{},[258,16061,3395],{},[332,16063,992],{},[258,16065,14572],{},[332,16067,5577],{},[337,16069,16070,16072],{},[258,16071,4494],{},[255,16073,16074,16076,16078],{},[258,16075,13161],{},[332,16077,987],{},[738,16079,745],{},[332,16081,16082],{},"⟶",[258,16084,14572],{},[332,16086,354],{"separator":269},[262,16088,16089],{"encoding":264},"(I-Q)\\sum_{k=0}^{n}Q^k=I-Q^{n+1}\\longrightarrow I,",[238,16091,16093,16114,16237,16255,16309],{"className":16092,"ariaHidden":269},[268],[238,16094,16096,16099,16102,16105,16108,16111],{"className":16095},[273],[238,16097],{"className":16098,"style":444},[277],[238,16100,335],{"className":16101},[381],[238,16103,14572],{"className":16104,"style":14619},[282,283],[238,16106],{"className":16107,"style":660},[429],[238,16109,5577],{"className":16110},[664],[238,16112],{"className":16113,"style":660},[429],[238,16115,16117,16120,16123,16126,16129,16196,16199,16228,16231,16234],{"className":16116},[273],[238,16118],{"className":16119,"style":3489},[277],[238,16121,4494],{"className":16122},[282,283],[238,16124,360],{"className":16125},[462],[238,16127],{"className":16128,"style":455},[429],[238,16130,16132],{"className":16131},[1943,3493],[238,16133,16135,16188],{"className":16134},[395,848],[238,16136,16138,16185],{"className":16137},[399],[238,16139,16141,16161,16171],{"className":16140,"style":3503},[403],[238,16142,16143,16146],{"style":3506},[238,16144],{"className":16145,"style":3510},[411],[238,16147,16149],{"className":16148},[416,417,418,419],[238,16150,16152,16155,16158],{"className":16151},[282,419],[238,16153,3395],{"className":16154,"style":3520},[282,283,419],[238,16156,992],{"className":16157},[434,419],[238,16159,740],{"className":16160},[282,419],[238,16162,16163,16166],{"style":3529},[238,16164],{"className":16165,"style":3510},[411],[238,16167,16168],{},[238,16169,3390],{"className":16170},[1943,3538,3539],[238,16172,16173,16176],{"style":3542},[238,16174],{"className":16175,"style":3510},[411],[238,16177,16179],{"className":16178},[416,417,418,419],[238,16180,16182],{"className":16181},[282,419],[238,16183,13161],{"className":16184},[282,283,419],[238,16186,872],{"className":16187},[871],[238,16189,16191],{"className":16190},[399],[238,16192,16194],{"className":16193,"style":3564},[403],[238,16195],{},[238,16197],{"className":16198,"style":455},[429],[238,16200,16202,16205],{"className":16201},[282],[238,16203,4494],{"className":16204},[282,283],[238,16206,16208],{"className":16207},[391],[238,16209,16211],{"className":16210},[395],[238,16212,16214],{"className":16213},[399],[238,16215,16217],{"className":16216,"style":3588},[403],[238,16218,16219,16222],{"style":1226},[238,16220],{"className":16221,"style":412},[411],[238,16223,16225],{"className":16224},[416,417,418,419],[238,16226,3395],{"className":16227,"style":3520},[282,283,419],[238,16229],{"className":16230,"style":430},[429],[238,16232,992],{"className":16233},[434],[238,16235],{"className":16236,"style":430},[429],[238,16238,16240,16243,16246,16249,16252],{"className":16239},[273],[238,16241],{"className":16242,"style":653},[277],[238,16244,14572],{"className":16245,"style":14619},[282,283],[238,16247],{"className":16248,"style":660},[429],[238,16250,5577],{"className":16251},[664],[238,16253],{"className":16254,"style":660},[429],[238,16256,16258,16262,16300,16303,16306],{"className":16257},[273],[238,16259],{"className":16260,"style":16261},[277],"height:1.0585em;vertical-align:-0.1944em;",[238,16263,16265,16268],{"className":16264},[282],[238,16266,4494],{"className":16267},[282,283],[238,16269,16271],{"className":16270},[391],[238,16272,16274],{"className":16273},[395],[238,16275,16277],{"className":16276},[399],[238,16278,16280],{"className":16279,"style":15139},[403],[238,16281,16282,16285],{"style":1226},[238,16283],{"className":16284,"style":412},[411],[238,16286,16288],{"className":16287},[416,417,418,419],[238,16289,16291,16294,16297],{"className":16290},[282,419],[238,16292,13161],{"className":16293},[282,283,419],[238,16295,987],{"className":16296},[664,419],[238,16298,745],{"className":16299},[282,419],[238,16301],{"className":16302,"style":430},[429],[238,16304,16082],{"className":16305},[434],[238,16307],{"className":16308,"style":430},[429],[238,16310,16312,16315,16318],{"className":16311},[273],[238,16313],{"className":16314,"style":6651},[277],[238,16316,14572],{"className":16317,"style":14619},[282,283],[238,16319,354],{"className":16320},[451],[11,16322,16323,16324,14913],{},"故 ",[238,16325,16327,16345],{"className":16326,"translate":242},[241],[238,16328,16330],{"className":16329},[246],[248,16331,16332],{"xmlns":250},[252,16333,16334,16342],{},[255,16335,16336,16338,16340],{},[258,16337,14572],{},[332,16339,5577],{},[258,16341,4494],{},[262,16343,16344],{"encoding":264},"I-Q",[238,16346,16348,16366],{"className":16347,"ariaHidden":269},[268],[238,16349,16351,16354,16357,16360,16363],{"className":16350},[273],[238,16352],{"className":16353,"style":653},[277],[238,16355,14572],{"className":16356,"style":14619},[282,283],[238,16358],{"className":16359,"style":660},[429],[238,16361,5577],{"className":16362},[664],[238,16364],{"className":16365,"style":660},[429],[238,16367,16369,16372],{"className":16368},[273],[238,16370],{"className":16371,"style":6651},[277],[238,16373,4494],{"className":16374},[282,283],[238,16376,16378],{"className":16377,"translate":242},[945],[238,16379,16381,16435],{"className":16380,"translate":242},[241],[238,16382,16384],{"className":16383},[246],[248,16385,16386],{"xmlns":250,"display":954},[252,16387,16388,16432],{},[255,16389,16390,16392,16394,16396,16398,16408,16410,16424,16430],{},[332,16391,335],{"stretchy":334},[258,16393,14572],{},[332,16395,5577],{},[258,16397,4494],{},[337,16399,16400,16402],{},[332,16401,360],{"stretchy":334},[255,16403,16404,16406],{},[332,16405,5577],{},[738,16407,745],{},[332,16409,992],{},[3386,16411,16412,16414,16422],{},[332,16413,3390],{},[255,16415,16416,16418,16420],{},[258,16417,3395],{},[332,16419,992],{},[738,16421,740],{},[258,16423,1895],{"mathvariant":344},[337,16425,16426,16428],{},[258,16427,4494],{},[258,16429,3395],{},[258,16431,3102],{"mathvariant":344},[262,16433,16434],{"encoding":264},"(I-Q)^{-1}=\\sum_{k=0}^{\\infty}Q^k.",[238,16436,16438,16459,16512],{"className":16437,"ariaHidden":269},[268],[238,16439,16441,16444,16447,16450,16453,16456],{"className":16440},[273],[238,16442],{"className":16443,"style":444},[277],[238,16445,335],{"className":16446},[381],[238,16448,14572],{"className":16449,"style":14619},[282,283],[238,16451],{"className":16452,"style":660},[429],[238,16454,5577],{"className":16455},[664],[238,16457],{"className":16458,"style":660},[429],[238,16460,16462,16465,16468,16503,16506,16509],{"className":16461},[273],[238,16463],{"className":16464,"style":15088},[277],[238,16466,4494],{"className":16467},[282,283],[238,16469,16471,16474],{"className":16470},[462],[238,16472,360],{"className":16473},[462],[238,16475,16477],{"className":16476},[391],[238,16478,16480],{"className":16479},[395],[238,16481,16483],{"className":16482},[399],[238,16484,16486],{"className":16485,"style":15139},[403],[238,16487,16488,16491],{"style":1226},[238,16489],{"className":16490,"style":412},[411],[238,16492,16494],{"className":16493},[416,417,418,419],[238,16495,16497,16500],{"className":16496},[282,419],[238,16498,5577],{"className":16499},[282,419],[238,16501,745],{"className":16502},[282,419],[238,16504],{"className":16505,"style":430},[429],[238,16507,992],{"className":16508},[434],[238,16510],{"className":16511,"style":430},[429],[238,16513,16515,16518,16585,16588,16617],{"className":16514},[273],[238,16516],{"className":16517,"style":3489},[277],[238,16519,16521],{"className":16520},[1943,3493],[238,16522,16524,16577],{"className":16523},[395,848],[238,16525,16527,16574],{"className":16526},[399],[238,16528,16530,16550,16560],{"className":16529,"style":3503},[403],[238,16531,16532,16535],{"style":3506},[238,16533],{"className":16534,"style":3510},[411],[238,16536,16538],{"className":16537},[416,417,418,419],[238,16539,16541,16544,16547],{"className":16540},[282,419],[238,16542,3395],{"className":16543,"style":3520},[282,283,419],[238,16545,992],{"className":16546},[434,419],[238,16548,740],{"className":16549},[282,419],[238,16551,16552,16555],{"style":3529},[238,16553],{"className":16554,"style":3510},[411],[238,16556,16557],{},[238,16558,3390],{"className":16559},[1943,3538,3539],[238,16561,16562,16565],{"style":3542},[238,16563],{"className":16564,"style":3510},[411],[238,16566,16568],{"className":16567},[416,417,418,419],[238,16569,16571],{"className":16570},[282,419],[238,16572,1895],{"className":16573},[282,419],[238,16575,872],{"className":16576},[871],[238,16578,16580],{"className":16579},[399],[238,16581,16583],{"className":16582,"style":3564},[403],[238,16584],{},[238,16586],{"className":16587,"style":455},[429],[238,16589,16591,16594],{"className":16590},[282],[238,16592,4494],{"className":16593},[282,283],[238,16595,16597],{"className":16596},[391],[238,16598,16600],{"className":16599},[395],[238,16601,16603],{"className":16602},[399],[238,16604,16606],{"className":16605,"style":3588},[403],[238,16607,16608,16611],{"style":1226},[238,16609],{"className":16610,"style":412},[411],[238,16612,16614],{"className":16613},[416,417,418,419],[238,16615,3395],{"className":16616,"style":3520},[282,283,419],[238,16618,3102],{"className":16619},[282],[11,16621,16622,16623,941,16627,16685,16686,16834],{},"由",[357,16624,16626],{"href":16625},"#theorem-bellman-expectation","定理 2.1",[238,16628,16630,16647],{"className":16629,"translate":242},[241],[238,16631,16633],{"className":16632},[246],[248,16634,16635],{"xmlns":250},[252,16636,16637,16645],{},[255,16638,16639],{},[337,16640,16641,16643],{},[258,16642,4429],{},[258,16644,2297],{},[262,16646,6581],{"encoding":264},[238,16648,16650],{"className":16649,"ariaHidden":269},[268],[238,16651,16653,16656],{"className":16652},[273],[238,16654],{"className":16655,"style":278},[277],[238,16657,16659,16662],{"className":16658},[282],[238,16660,4429],{"className":16661,"style":660},[282,283],[238,16663,16665],{"className":16664},[391],[238,16666,16668],{"className":16667},[395],[238,16669,16671],{"className":16670},[399],[238,16672,16674],{"className":16673,"style":6609},[403],[238,16675,16676,16679],{"style":407},[238,16677],{"className":16678,"style":412},[411],[238,16680,16682],{"className":16681},[416,417,418,419],[238,16683,2297],{"className":16684,"style":2313},[282,283,419]," 满足 ",[238,16687,16689,16725],{"className":16688,"translate":242},[241],[238,16690,16692],{"className":16691},[246],[248,16693,16694],{"xmlns":250},[252,16695,16696,16722],{},[255,16697,16698,16700,16702,16704,16706,16708,16714,16716],{},[332,16699,335],{"stretchy":334},[258,16701,14572],{},[332,16703,5577],{},[258,16705,4494],{},[332,16707,360],{"stretchy":334},[337,16709,16710,16712],{},[258,16711,4429],{},[258,16713,2297],{},[332,16715,992],{},[337,16717,16718,16720],{},[258,16719,600],{},[258,16721,2297],{},[262,16723,16724],{"encoding":264},"(I-Q)V^\\pi=r^\\pi",[238,16726,16728,16749,16799],{"className":16727,"ariaHidden":269},[268],[238,16729,16731,16734,16737,16740,16743,16746],{"className":16730},[273],[238,16732],{"className":16733,"style":444},[277],[238,16735,335],{"className":16736},[381],[238,16738,14572],{"className":16739,"style":14619},[282,283],[238,16741],{"className":16742,"style":660},[429],[238,16744,5577],{"className":16745},[664],[238,16747],{"className":16748,"style":660},[429],[238,16750,16752,16755,16758,16761,16790,16793,16796],{"className":16751},[273],[238,16753],{"className":16754,"style":444},[277],[238,16756,4494],{"className":16757},[282,283],[238,16759,360],{"className":16760},[462],[238,16762,16764,16767],{"className":16763},[282],[238,16765,4429],{"className":16766,"style":660},[282,283],[238,16768,16770],{"className":16769},[391],[238,16771,16773],{"className":16772},[395],[238,16774,16776],{"className":16775},[399],[238,16777,16779],{"className":16778,"style":6609},[403],[238,16780,16781,16784],{"style":407},[238,16782],{"className":16783,"style":412},[411],[238,16785,16787],{"className":16786},[416,417,418,419],[238,16788,2297],{"className":16789,"style":2313},[282,283,419],[238,16791],{"className":16792,"style":430},[429],[238,16794,992],{"className":16795},[434],[238,16797],{"className":16798,"style":430},[429],[238,16800,16802,16805],{"className":16801},[273],[238,16803],{"className":16804,"style":6609},[277],[238,16806,16808,16811],{"className":16807},[282],[238,16809,600],{"className":16810,"style":637},[282,283],[238,16812,16814],{"className":16813},[391],[238,16815,16817],{"className":16816},[395],[238,16818,16820],{"className":16819},[399],[238,16821,16823],{"className":16822,"style":6609},[403],[238,16824,16825,16828],{"style":407},[238,16826],{"className":16827,"style":412},[411],[238,16829,16831],{"className":16830},[416,417,418,419],[238,16832,2297],{"className":16833,"style":2313},[282,283,419],"，于是",[238,16836,16838],{"className":16837,"translate":242},[945],[238,16839,16841,16959],{"className":16840,"translate":242},[241],[238,16842,16844],{"className":16843},[246],[248,16845,16846],{"xmlns":250,"display":954},[252,16847,16848,16956],{},[958,16849,16850,16896],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,16851,16852,16862],{},[966,16853,16854],{},[969,16855,16856],{"scriptlevel":740,"displaystyle":269},[337,16857,16858,16860],{},[258,16859,4429],{},[258,16861,2297],{},[966,16863,16864],{},[969,16865,16866],{"scriptlevel":740,"displaystyle":269},[255,16867,16868,16870,16872,16874,16876,16878,16880,16890],{},[255,16869],{},[332,16871,992],{},[332,16873,335],{"stretchy":334},[258,16875,14572],{},[332,16877,5577],{},[258,16879,4494],{},[337,16881,16882,16884],{},[332,16883,360],{"stretchy":334},[255,16885,16886,16888],{},[332,16887,5577],{},[738,16889,745],{},[337,16891,16892,16894],{},[258,16893,600],{},[258,16895,2297],{},[963,16897,16898,16904],{},[966,16899,16900],{},[969,16901,16902],{"scriptlevel":740,"displaystyle":269},[255,16903],{},[966,16905,16906],{},[969,16907,16908],{"scriptlevel":740,"displaystyle":269},[255,16909,16910,16912,16914,16928,16934,16936,16942,16948,16954],{},[255,16911],{},[332,16913,992],{},[3386,16915,16916,16918,16926],{},[332,16917,3390],{},[255,16919,16920,16922,16924],{},[258,16921,3395],{},[332,16923,992],{},[738,16925,740],{},[258,16927,1895],{"mathvariant":344},[337,16929,16930,16932],{},[258,16931,730],{},[258,16933,3395],{},[332,16935,335],{"stretchy":334},[337,16937,16938,16940],{},[258,16939,330],{},[258,16941,2297],{},[337,16943,16944,16946],{},[332,16945,360],{"stretchy":334},[258,16947,3395],{},[337,16949,16950,16952],{},[258,16951,600],{},[258,16953,2297],{},[258,16955,3102],{"mathvariant":344},[262,16957,16958],{"encoding":264},"\\begin{aligned}\nV^\\pi&=(I-Q)^{-1}r^\\pi\\\\\n&=\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi.\n\\end{aligned}",[238,16960,16962],{"className":16961,"ariaHidden":269},[268],[238,16963,16965,16969],{"className":16964},[273],[238,16966],{"className":16967,"style":16968},[277],"height:4.4776em;vertical-align:-1.9888em;",[238,16970,16972],{"className":16971},[282],[238,16973,16975,17047],{"className":16974},[958],[238,16976,16978],{"className":16977},[1117],[238,16979,16981,17038],{"className":16980},[395,848],[238,16982,16984,17035],{"className":16983},[399],[238,16985,16988,17026],{"className":16986,"style":16987},[403],"height:2.4888em;",[238,16989,16991,16994],{"style":16990},"top:-5.2761em;",[238,16992],{"className":16993,"style":5981},[411],[238,16995,16997],{"className":16996},[282],[238,16998,17000,17003],{"className":16999},[282],[238,17001,4429],{"className":17002,"style":660},[282,283],[238,17004,17006],{"className":17005},[391],[238,17007,17009],{"className":17008},[395],[238,17010,17012],{"className":17011},[399],[238,17013,17015],{"className":17014,"style":4620},[403],[238,17016,17017,17020],{"style":1226},[238,17018],{"className":17019,"style":412},[411],[238,17021,17023],{"className":17022},[416,417,418,419],[238,17024,2297],{"className":17025,"style":2313},[282,283,419],[238,17027,17029,17032],{"style":17028},"top:-2.9647em;",[238,17030],{"className":17031,"style":5981},[411],[238,17033],{"className":17034},[282],[238,17036,872],{"className":17037},[871],[238,17039,17041],{"className":17040},[399],[238,17042,17045],{"className":17043,"style":17044},[403],"height:1.9888em;",[238,17046],{},[238,17048,17050],{"className":17049},[1422],[238,17051,17053,17376],{"className":17052},[395,848],[238,17054,17056,17373],{"className":17055},[399],[238,17057,17059,17161],{"className":17058,"style":16987},[403],[238,17060,17061,17064],{"style":16990},[238,17062],{"className":17063,"style":5981},[411],[238,17065,17067,17070,17073,17076,17079,17082,17085,17088,17091,17094,17097,17132],{"className":17066},[282],[238,17068],{"className":17069},[282],[238,17071],{"className":17072,"style":430},[429],[238,17074,992],{"className":17075},[434],[238,17077],{"className":17078,"style":430},[429],[238,17080,335],{"className":17081},[381],[238,17083,14572],{"className":17084,"style":14619},[282,283],[238,17086],{"className":17087,"style":660},[429],[238,17089,5577],{"className":17090},[664],[238,17092],{"className":17093,"style":660},[429],[238,17095,4494],{"className":17096},[282,283],[238,17098,17100,17103],{"className":17099},[462],[238,17101,360],{"className":17102},[462],[238,17104,17106],{"className":17105},[391],[238,17107,17109],{"className":17108},[395],[238,17110,17112],{"className":17111},[399],[238,17113,17115],{"className":17114,"style":15139},[403],[238,17116,17117,17120],{"style":1226},[238,17118],{"className":17119,"style":412},[411],[238,17121,17123],{"className":17122},[416,417,418,419],[238,17124,17126,17129],{"className":17125},[282,419],[238,17127,5577],{"className":17128},[282,419],[238,17130,745],{"className":17131},[282,419],[238,17133,17135,17138],{"className":17134},[282],[238,17136,600],{"className":17137,"style":637},[282,283],[238,17139,17141],{"className":17140},[391],[238,17142,17144],{"className":17143},[395],[238,17145,17147],{"className":17146},[399],[238,17148,17150],{"className":17149,"style":4620},[403],[238,17151,17152,17155],{"style":1226},[238,17153],{"className":17154,"style":412},[411],[238,17156,17158],{"className":17157},[416,417,418,419],[238,17159,2297],{"className":17160,"style":2313},[282,283,419],[238,17162,17163,17166],{"style":17028},[238,17164],{"className":17165,"style":5981},[411],[238,17167,17169,17172,17175,17178,17181,17248,17251,17280,17283,17312,17341,17370],{"className":17168},[282],[238,17170],{"className":17171},[282],[238,17173],{"className":17174,"style":430},[429],[238,17176,992],{"className":17177},[434],[238,17179],{"className":17180,"style":430},[429],[238,17182,17184],{"className":17183},[1943,3493],[238,17185,17187,17240],{"className":17186},[395,848],[238,17188,17190,17237],{"className":17189},[399],[238,17191,17193,17213,17223],{"className":17192,"style":3503},[403],[238,17194,17195,17198],{"style":3506},[238,17196],{"className":17197,"style":3510},[411],[238,17199,17201],{"className":17200},[416,417,418,419],[238,17202,17204,17207,17210],{"className":17203},[282,419],[238,17205,3395],{"className":17206,"style":3520},[282,283,419],[238,17208,992],{"className":17209},[434,419],[238,17211,740],{"className":17212},[282,419],[238,17214,17215,17218],{"style":3529},[238,17216],{"className":17217,"style":3510},[411],[238,17219,17220],{},[238,17221,3390],{"className":17222},[1943,3538,3539],[238,17224,17225,17228],{"style":3542},[238,17226],{"className":17227,"style":3510},[411],[238,17229,17231],{"className":17230},[416,417,418,419],[238,17232,17234],{"className":17233},[282,419],[238,17235,1895],{"className":17236},[282,419],[238,17238,872],{"className":17239},[871],[238,17241,17243],{"className":17242},[399],[238,17244,17246],{"className":17245,"style":3564},[403],[238,17247],{},[238,17249],{"className":17250,"style":455},[429],[238,17252,17254,17257],{"className":17253},[282],[238,17255,730],{"className":17256,"style":764},[282,283],[238,17258,17260],{"className":17259},[391],[238,17261,17263],{"className":17262},[395],[238,17264,17266],{"className":17265},[399],[238,17267,17269],{"className":17268,"style":3588},[403],[238,17270,17271,17274],{"style":1226},[238,17272],{"className":17273,"style":412},[411],[238,17275,17277],{"className":17276},[416,417,418,419],[238,17278,3395],{"className":17279,"style":3520},[282,283,419],[238,17281,335],{"className":17282},[381],[238,17284,17286,17289],{"className":17285},[282],[238,17287,330],{"className":17288,"style":377},[282,283],[238,17290,17292],{"className":17291},[391],[238,17293,17295],{"className":17294},[395],[238,17296,17298],{"className":17297},[399],[238,17299,17301],{"className":17300,"style":4620},[403],[238,17302,17303,17306],{"style":1226},[238,17304],{"className":17305,"style":412},[411],[238,17307,17309],{"className":17308},[416,417,418,419],[238,17310,2297],{"className":17311,"style":2313},[282,283,419],[238,17313,17315,17318],{"className":17314},[462],[238,17316,360],{"className":17317},[462],[238,17319,17321],{"className":17320},[391],[238,17322,17324],{"className":17323},[395],[238,17325,17327],{"className":17326},[399],[238,17328,17330],{"className":17329,"style":3588},[403],[238,17331,17332,17335],{"style":1226},[238,17333],{"className":17334,"style":412},[411],[238,17336,17338],{"className":17337},[416,417,418,419],[238,17339,3395],{"className":17340,"style":3520},[282,283,419],[238,17342,17344,17347],{"className":17343},[282],[238,17345,600],{"className":17346,"style":637},[282,283],[238,17348,17350],{"className":17349},[391],[238,17351,17353],{"className":17352},[395],[238,17354,17356],{"className":17355},[399],[238,17357,17359],{"className":17358,"style":4620},[403],[238,17360,17361,17364],{"style":1226},[238,17362],{"className":17363,"style":412},[411],[238,17365,17367],{"className":17366},[416,417,418,419],[238,17368,2297],{"className":17369,"style":2313},[282,283,419],[238,17371,3102],{"className":17372},[282],[238,17374,872],{"className":17375},[871],[238,17377,17379],{"className":17378},[399],[238,17380,17382],{"className":17381,"style":17044},[403],[238,17383],{},[11,17385,11882],{},[11,17387,17388,17389,17474,17475,17526],{},"矩阵解给出了一个重要的结构性事实：在有限状态与固定策略下，贝尔曼方程是一个线性方程组，其系数矩阵为 ",[238,17390,17392,17415],{"className":17391,"translate":242},[241],[238,17393,17395],{"className":17394},[246],[248,17396,17397],{"xmlns":250},[252,17398,17399,17413],{},[255,17400,17401,17403,17405,17407],{},[258,17402,14572],{},[332,17404,5577],{},[258,17406,730],{},[337,17408,17409,17411],{},[258,17410,330],{},[258,17412,2297],{},[262,17414,14853],{"encoding":264},[238,17416,17418,17436],{"className":17417,"ariaHidden":269},[268],[238,17419,17421,17424,17427,17430,17433],{"className":17420},[273],[238,17422],{"className":17423,"style":653},[277],[238,17425,14572],{"className":17426,"style":14619},[282,283],[238,17428],{"className":17429,"style":660},[429],[238,17431,5577],{"className":17432},[664],[238,17434],{"className":17435,"style":660},[429],[238,17437,17439,17442,17445],{"className":17438},[273],[238,17440],{"className":17441,"style":6651},[277],[238,17443,730],{"className":17444,"style":764},[282,283],[238,17446,17448,17451],{"className":17447},[282],[238,17449,330],{"className":17450,"style":377},[282,283],[238,17452,17454],{"className":17453},[391],[238,17455,17457],{"className":17456},[395],[238,17458,17460],{"className":17459},[399],[238,17461,17463],{"className":17462,"style":6609},[403],[238,17464,17465,17468],{"style":407},[238,17466],{"className":17467,"style":412},[411],[238,17469,17471],{"className":17470},[416,417,418,419],[238,17472,2297],{"className":17473,"style":2313},[282,283,419],"。折扣因子 ",[238,17476,17478,17496],{"className":17477,"translate":242},[241],[238,17479,17481],{"className":17480},[246],[248,17482,17483],{"xmlns":250},[252,17484,17485,17493],{},[255,17486,17487,17489,17491],{},[258,17488,730],{},[332,17490,1892],{},[738,17492,745],{},[262,17494,17495],{"encoding":264},"\\gamma\u003C1",[238,17497,17499,17517],{"className":17498,"ariaHidden":269},[268],[238,17500,17502,17505,17508,17511,17514],{"className":17501},[273],[238,17503],{"className":17504,"style":760},[277],[238,17506,730],{"className":17507,"style":764},[282,283],[238,17509],{"className":17510,"style":430},[429],[238,17512,1892],{"className":17513},[434],[238,17515],{"className":17516,"style":430},[429],[238,17518,17520,17523],{"className":17519},[273],[238,17521],{"className":17522,"style":2672},[277],[238,17524,745],{"className":17525},[282]," 保证该矩阵可逆，因此解存在且唯一。对于一般状态空间，矩阵不再有限，但同样的结论可以由第 3 节的压缩映射论证得到。",[225,17528,17530],{"id":17529},"_24-例两状态马尔可夫链","2.4 例：两状态马尔可夫链",[11,17532,17533,17534,17609],{},"考虑两个状态 ",[238,17535,17537,17564],{"className":17536,"translate":242},[241],[238,17538,17540],{"className":17539},[246],[248,17541,17542],{"xmlns":250},[252,17543,17544,17561],{},[255,17545,17546,17548,17550,17552,17554,17556,17559],{},[258,17547,260],{},[332,17549,992],{},[332,17551,13150],{"stretchy":334},[738,17553,745],{},[332,17555,354],{"separator":269},[738,17557,17558],{},"2",[332,17560,13164],{"stretchy":334},[262,17562,17563],{"encoding":264},"S=\\{1,2\\}",[238,17565,17567,17585],{"className":17566,"ariaHidden":269},[268],[238,17568,17570,17573,17576,17579,17582],{"className":17569},[273],[238,17571],{"className":17572,"style":278},[277],[238,17574,260],{"className":17575,"style":284},[282,283],[238,17577],{"className":17578,"style":430},[429],[238,17580,992],{"className":17581},[434],[238,17583],{"className":17584,"style":430},[429],[238,17586,17588,17591,17594,17597,17600,17603,17606],{"className":17587},[273],[238,17589],{"className":17590,"style":444},[277],[238,17592,13150],{"className":17593},[381],[238,17595,745],{"className":17596},[282],[238,17598,354],{"className":17599},[451],[238,17601],{"className":17602,"style":455},[429],[238,17604,17558],{"className":17605},[282],[238,17607,13164],{"className":17608},[462],"，只有一个动作。设转移关系为",[238,17611,17613],{"className":17612,"translate":242},[945],[238,17614,17616,17666],{"className":17615,"translate":242},[241],[238,17617,17619],{"className":17618},[246],[248,17620,17621],{"xmlns":250,"display":954},[252,17622,17623,17663],{},[255,17624,17625,17627,17629,17631,17633,17635,17637,17639,17641,17643,17645,17647,17649,17651,17653,17655,17657,17659,17661],{},[258,17626,330],{},[332,17628,335],{"stretchy":334},[738,17630,17558],{},[332,17632,349],{},[738,17634,745],{},[332,17636,360],{"stretchy":334},[332,17638,992],{},[738,17640,745],{},[332,17642,354],{"separator":269},[429,17644],{"width":10450},[258,17646,330],{},[332,17648,335],{"stretchy":334},[738,17650,745],{},[332,17652,349],{},[738,17654,17558],{},[332,17656,360],{"stretchy":334},[332,17658,992],{},[738,17660,745],{},[332,17662,354],{"separator":269},[262,17664,17665],{"encoding":264},"P(2\\mid 1)=1,\\qquad P(1\\mid 2)=1,",[238,17667,17669,17693,17714,17750,17771],{"className":17668,"ariaHidden":269},[268],[238,17670,17672,17675,17678,17681,17684,17687,17690],{"className":17671},[273],[238,17673],{"className":17674,"style":444},[277],[238,17676,330],{"className":17677,"style":377},[282,283],[238,17679,335],{"className":17680},[381],[238,17682,17558],{"className":17683},[282],[238,17685],{"className":17686,"style":430},[429],[238,17688,349],{"className":17689},[434],[238,17691],{"className":17692,"style":430},[429],[238,17694,17696,17699,17702,17705,17708,17711],{"className":17695},[273],[238,17697],{"className":17698,"style":444},[277],[238,17700,745],{"className":17701},[282],[238,17703,360],{"className":17704},[462],[238,17706],{"className":17707,"style":430},[429],[238,17709,992],{"className":17710},[434],[238,17712],{"className":17713,"style":430},[429],[238,17715,17717,17720,17723,17726,17729,17732,17735,17738,17741,17744,17747],{"className":17716},[273],[238,17718],{"className":17719,"style":444},[277],[238,17721,745],{"className":17722},[282],[238,17724,354],{"className":17725},[451],[238,17727],{"className":17728,"style":10986},[429],[238,17730],{"className":17731,"style":455},[429],[238,17733,330],{"className":17734,"style":377},[282,283],[238,17736,335],{"className":17737},[381],[238,17739,745],{"className":17740},[282],[238,17742],{"className":17743,"style":430},[429],[238,17745,349],{"className":17746},[434],[238,17748],{"className":17749,"style":430},[429],[238,17751,17753,17756,17759,17762,17765,17768],{"className":17752},[273],[238,17754],{"className":17755,"style":444},[277],[238,17757,17558],{"className":17758},[282],[238,17760,360],{"className":17761},[462],[238,17763],{"className":17764,"style":430},[429],[238,17766,992],{"className":17767},[434],[238,17769],{"className":17770,"style":430},[429],[238,17772,17774,17778,17781],{"className":17773},[273],[238,17775],{"className":17776,"style":17777},[277],"height:0.8389em;vertical-align:-0.1944em;",[238,17779,745],{"className":17780},[282],[238,17782,354],{"className":17783},[451],[11,17785,17786,17787,941,17853,17919,17920,17972],{},"奖励为 ",[238,17788,17790,17814],{"className":17789,"translate":242},[241],[238,17791,17793],{"className":17792},[246],[248,17794,17795],{"xmlns":250},[252,17796,17797,17811],{},[255,17798,17799,17801,17803,17805,17807,17809],{},[258,17800,600],{},[332,17802,335],{"stretchy":334},[738,17804,745],{},[332,17806,360],{"stretchy":334},[332,17808,992],{},[738,17810,745],{},[262,17812,17813],{"encoding":264},"r(1)=1",[238,17815,17817,17844],{"className":17816,"ariaHidden":269},[268],[238,17818,17820,17823,17826,17829,17832,17835,17838,17841],{"className":17819},[273],[238,17821],{"className":17822,"style":444},[277],[238,17824,600],{"className":17825,"style":637},[282,283],[238,17827,335],{"className":17828},[381],[238,17830,745],{"className":17831},[282],[238,17833,360],{"className":17834},[462],[238,17836],{"className":17837,"style":430},[429],[238,17839,992],{"className":17840},[434],[238,17842],{"className":17843,"style":430},[429],[238,17845,17847,17850],{"className":17846},[273],[238,17848],{"className":17849,"style":2672},[277],[238,17851,745],{"className":17852},[282],[238,17854,17856,17880],{"className":17855,"translate":242},[241],[238,17857,17859],{"className":17858},[246],[248,17860,17861],{"xmlns":250},[252,17862,17863,17877],{},[255,17864,17865,17867,17869,17871,17873,17875],{},[258,17866,600],{},[332,17868,335],{"stretchy":334},[738,17870,17558],{},[332,17872,360],{"stretchy":334},[332,17874,992],{},[738,17876,740],{},[262,17878,17879],{"encoding":264},"r(2)=0",[238,17881,17883,17910],{"className":17882,"ariaHidden":269},[268],[238,17884,17886,17889,17892,17895,17898,17901,17904,17907],{"className":17885},[273],[238,17887],{"className":17888,"style":444},[277],[238,17890,600],{"className":17891,"style":637},[282,283],[238,17893,335],{"className":17894},[381],[238,17896,17558],{"className":17897},[282],[238,17899,360],{"className":17900},[462],[238,17902],{"className":17903,"style":430},[429],[238,17905,992],{"className":17906},[434],[238,17908],{"className":17909,"style":430},[429],[238,17911,17913,17916],{"className":17912},[273],[238,17914],{"className":17915,"style":2672},[277],[238,17917,740],{"className":17918},[282],"，折扣因子 ",[238,17921,17923,17942],{"className":17922,"translate":242},[241],[238,17924,17926],{"className":17925},[246],[248,17927,17928],{"xmlns":250},[252,17929,17930,17939],{},[255,17931,17932,17934,17936],{},[258,17933,730],{},[332,17935,992],{},[738,17937,17938],{},"0.9",[262,17940,17941],{"encoding":264},"\\gamma=0.9",[238,17943,17945,17963],{"className":17944,"ariaHidden":269},[268],[238,17946,17948,17951,17954,17957,17960],{"className":17947},[273],[238,17949],{"className":17950,"style":2853},[277],[238,17952,730],{"className":17953,"style":764},[282,283],[238,17955],{"className":17956,"style":430},[429],[238,17958,992],{"className":17959},[434],[238,17961],{"className":17962,"style":430},[429],[238,17964,17966,17969],{"className":17965},[273],[238,17967],{"className":17968,"style":2672},[277],[238,17970,17938],{"className":17971},[282],"。贝尔曼方程为",[238,17974,17976],{"className":17975,"translate":242},[945],[238,17977,17979,18045],{"className":17978,"translate":242},[241],[238,17980,17982],{"className":17981},[246],[248,17983,17984],{"xmlns":250,"display":954},[252,17985,17986,18042],{},[255,17987,17988,17990,17992,17994,17996,17998,18000,18002,18004,18006,18008,18010,18012,18014,18016,18018,18020,18022,18024,18026,18028,18030,18032,18034,18036,18038,18040],{},[258,17989,4429],{},[332,17991,335],{"stretchy":334},[738,17993,745],{},[332,17995,360],{"stretchy":334},[332,17997,992],{},[738,17999,745],{},[332,18001,987],{},[738,18003,17938],{},[258,18005,4429],{},[332,18007,335],{"stretchy":334},[738,18009,17558],{},[332,18011,360],{"stretchy":334},[332,18013,354],{"separator":269},[429,18015],{"width":10450},[258,18017,4429],{},[332,18019,335],{"stretchy":334},[738,18021,17558],{},[332,18023,360],{"stretchy":334},[332,18025,992],{},[738,18027,740],{},[332,18029,987],{},[738,18031,17938],{},[258,18033,4429],{},[332,18035,335],{"stretchy":334},[738,18037,745],{},[332,18039,360],{"stretchy":334},[258,18041,3102],{"mathvariant":344},[262,18043,18044],{"encoding":264},"V(1)=1+0.9V(2),\\qquad V(2)=0+0.9V(1).",[238,18046,18048,18075,18094,18145,18163],{"className":18047,"ariaHidden":269},[268],[238,18049,18051,18054,18057,18060,18063,18066,18069,18072],{"className":18050},[273],[238,18052],{"className":18053,"style":444},[277],[238,18055,4429],{"className":18056,"style":660},[282,283],[238,18058,335],{"className":18059},[381],[238,18061,745],{"className":18062},[282],[238,18064,360],{"className":18065},[462],[238,18067],{"className":18068,"style":430},[429],[238,18070,992],{"className":18071},[434],[238,18073],{"className":18074,"style":430},[429],[238,18076,18078,18082,18085,18088,18091],{"className":18077},[273],[238,18079],{"className":18080,"style":18081},[277],"height:0.7278em;vertical-align:-0.0833em;",[238,18083,745],{"className":18084},[282],[238,18086],{"className":18087,"style":660},[429],[238,18089,987],{"className":18090},[664],[238,18092],{"className":18093,"style":660},[429],[238,18095,18097,18100,18103,18106,18109,18112,18115,18118,18121,18124,18127,18130,18133,18136,18139,18142],{"className":18096},[273],[238,18098],{"className":18099,"style":444},[277],[238,18101,17938],{"className":18102},[282],[238,18104,4429],{"className":18105,"style":660},[282,283],[238,18107,335],{"className":18108},[381],[238,18110,17558],{"className":18111},[282],[238,18113,360],{"className":18114},[462],[238,18116,354],{"className":18117},[451],[238,18119],{"className":18120,"style":10986},[429],[238,18122],{"className":18123,"style":455},[429],[238,18125,4429],{"className":18126,"style":660},[282,283],[238,18128,335],{"className":18129},[381],[238,18131,17558],{"className":18132},[282],[238,18134,360],{"className":18135},[462],[238,18137],{"className":18138,"style":430},[429],[238,18140,992],{"className":18141},[434],[238,18143],{"className":18144,"style":430},[429],[238,18146,18148,18151,18154,18157,18160],{"className":18147},[273],[238,18149],{"className":18150,"style":18081},[277],[238,18152,740],{"className":18153},[282],[238,18155],{"className":18156,"style":660},[429],[238,18158,987],{"className":18159},[664],[238,18161],{"className":18162,"style":660},[429],[238,18164,18166,18169,18172,18175,18178,18181,18184],{"className":18165},[273],[238,18167],{"className":18168,"style":444},[277],[238,18170,17938],{"className":18171},[282],[238,18173,4429],{"className":18174,"style":660},[282,283],[238,18176,335],{"className":18177},[381],[238,18179,745],{"className":18180},[282],[238,18182,360],{"className":18183},[462],[238,18185,3102],{"className":18186},[282],[11,18188,18189],{},"将第二式代入第一式，得",[238,18191,18193],{"className":18192,"translate":242},[945],[238,18194,18196,18235],{"className":18195,"translate":242},[241],[238,18197,18199],{"className":18198},[246],[248,18200,18201],{"xmlns":250,"display":954},[252,18202,18203,18232],{},[255,18204,18205,18207,18209,18211,18213,18215,18217,18219,18222,18224,18226,18228,18230],{},[258,18206,4429],{},[332,18208,335],{"stretchy":334},[738,18210,745],{},[332,18212,360],{"stretchy":334},[332,18214,992],{},[738,18216,745],{},[332,18218,987],{},[738,18220,18221],{},"0.81",[258,18223,4429],{},[332,18225,335],{"stretchy":334},[738,18227,745],{},[332,18229,360],{"stretchy":334},[332,18231,354],{"separator":269},[262,18233,18234],{"encoding":264},"V(1)=1+0.81V(1),",[238,18236,18238,18265,18283],{"className":18237,"ariaHidden":269},[268],[238,18239,18241,18244,18247,18250,18253,18256,18259,18262],{"className":18240},[273],[238,18242],{"className":18243,"style":444},[277],[238,18245,4429],{"className":18246,"style":660},[282,283],[238,18248,335],{"className":18249},[381],[238,18251,745],{"className":18252},[282],[238,18254,360],{"className":18255},[462],[238,18257],{"className":18258,"style":430},[429],[238,18260,992],{"className":18261},[434],[238,18263],{"className":18264,"style":430},[429],[238,18266,18268,18271,18274,18277,18280],{"className":18267},[273],[238,18269],{"className":18270,"style":18081},[277],[238,18272,745],{"className":18273},[282],[238,18275],{"className":18276,"style":660},[429],[238,18278,987],{"className":18279},[664],[238,18281],{"className":18282,"style":660},[429],[238,18284,18286,18289,18292,18295,18298,18301,18304],{"className":18285},[273],[238,18287],{"className":18288,"style":444},[277],[238,18290,18221],{"className":18291},[282],[238,18293,4429],{"className":18294,"style":660},[282,283],[238,18296,335],{"className":18297},[381],[238,18299,745],{"className":18300},[282],[238,18302,360],{"className":18303},[462],[238,18305,354],{"className":18306},[451],[11,18308,18309],{},"因此",[238,18311,18313],{"className":18312,"translate":242},[945],[238,18314,18316,18384],{"className":18315,"translate":242},[241],[238,18317,18319],{"className":18318},[246],[248,18320,18321],{"xmlns":250,"display":954},[252,18322,18323,18381],{},[255,18324,18325,18327,18329,18331,18333,18335,18347,18349,18352,18354,18356,18358,18360,18362,18364,18366,18368,18370,18372,18374,18376,18378],{},[258,18326,4429],{},[332,18328,335],{"stretchy":334},[738,18330,745],{},[332,18332,360],{"stretchy":334},[332,18334,992],{},[5561,18336,18337,18339],{},[738,18338,745],{},[255,18340,18341,18343,18345],{},[738,18342,745],{},[332,18344,5577],{},[738,18346,18221],{},[332,18348,992],{},[738,18350,18351],{},"5.2632",[332,18353,354],{"separator":269},[429,18355],{"width":10450},[258,18357,4429],{},[332,18359,335],{"stretchy":334},[738,18361,17558],{},[332,18363,360],{"stretchy":334},[332,18365,992],{},[738,18367,17938],{},[258,18369,4429],{},[332,18371,335],{"stretchy":334},[738,18373,745],{},[332,18375,360],{"stretchy":334},[332,18377,992],{},[738,18379,18380],{},"4.7368.",[262,18382,18383],{"encoding":264},"V(1)=\\frac{1}{1-0.81}=5.2632,\\qquad\nV(2)=0.9V(1)=4.7368.",[238,18385,18387,18414,18506,18545,18575],{"className":18386,"ariaHidden":269},[268],[238,18388,18390,18393,18396,18399,18402,18405,18408,18411],{"className":18389},[273],[238,18391],{"className":18392,"style":444},[277],[238,18394,4429],{"className":18395,"style":660},[282,283],[238,18397,335],{"className":18398},[381],[238,18400,745],{"className":18401},[282],[238,18403,360],{"className":18404},[462],[238,18406],{"className":18407,"style":430},[429],[238,18409,992],{"className":18410},[434],[238,18412],{"className":18413,"style":430},[429],[238,18415,18417,18421,18497,18500,18503],{"className":18416},[273],[238,18418],{"className":18419,"style":18420},[277],"height:2.0908em;vertical-align:-0.7693em;",[238,18422,18424,18427,18494],{"className":18423},[282],[238,18425],{"className":18426},[381,5662],[238,18428,18430],{"className":18429},[5561],[238,18431,18433,18485],{"className":18432},[395,848],[238,18434,18436,18482],{"className":18435},[399],[238,18437,18440,18463,18471],{"className":18438,"style":18439},[403],"height:1.3214em;",[238,18441,18442,18445],{"style":5678},[238,18443],{"className":18444,"style":1134},[411],[238,18446,18448,18451,18454,18457,18460],{"className":18447},[282],[238,18449,745],{"className":18450},[282],[238,18452],{"className":18453,"style":660},[429],[238,18455,5577],{"className":18456},[664],[238,18458],{"className":18459,"style":660},[429],[238,18461,18221],{"className":18462},[282],[238,18464,18465,18468],{"style":5702},[238,18466],{"className":18467,"style":1134},[411],[238,18469],{"className":18470,"style":5710},[5709],[238,18472,18473,18476],{"style":5713},[238,18474],{"className":18475,"style":1134},[411],[238,18477,18479],{"className":18478},[282],[238,18480,745],{"className":18481},[282],[238,18483,872],{"className":18484},[871],[238,18486,18488],{"className":18487},[399],[238,18489,18492],{"className":18490,"style":18491},[403],"height:0.7693em;",[238,18493],{},[238,18495],{"className":18496},[462,5662],[238,18498],{"className":18499,"style":430},[429],[238,18501,992],{"className":18502},[434],[238,18504],{"className":18505,"style":430},[429],[238,18507,18509,18512,18515,18518,18521,18524,18527,18530,18533,18536,18539,18542],{"className":18508},[273],[238,18510],{"className":18511,"style":444},[277],[238,18513,18351],{"className":18514},[282],[238,18516,354],{"className":18517},[451],[238,18519],{"className":18520,"style":10986},[429],[238,18522],{"className":18523,"style":455},[429],[238,18525,4429],{"className":18526,"style":660},[282,283],[238,18528,335],{"className":18529},[381],[238,18531,17558],{"className":18532},[282],[238,18534,360],{"className":18535},[462],[238,18537],{"className":18538,"style":430},[429],[238,18540,992],{"className":18541},[434],[238,18543],{"className":18544,"style":430},[429],[238,18546,18548,18551,18554,18557,18560,18563,18566,18569,18572],{"className":18547},[273],[238,18549],{"className":18550,"style":444},[277],[238,18552,17938],{"className":18553},[282],[238,18555,4429],{"className":18556,"style":660},[282,283],[238,18558,335],{"className":18559},[381],[238,18561,745],{"className":18562},[282],[238,18564,360],{"className":18565},[462],[238,18567],{"className":18568,"style":430},[429],[238,18570,992],{"className":18571},[434],[238,18573],{"className":18574,"style":430},[429],[238,18576,18578,18581],{"className":18577},[273],[238,18579],{"className":18580,"style":2672},[277],[238,18582,18380],{"className":18583},[282],[11,18585,18586],{},"用矩阵形式验证，转移矩阵与奖励向量为",[238,18588,18590],{"className":18589,"translate":242},[945],[238,18591,18593,18682],{"className":18592,"translate":242},[241],[238,18594,18596],{"className":18595},[246],[248,18597,18598],{"xmlns":250,"display":954},[252,18599,18600,18679],{},[255,18601,18602,18604,18606,18644,18646,18648,18650,18652,18677],{},[258,18603,330],{},[332,18605,992],{},[255,18607,18608,18610,18642],{},[332,18609,335],{"fence":269},[958,18611,18614,18628],{"rowspacing":18612,"columnalign":18613,"columnspacing":7090},"0.16em","center 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"className":19793},[399],[238,19795,19797,19808],{"className":19796,"style":18739},[403],[238,19798,19799,19802],{"style":18742},[238,19800],{"className":19801,"style":1134},[411],[238,19803,19805],{"className":19804},[282],[238,19806,18351],{"className":19807},[282],[238,19809,19810,19813],{"style":18754},[238,19811],{"className":19812,"style":1134},[411],[238,19814,19816],{"className":19815},[282],[238,19817,19447],{"className":19818},[282],[238,19820,872],{"className":19821},[871],[238,19823,19825],{"className":19824},[399],[238,19826,19828],{"className":19827,"style":18773},[403],[238,19829],{},[238,19831,19833],{"className":19832,"style":4791},[462,4790],[238,19834,360],{"className":19835},[18719,418],[238,19837],{"className":19838,"style":455},[429],[238,19840,3102],{"className":19841},[282],[238,19843,872],{"className":19844},[871],[238,19846,19848],{"className":19847},[399],[238,19849,19851],{"className":19850,"style":19515},[403],[238,19852],{},[11,19854,19855,19856,19939],{},"这个例子说明，即使状态每两步就回到原处，只要 ",[238,19857,19859,19881],{"className":19858,"translate":242},[241],[238,19860,19862],{"className":19861},[246],[248,19863,19864],{"xmlns":250},[252,19865,19866,19878],{},[255,19867,19868,19874,19876],{},[337,19869,19870,19872],{},[258,19871,730],{},[738,19873,17558],{},[332,19875,1892],{},[738,19877,745],{},[262,19879,19880],{"encoding":264},"\\gamma^2\u003C1",[238,19882,19884,19930],{"className":19883,"ariaHidden":269},[268],[238,19885,19887,19891,19921,19924,19927],{"className":19886},[273],[238,19888],{"className":19889,"style":19890},[277],"height:1.0085em;vertical-align:-0.1944em;",[238,19892,19894,19897],{"className":19893},[282],[238,19895,730],{"className":19896,"style":764},[282,283],[238,19898,19900],{"className":19899},[391],[238,19901,19903],{"className":19902},[395],[238,19904,19906],{"className":19905},[399],[238,19907,19910],{"className":19908,"style":19909},[403],"height:0.8141em;",[238,19911,19912,19915],{"style":407},[238,19913],{"className":19914,"style":412},[411],[238,19916,19918],{"className":19917},[416,417,418,419],[238,19919,17558],{"className":19920},[282,419],[238,19922],{"className":19923,"style":430},[429],[238,19925,1892],{"className":19926},[434],[238,19928],{"className":19929,"style":430},[429],[238,19931,19933,19936],{"className":19932},[273],[238,19934],{"className":19935,"style":2672},[277],[238,19937,745],{"className":19938},[282],"，折扣回报的期望仍然是有限的。值函数不是把无限多项直接相加，而是通过解线性方程组把无限和压缩成一个有限量。",[225,19941,19943],{"id":19942},"_25-方程与回报定义的等价性","2.5 方程与回报定义的等价性",[11,19945,19946],{},"贝尔曼方程不仅是值函数的必要条件，也是充分条件。下面的命题说明，任何有界解都必然等于由折扣回报定义的值函数。",[230,19948,19951],{"id":19949,"title":19950},"proposition-policy-verification","命题 2.3（策略评估的验证定理）",[11,19952,236,19953,16685,20021,14826,20107,20188,20189,20247],{},[238,19954,19956,19981],{"className":19955,"translate":242},[241],[238,19957,19959],{"className":19958},[246],[248,19960,19961],{"xmlns":250},[252,19962,19963,19978],{},[255,19964,19965,19967,19969,19972,19974,19976],{},[258,19966,4429],{},[332,19968,733],{},[258,19970,19971],{},"B",[332,19973,335],{"stretchy":334},[258,19975,260],{},[332,19977,360],{"stretchy":334},[262,19979,19980],{"encoding":264},"V\\in B(S)",[238,19982,19984,20002],{"className":19983,"ariaHidden":269},[268],[238,19985,19987,19990,19993,19996,19999],{"className":19986},[273],[238,19988],{"className":19989,"style":13260},[277],[238,19991,4429],{"className":19992,"style":660},[282,283],[238,19994],{"className":19995,"style":430},[429],[238,19997,733],{"className":19998},[434],[238,20000],{"className":20001,"style":430},[429],[238,20003,20005,20008,20012,20015,20018],{"className":20004},[273],[238,20006],{"className":20007,"style":444},[277],[238,20009,19971],{"className":20010,"style":20011},[282,283],"margin-right:0.0502em;",[238,20013,335],{"className":20014},[381],[238,20016,260],{"className":20017,"style":284},[282,283],[238,20019,360],{"className":20020},[462],[238,20022,20024,20048],{"className":20023,"translate":242},[241],[238,20025,20027],{"className":20026},[246],[248,20028,20029],{"xmlns":250},[252,20030,20031,20045],{},[255,20032,20033,20035,20037,20043],{},[258,20034,4429],{},[332,20036,992],{},[337,20038,20039,20041],{},[258,20040,11971],{"mathvariant":816},[258,20042,2297],{},[258,20044,4429],{},[262,20046,20047],{"encoding":264},"V=\\mathcal{T}^\\pi V",[238,20049,20051,20069],{"className":20050,"ariaHidden":269},[268],[238,20052,20054,20057,20060,20063,20066],{"className":20053},[273],[238,20055],{"className":20056,"style":278},[277],[238,20058,4429],{"className":20059,"style":660},[282,283],[238,20061],{"className":20062,"style":430},[429],[238,20064,992],{"className":20065},[434],[238,20067],{"className":20068,"style":430},[429],[238,20070,20072,20075,20104],{"className":20071},[273],[238,20073],{"className":20074,"style":278},[277],[238,20076,20078,20081],{"className":20077},[282],[238,20079,11971],{"className":20080,"style":11992},[282,840],[238,20082,20084],{"className":20083},[391],[238,20085,20087],{"className":20086},[395],[238,20088,20090],{"className":20089},[399],[238,20091,20093],{"className":20092,"style":6609},[403],[238,20094,20095,20098],{"style":407},[238,20096],{"className":20097,"style":412},[411],[238,20099,20101],{"className":20100},[416,417,418,419],[238,20102,2297],{"className":20103,"style":2313},[282,283,419],[238,20105,4429],{"className":20106,"style":660},[282,283],[238,20108,20110,20132],{"className":20109,"translate":242},[241],[238,20111,20113],{"className":20112},[246],[248,20114,20115],{"xmlns":250},[252,20116,20117,20129],{},[255,20118,20119,20121,20123],{},[258,20120,4429],{},[332,20122,992],{},[337,20124,20125,20127],{},[258,20126,4429],{},[258,20128,2297],{},[262,20130,20131],{"encoding":264},"V=V^\\pi",[238,20133,20135,20153],{"className":20134,"ariaHidden":269},[268],[238,20136,20138,20141,20144,20147,20150],{"className":20137},[273],[238,20139],{"className":20140,"style":278},[277],[238,20142,4429],{"className":20143,"style":660},[282,283],[238,20145],{"className":20146,"style":430},[429],[238,20148,992],{"className":20149},[434],[238,20151],{"className":20152,"style":430},[429],[238,20154,20156,20159],{"className":20155},[273],[238,20157],{"className":20158,"style":278},[277],[238,20160,20162,20165],{"className":20161},[282],[238,20163,4429],{"className":20164,"style":660},[282,283],[238,20166,20168],{"className":20167},[391],[238,20169,20171],{"className":20170},[395],[238,20172,20174],{"className":20173},[399],[238,20175,20177],{"className":20176,"style":6609},[403],[238,20178,20179,20182],{"style":407},[238,20180],{"className":20181,"style":412},[411],[238,20183,20185],{"className":20184},[416,417,418,419],[238,20186,2297],{"className":20187,"style":2313},[282,283,419],"。特别地，",[238,20190,20192,20209],{"className":20191,"translate":242},[241],[238,20193,20195],{"className":20194},[246],[248,20196,20197],{"xmlns":250},[252,20198,20199,20207],{},[255,20200,20201],{},[337,20202,20203,20205],{},[258,20204,11971],{"mathvariant":816},[258,20206,2297],{},[262,20208,11976],{"encoding":264},[238,20210,20212],{"className":20211,"ariaHidden":269},[268],[238,20213,20215,20218],{"className":20214},[273],[238,20216],{"className":20217,"style":278},[277],[238,20219,20221,20224],{"className":20220},[282],[238,20222,11971],{"className":20223,"style":11992},[282,840],[238,20225,20227],{"className":20226},[391],[238,20228,20230],{"className":20229},[395],[238,20231,20233],{"className":20232},[399],[238,20234,20236],{"className":20235,"style":6609},[403],[238,20237,20238,20241],{"style":407},[238,20239],{"className":20240,"style":412},[411],[238,20242,20244],{"className":20243},[416,417,418,419],[238,20245,2297],{"className":20246,"style":2313},[282,283,419]," 在有界函数空间中至多有一个不动点。",[11,20249,20250,20252,20253,20338],{},[1688,20251,5794],{}," 设 ",[238,20254,20256,20279],{"className":20255,"translate":242},[241],[238,20257,20259],{"className":20258},[246],[248,20260,20261],{"xmlns":250},[252,20262,20263,20277],{},[255,20264,20265,20267,20269,20275],{},[258,20266,4429],{},[332,20268,992],{},[337,20270,20271,20273],{},[258,20272,11971],{"mathvariant":816},[258,20274,2297],{},[258,20276,4429],{},[262,20278,20047],{"encoding":264},[238,20280,20282,20300],{"className":20281,"ariaHidden":269},[268],[238,20283,20285,20288,20291,20294,20297],{"className":20284},[273],[238,20286],{"className":20287,"style":278},[277],[238,20289,4429],{"className":20290,"style":660},[282,283],[238,20292],{"className":20293,"style":430},[429],[238,20295,992],{"className":20296},[434],[238,20298],{"className":20299,"style":430},[429],[238,20301,20303,20306,20335],{"className":20302},[273],[238,20304],{"className":20305,"style":278},[277],[238,20307,20309,20312],{"className":20308},[282],[238,20310,11971],{"className":20311,"style":11992},[282,840],[238,20313,20315],{"className":20314},[391],[238,20316,20318],{"className":20317},[395],[238,20319,20321],{"className":20320},[399],[238,20322,20324],{"className":20323,"style":6609},[403],[238,20325,20326,20329],{"style":407},[238,20327],{"className":20328,"style":412},[411],[238,20330,20332],{"className":20331},[416,417,418,419],[238,20333,2297],{"className":20334,"style":2313},[282,283,419],[238,20336,4429],{"className":20337,"style":660},[282,283],"。反复代入得",[238,20340,20342],{"className":20341,"translate":242},[945],[238,20343,20345,20433],{"className":20344,"translate":242},[241],[238,20346,20348],{"className":20347},[246],[248,20349,20350],{"xmlns":250,"display":954},[252,20351,20352,20430],{},[255,20353,20354,20356,20358,20378,20384,20386,20392,20398,20404,20406,20412,20414,20420,20426,20428],{},[258,20355,4429],{},[332,20357,992],{},[3386,20359,20360,20362,20370],{},[332,20361,3390],{},[255,20363,20364,20366,20368],{},[258,20365,3395],{},[332,20367,992],{},[738,20369,740],{},[255,20371,20372,20374,20376],{},[258,20373,13161],{},[332,20375,5577],{},[738,20377,745],{},[337,20379,20380,20382],{},[258,20381,730],{},[258,20383,3395],{},[332,20385,335],{"stretchy":334},[337,20387,20388,20390],{},[258,20389,330],{},[258,20391,2297],{},[337,20393,20394,20396],{},[332,20395,360],{"stretchy":334},[258,20397,3395],{},[337,20399,20400,20402],{},[258,20401,600],{},[258,20403,2297],{},[332,20405,987],{},[337,20407,20408,20410],{},[258,20409,730],{},[258,20411,13161],{},[332,20413,335],{"stretchy":334},[337,20415,20416,20418],{},[258,20417,330],{},[258,20419,2297],{},[337,20421,20422,20424],{},[332,20423,360],{"stretchy":334},[258,20425,13161],{},[258,20427,4429],{},[258,20429,3102],{"mathvariant":344},[262,20431,20432],{"encoding":264},"V=\\sum_{k=0}^{n-1}\\gamma^k(P^\\pi)^k r^\\pi\n+\\gamma^n(P^\\pi)^n V.",[238,20434,20436,20454,20666],{"className":20435,"ariaHidden":269},[268],[238,20437,20439,20442,20445,20448,20451],{"className":20438},[273],[238,20440],{"className":20441,"style":278},[277],[238,20443,4429],{"className":20444,"style":660},[282,283],[238,20446],{"className":20447,"style":430},[429],[238,20449,992],{"className":20450},[434],[238,20452],{"className":20453,"style":430},[429],[238,20455,20457,20461,20535,20538,20567,20570,20599,20628,20657,20660,20663],{"className":20456},[273],[238,20458],{"className":20459,"style":20460},[277],"height:3.1032em;vertical-align:-1.3021em;",[238,20462,20464],{"className":20463},[1943,3493],[238,20465,20467,20527],{"className":20466},[395,848],[238,20468,20470,20524],{"className":20469},[399],[238,20471,20474,20494,20504],{"className":20472,"style":20473},[403],"height:1.8011em;",[238,20475,20476,20479],{"style":3506},[238,20477],{"className":20478,"style":3510},[411],[238,20480,20482],{"className":20481},[416,417,418,419],[238,20483,20485,20488,20491],{"className":20484},[282,419],[238,20486,3395],{"className":20487,"style":3520},[282,283,419],[238,20489,992],{"className":20490},[434,419],[238,20492,740],{"className":20493},[282,419],[238,20495,20496,20499],{"style":3529},[238,20497],{"className":20498,"style":3510},[411],[238,20500,20501],{},[238,20502,3390],{"className":20503},[1943,3538,3539],[238,20505,20506,20509],{"style":3542},[238,20507],{"className":20508,"style":3510},[411],[238,20510,20512],{"className":20511},[416,417,418,419],[238,20513,20515,20518,20521],{"className":20514},[282,419],[238,20516,13161],{"className":20517},[282,283,419],[238,20519,5577],{"className":20520},[664,419],[238,20522,745],{"className":20523},[282,419],[238,20525,872],{"className":20526},[871],[238,20528,20530],{"className":20529},[399],[238,20531,20533],{"className":20532,"style":3564},[403],[238,20534],{},[238,20536],{"className":20537,"style":455},[429],[238,20539,20541,20544],{"className":20540},[282],[238,20542,730],{"className":20543,"style":764},[282,283],[238,20545,20547],{"className":20546},[391],[238,20548,20550],{"className":20549},[395],[238,20551,20553],{"className":20552},[399],[238,20554,20556],{"className":20555,"style":3588},[403],[238,20557,20558,20561],{"style":1226},[238,20559],{"className":20560,"style":412},[411],[238,20562,20564],{"className":20563},[416,417,418,419],[238,20565,3395],{"className":20566,"style":3520},[282,283,419],[238,20568,335],{"className":20569},[381],[238,20571,20573,20576],{"className":20572},[282],[238,20574,330],{"className":20575,"style":377},[282,283],[238,20577,20579],{"className":20578},[391],[238,20580,20582],{"className":20581},[395],[238,20583,20585],{"className":20584},[399],[238,20586,20588],{"className":20587,"style":4620},[403],[238,20589,20590,20593],{"style":1226},[238,20591],{"className":20592,"style":412},[411],[238,20594,20596],{"className":20595},[416,417,418,419],[238,20597,2297],{"className":20598,"style":2313},[282,283,419],[238,20600,20602,20605],{"className":20601},[462],[238,20603,360],{"className":20604},[462],[238,20606,20608],{"className":20607},[391],[238,20609,20611],{"className":20610},[395],[238,20612,20614],{"className":20613},[399],[238,20615,20617],{"className":20616,"style":3588},[403],[238,20618,20619,20622],{"style":1226},[238,20620],{"className":20621,"style":412},[411],[238,20623,20625],{"className":20624},[416,417,418,419],[238,20626,3395],{"className":20627,"style":3520},[282,283,419],[238,20629,20631,20634],{"className":20630},[282],[238,20632,600],{"className":20633,"style":637},[282,283],[238,20635,20637],{"className":20636},[391],[238,20638,20640],{"className":20639},[395],[238,20641,20643],{"className":20642},[399],[238,20644,20646],{"className":20645,"style":4620},[403],[238,20647,20648,20651],{"style":1226},[238,20649],{"className":20650,"style":412},[411],[238,20652,20654],{"className":20653},[416,417,418,419],[238,20655,2297],{"className":20656,"style":2313},[282,283,419],[238,20658],{"className":20659,"style":660},[429],[238,20661,987],{"className":20662},[664],[238,20664],{"className":20665,"style":660},[429],[238,20667,20669,20672,20701,20704,20733,20762,20765],{"className":20668},[273],[238,20670],{"className":20671,"style":444},[277],[238,20673,20675,20678],{"className":20674},[282],[238,20676,730],{"className":20677,"style":764},[282,283],[238,20679,20681],{"className":20680},[391],[238,20682,20684],{"className":20683},[395],[238,20685,20687],{"className":20686},[399],[238,20688,20690],{"className":20689,"style":4620},[403],[238,20691,20692,20695],{"style":1226},[238,20693],{"className":20694,"style":412},[411],[238,20696,20698],{"className":20697},[416,417,418,419],[238,20699,13161],{"className":20700},[282,283,419],[238,20702,335],{"className":20703},[381],[238,20705,20707,20710],{"className":20706},[282],[238,20708,330],{"className":20709,"style":377},[282,283],[238,20711,20713],{"className":20712},[391],[238,20714,20716],{"className":20715},[395],[238,20717,20719],{"className":20718},[399],[238,20720,20722],{"className":20721,"style":4620},[403],[238,20723,20724,20727],{"style":1226},[238,20725],{"className":20726,"style":412},[411],[238,20728,20730],{"className":20729},[416,417,418,419],[238,20731,2297],{"className":20732,"style":2313},[282,283,419],[238,20734,20736,20739],{"className":20735},[462],[238,20737,360],{"className":20738},[462],[238,20740,20742],{"className":20741},[391],[238,20743,20745],{"className":20744},[395],[238,20746,20748],{"className":20747},[399],[238,20749,20751],{"className":20750,"style":4620},[403],[238,20752,20753,20756],{"style":1226},[238,20754],{"className":20755,"style":412},[411],[238,20757,20759],{"className":20758},[416,417,418,419],[238,20760,13161],{"className":20761},[282,283,419],[238,20763,4429],{"className":20764,"style":660},[282,283],[238,20766,3102],{"className":20767},[282],[11,20769,20770,20771,20999,21000,21028],{},"由于 ",[238,20772,20774,20822],{"className":20773,"translate":242},[241],[238,20775,20777],{"className":20776},[246],[248,20778,20779],{"xmlns":250},[252,20780,20781,20819],{},[255,20782,20783,20785,20787,20793,20799,20801,20807,20809,20811,20813],{},[258,20784,15588],{"mathvariant":344},[332,20786,335],{"stretchy":334},[337,20788,20789,20791],{},[258,20790,330],{},[258,20792,2297],{},[337,20794,20795,20797],{},[332,20796,360],{"stretchy":334},[258,20798,13161],{},[258,20800,4429],{},[812,20802,20803,20805],{},[258,20804,15588],{"mathvariant":344},[258,20806,1895],{"mathvariant":344},[332,20808,2023],{},[258,20810,15588],{"mathvariant":344},[258,20812,4429],{},[812,20814,20815,20817],{},[258,20816,15588],{"mathvariant":344},[258,20818,1895],{"mathvariant":344},[262,20820,20821],{"encoding":264},"\\|(P^\\pi)^n V\\|_\\infty\\le\\|V\\|_\\infty",[238,20823,20825,20947],{"className":20824,"ariaHidden":269},[268],[238,20826,20828,20831,20834,20837,20866,20895,20898,20938,20941,20944],{"className":20827},[273],[238,20829],{"className":20830,"style":444},[277],[238,20832,15588],{"className":20833},[282],[238,20835,335],{"className":20836},[381],[238,20838,20840,20843],{"className":20839},[282],[238,20841,330],{"className":20842,"style":377},[282,283],[238,20844,20846],{"className":20845},[391],[238,20847,20849],{"className":20848},[395],[238,20850,20852],{"className":20851},[399],[238,20853,20855],{"className":20854,"style":6609},[403],[238,20856,20857,20860],{"style":407},[238,20858],{"className":20859,"style":412},[411],[238,20861,20863],{"className":20862},[416,417,418,419],[238,20864,2297],{"className":20865,"style":2313},[282,283,419],[238,20867,20869,20872],{"className":20868},[462],[238,20870,360],{"className":20871},[462],[238,20873,20875],{"className":20874},[391],[238,20876,20878],{"className":20877},[395],[238,20879,20881],{"className":20880},[399],[238,20882,20884],{"className":20883,"style":6609},[403],[238,20885,20886,20889],{"style":407},[238,20887],{"className":20888,"style":412},[411],[238,20890,20892],{"className":20891},[416,417,418,419],[238,20893,13161],{"className":20894},[282,283,419],[238,20896,4429],{"className":20897,"style":660},[282,283],[238,20899,20901,20904],{"className":20900},[282],[238,20902,15588],{"className":20903},[282],[238,20905,20907],{"className":20906},[391],[238,20908,20910,20930],{"className":20909},[395,848],[238,20911,20913,20927],{"className":20912},[399],[238,20914,20916],{"className":20915,"style":1927},[403],[238,20917,20918,20921],{"style":1165},[238,20919],{"className":20920,"style":412},[411],[238,20922,20924],{"className":20923},[416,417,418,419],[238,20925,1895],{"className":20926},[282,419],[238,20928,872],{"className":20929},[871],[238,20931,20933],{"className":20932},[399],[238,20934,20936],{"className":20935,"style":879},[403],[238,20937],{},[238,20939],{"className":20940,"style":430},[429],[238,20942,2023],{"className":20943},[434],[238,20945],{"className":20946,"style":430},[429],[238,20948,20950,20953,20956,20959],{"className":20949},[273],[238,20951],{"className":20952,"style":444},[277],[238,20954,15588],{"className":20955},[282],[238,20957,4429],{"className":20958,"style":660},[282,283],[238,20960,20962,20965],{"className":20961},[282],[238,20963,15588],{"className":20964},[282],[238,20966,20968],{"className":20967},[391],[238,20969,20971,20991],{"className":20970},[395,848],[238,20972,20974,20988],{"className":20973},[399],[238,20975,20977],{"className":20976,"style":1927},[403],[238,20978,20979,20982],{"style":1165},[238,20980],{"className":20981,"style":412},[411],[238,20983,20985],{"className":20984},[416,417,418,419],[238,20986,1895],{"className":20987},[282,419],[238,20989,872],{"className":20990},[871],[238,20992,20994],{"className":20993},[399],[238,20995,20997],{"className":20996,"style":879},[403],[238,20998],{},"，最后一项在 sup 范数下趋于 ",[238,21001,21003,21016],{"className":21002,"translate":242},[241],[238,21004,21006],{"className":21005},[246],[248,21007,21008],{"xmlns":250},[252,21009,21010,21014],{},[255,21011,21012],{},[738,21013,740],{},[262,21015,740],{"encoding":264},[238,21017,21019],{"className":21018,"ariaHidden":269},[268],[238,21020,21022,21025],{"className":21021},[273],[238,21023],{"className":21024,"style":2672},[277],[238,21026,740],{"className":21027},[282],"。另一方面，",[238,21030,21032],{"className":21031,"translate":242},[945],[238,21033,21035,21087],{"className":21034,"translate":242},[241],[238,21036,21038],{"className":21037},[246],[248,21039,21040],{"xmlns":250,"display":954},[252,21041,21042,21084],{},[255,21043,21044,21058,21064,21066,21072,21078],{},[3386,21045,21046,21048,21056],{},[332,21047,3390],{},[255,21049,21050,21052,21054],{},[258,21051,3395],{},[332,21053,992],{},[738,21055,740],{},[258,21057,1895],{"mathvariant":344},[337,21059,21060,21062],{},[258,21061,730],{},[258,21063,3395],{},[332,21065,335],{"stretchy":334},[337,21067,21068,21070],{},[258,21069,330],{},[258,21071,2297],{},[337,21073,21074,21076],{},[332,21075,360],{"stretchy":334},[258,21077,3395],{},[337,21079,21080,21082],{},[258,21081,600],{},[258,21083,2297],{},[262,21085,21086],{"encoding":264},"\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi",[238,21088,21090],{"className":21089,"ariaHidden":269},[268],[238,21091,21093,21096,21163,21166,21195,21198,21227,21256],{"className":21092},[273],[238,21094],{"className":21095,"style":3489},[277],[238,21097,21099],{"className":21098},[1943,3493],[238,21100,21102,21155],{"className":21101},[395,848],[238,21103,21105,21152],{"className":21104},[399],[238,21106,21108,21128,21138],{"className":21107,"style":3503},[403],[238,21109,21110,21113],{"style":3506},[238,21111],{"className":21112,"style":3510},[411],[238,21114,21116],{"className":21115},[416,417,418,419],[238,21117,21119,21122,21125],{"className":21118},[282,419],[238,21120,3395],{"className":21121,"style":3520},[282,283,419],[238,21123,992],{"className":21124},[434,419],[238,21126,740],{"className":21127},[282,419],[238,21129,21130,21133],{"style":3529},[238,21131],{"className":21132,"style":3510},[411],[238,21134,21135],{},[238,21136,3390],{"className":21137},[1943,3538,3539],[238,21139,21140,21143],{"style":3542},[238,21141],{"className":21142,"style":3510},[411],[238,21144,21146],{"className":21145},[416,417,418,419],[238,21147,21149],{"className":21148},[282,419],[238,21150,1895],{"className":21151},[282,419],[238,21153,872],{"className":21154},[871],[238,21156,21158],{"className":21157},[399],[238,21159,21161],{"className":21160,"style":3564},[403],[238,21162],{},[238,21164],{"className":21165,"style":455},[429],[238,21167,21169,21172],{"className":21168},[282],[238,21170,730],{"className":21171,"style":764},[282,283],[238,21173,21175],{"className":21174},[391],[238,21176,21178],{"className":21177},[395],[238,21179,21181],{"className":21180},[399],[238,21182,21184],{"className":21183,"style":3588},[403],[238,21185,21186,21189],{"style":1226},[238,21187],{"className":21188,"style":412},[411],[238,21190,21192],{"className":21191},[416,417,418,419],[238,21193,3395],{"className":21194,"style":3520},[282,283,419],[238,21196,335],{"className":21197},[381],[238,21199,21201,21204],{"className":21200},[282],[238,21202,330],{"className":21203,"style":377},[282,283],[238,21205,21207],{"className":21206},[391],[238,21208,21210],{"className":21209},[395],[238,21211,21213],{"className":21212},[399],[238,21214,21216],{"className":21215,"style":4620},[403],[238,21217,21218,21221],{"style":1226},[238,21219],{"className":21220,"style":412},[411],[238,21222,21224],{"className":21223},[416,417,418,419],[238,21225,2297],{"className":21226,"style":2313},[282,283,419],[238,21228,21230,21233],{"className":21229},[462],[238,21231,360],{"className":21232},[462],[238,21234,21236],{"className":21235},[391],[238,21237,21239],{"className":21238},[395],[238,21240,21242],{"className":21241},[399],[238,21243,21245],{"className":21244,"style":3588},[403],[238,21246,21247,21250],{"style":1226},[238,21248],{"className":21249,"style":412},[411],[238,21251,21253],{"className":21252},[416,417,418,419],[238,21254,3395],{"className":21255,"style":3520},[282,283,419],[238,21257,21259,21262],{"className":21258},[282],[238,21260,600],{"className":21261,"style":637},[282,283],[238,21263,21265],{"className":21264},[391],[238,21266,21268],{"className":21267},[395],[238,21269,21271],{"className":21270},[399],[238,21272,21274],{"className":21273,"style":4620},[403],[238,21275,21276,21279],{"style":1226},[238,21277],{"className":21278,"style":412},[411],[238,21280,21282],{"className":21281},[416,417,418,419],[238,21283,2297],{"className":21284,"style":2313},[282,283,419],[11,21286,21287,21288,21346,21347,21427],{},"正是折扣回报的期望，即 ",[238,21289,21291,21308],{"className":21290,"translate":242},[241],[238,21292,21294],{"className":21293},[246],[248,21295,21296],{"xmlns":250},[252,21297,21298,21306],{},[255,21299,21300],{},[337,21301,21302,21304],{},[258,21303,4429],{},[258,21305,2297],{},[262,21307,6581],{"encoding":264},[238,21309,21311],{"className":21310,"ariaHidden":269},[268],[238,21312,21314,21317],{"className":21313},[273],[238,21315],{"className":21316,"style":278},[277],[238,21318,21320,21323],{"className":21319},[282],[238,21321,4429],{"className":21322,"style":660},[282,283],[238,21324,21326],{"className":21325},[391],[238,21327,21329],{"className":21328},[395],[238,21330,21332],{"className":21331},[399],[238,21333,21335],{"className":21334,"style":6609},[403],[238,21336,21337,21340],{"style":407},[238,21338],{"className":21339,"style":412},[411],[238,21341,21343],{"className":21342},[416,417,418,419],[238,21344,2297],{"className":21345,"style":2313},[282,283,419],"。故 ",[238,21348,21350,21371],{"className":21349,"translate":242},[241],[238,21351,21353],{"className":21352},[246],[248,21354,21355],{"xmlns":250},[252,21356,21357,21369],{},[255,21358,21359,21361,21363],{},[258,21360,4429],{},[332,21362,992],{},[337,21364,21365,21367],{},[258,21366,4429],{},[258,21368,2297],{},[262,21370,20131],{"encoding":264},[238,21372,21374,21392],{"className":21373,"ariaHidden":269},[268],[238,21375,21377,21380,21383,21386,21389],{"className":21376},[273],[238,21378],{"className":21379,"style":278},[277],[238,21381,4429],{"className":21382,"style":660},[282,283],[238,21384],{"className":21385,"style":430},[429],[238,21387,992],{"className":21388},[434],[238,21390],{"className":21391,"style":430},[429],[238,21393,21395,21398],{"className":21394},[273],[238,21396],{"className":21397,"style":278},[277],[238,21399,21401,21404],{"className":21400},[282],[238,21402,4429],{"className":21403,"style":660},[282,283],[238,21405,21407],{"className":21406},[391],[238,21408,21410],{"className":21409},[395],[238,21411,21413],{"className":21412},[399],[238,21414,21416],{"className":21415,"style":6609},[403],[238,21417,21418,21421],{"style":407},[238,21419],{"className":21420,"style":412},[411],[238,21422,21424],{"className":21423},[416,417,418,419],[238,21425,2297],{"className":21426,"style":2313},[282,283,419],"。证毕。",[11,21429,21430,21431,21489,21490,21494,21495,21545],{},"这个证明不依赖矩阵的有限维结构，只用到 ",[238,21432,21434,21451],{"className":21433,"translate":242},[241],[238,21435,21437],{"className":21436},[246],[248,21438,21439],{"xmlns":250},[252,21440,21441,21449],{},[255,21442,21443],{},[337,21444,21445,21447],{},[258,21446,330],{},[258,21448,2297],{},[262,21450,15505],{"encoding":264},[238,21452,21454],{"className":21453,"ariaHidden":269},[268],[238,21455,21457,21460],{"className":21456},[273],[238,21458],{"className":21459,"style":278},[277],[238,21461,21463,21466],{"className":21462},[282],[238,21464,330],{"className":21465,"style":377},[282,283],[238,21467,21469],{"className":21468},[391],[238,21470,21472],{"className":21471},[395],[238,21473,21475],{"className":21474},[399],[238,21476,21478],{"className":21477,"style":6609},[403],[238,21479,21480,21483],{"style":407},[238,21481],{"className":21482,"style":412},[411],[238,21484,21486],{"className":21485},[416,417,418,419],[238,21487,2297],{"className":21488,"style":2313},[282,283,419]," 是随机核这一事实。因此它可以直接推广到一般状态空间。",[357,21491,21493],{"href":21492},"#proposition-policy-verification","命题 2.3"," 也说明了贝尔曼方程的一个常见误读：它并不是「用未来价值近似当前价值」，而是在精确地刻画值函数。只要 ",[238,21496,21498,21515],{"className":21497,"translate":242},[241],[238,21499,21501],{"className":21500},[246],[248,21502,21503],{"xmlns":250},[252,21504,21505,21513],{},[255,21506,21507,21509,21511],{},[258,21508,730],{},[332,21510,1892],{},[738,21512,745],{},[262,21514,17495],{"encoding":264},[238,21516,21518,21536],{"className":21517,"ariaHidden":269},[268],[238,21519,21521,21524,21527,21530,21533],{"className":21520},[273],[238,21522],{"className":21523,"style":760},[277],[238,21525,730],{"className":21526,"style":764},[282,283],[238,21528],{"className":21529,"style":430},[429],[238,21531,1892],{"className":21532},[434],[238,21534],{"className":21535,"style":430},[429],[238,21537,21539,21542],{"className":21538},[273],[238,21540],{"className":21541,"style":2672},[277],[238,21543,745],{"className":21544},[282]," 且奖励有界，满足方程的函数就是值函数本身。",[220,21547,21549],{"id":21548},"三贝尔曼算子与压缩映射","三、贝尔曼算子与压缩映射",[225,21551,21553],{"id":21552},"_31-有界函数空间","3.1 有界函数空间",[11,21555,21556,21557,21601,21602,21630],{},"记 ",[238,21558,21560,21580],{"className":21559,"translate":242},[241],[238,21561,21563],{"className":21562},[246],[248,21564,21565],{"xmlns":250},[252,21566,21567,21577],{},[255,21568,21569,21571,21573,21575],{},[258,21570,19971],{},[332,21572,335],{"stretchy":334},[258,21574,260],{},[332,21576,360],{"stretchy":334},[262,21578,21579],{"encoding":264},"B(S)",[238,21581,21583],{"className":21582,"ariaHidden":269},[268],[238,21584,21586,21589,21592,21595,21598],{"className":21585},[273],[238,21587],{"className":21588,"style":444},[277],[238,21590,19971],{"className":21591,"style":20011},[282,283],[238,21593,335],{"className":21594},[381],[238,21596,260],{"className":21597,"style":284},[282,283],[238,21599,360],{"className":21600},[462]," 为 ",[238,21603,21605,21618],{"className":21604,"translate":242},[241],[238,21606,21608],{"className":21607},[246],[248,21609,21610],{"xmlns":250},[252,21611,21612,21616],{},[255,21613,21614],{},[258,21615,260],{},[262,21617,260],{"encoding":264},[238,21619,21621],{"className":21620,"ariaHidden":269},[268],[238,21622,21624,21627],{"className":21623},[273],[238,21625],{"className":21626,"style":278},[277],[238,21628,260],{"className":21629,"style":284},[282,283]," 上全体有界实值函数构成的空间，并赋予 sup 范数",[238,21632,21634],{"className":21633,"translate":242},[945],[238,21635,21637,21692],{"className":21636,"translate":242},[241],[238,21638,21640],{"className":21639},[246],[248,21641,21642],{"xmlns":250,"display":954},[252,21643,21644,21689],{},[255,21645,21646,21648,21650,21656,21658,21675,21677,21679,21681,21683,21685,21687],{},[258,21647,15588],{"mathvariant":344},[258,21649,4429],{},[812,21651,21652,21654],{},[258,21653,15588],{"mathvariant":344},[258,21655,1895],{"mathvariant":344},[332,21657,992],{},[6755,21659,21660,21667],{},[255,21661,21662,21665],{},[258,21663,21664],{},"sup",[332,21666,1889],{},[255,21668,21669,21671,21673],{},[258,21670,341],{},[332,21672,733],{},[258,21674,260],{},[258,21676,349],{"mathvariant":344},[258,21678,4429],{},[332,21680,335],{"stretchy":334},[258,21682,341],{},[332,21684,360],{"stretchy":334},[258,21686,349],{"mathvariant":344},[258,21688,3102],{"mathvariant":344},[262,21690,21691],{"encoding":264},"\\|V\\|_\\infty=\\sup_{s\\in S}|V(s)|.",[238,21693,21695,21756],{"className":21694,"ariaHidden":269},[268],[238,21696,21698,21701,21704,21707,21747,21750,21753],{"className":21697},[273],[238,21699],{"className":21700,"style":444},[277],[238,21702,15588],{"className":21703},[282],[238,21705,4429],{"className":21706,"style":660},[282,283],[238,21708,21710,21713],{"className":21709},[282],[238,21711,15588],{"className":21712},[282],[238,21714,21716],{"className":21715},[391],[238,21717,21719,21739],{"className":21718},[395,848],[238,21720,21722,21736],{"className":21721},[399],[238,21723,21725],{"className":21724,"style":1927},[403],[238,21726,21727,21730],{"style":1165},[238,21728],{"className":21729,"style":412},[411],[238,21731,21733],{"className":21732},[416,417,418,419],[238,21734,1895],{"className":21735},[282,419],[238,21737,872],{"className":21738},[871],[238,21740,21742],{"className":21741},[399],[238,21743,21745],{"className":21744,"style":879},[403],[238,21746],{},[238,21748],{"className":21749,"style":430},[429],[238,21751,992],{"className":21752},[434],[238,21754],{"className":21755,"style":430},[429],[238,21757,21759,21763,21819,21822,21825,21828,21831,21834,21837],{"className":21758},[273],[238,21760],{"className":21761,"style":21762},[277],"height:1.7161em;vertical-align:-0.9661em;",[238,21764,21766],{"className":21765},[1943,3493],[238,21767,21769,21810],{"className":21768},[395,848],[238,21770,21772,21807],{"className":21771},[399],[238,21773,21775,21796],{"className":21774,"style":489},[403],[238,21776,21778,21781],{"style":21777},"top:-2.1612em;margin-left:0em;",[238,21779],{"className":21780,"style":1134},[411],[238,21782,21784],{"className":21783},[416,417,418,419],[238,21785,21787,21790,21793],{"className":21786},[282,419],[238,21788,341],{"className":21789},[282,283,419],[238,21791,733],{"className":21792},[434,419],[238,21794,260],{"className":21795,"style":284},[282,283,419],[238,21797,21799,21802],{"style":21798},"top:-3em;",[238,21800],{"className":21801,"style":1134},[411],[238,21803,21804],{},[238,21805,21664],{"className":21806},[1943],[238,21808,872],{"className":21809},[871],[238,21811,21813],{"className":21812},[399],[238,21814,21817],{"className":21815,"style":21816},[403],"height:0.9661em;",[238,21818],{},[238,21820],{"className":21821,"style":455},[429],[238,21823,349],{"className":21824},[282],[238,21826,4429],{"className":21827,"style":660},[282,283],[238,21829,335],{"className":21830},[381],[238,21832,341],{"className":21833},[282,283],[238,21835,360],{"className":21836},[462],[238,21838,21840],{"className":21839},[282],"∣.",[11,21842,13131,21843,21871,21872,22002],{},[238,21844,21846,21859],{"className":21845,"translate":242},[241],[238,21847,21849],{"className":21848},[246],[248,21850,21851],{"xmlns":250},[252,21852,21853,21857],{},[255,21854,21855],{},[258,21856,260],{},[262,21858,260],{"encoding":264},[238,21860,21862],{"className":21861,"ariaHidden":269},[268],[238,21863,21865,21868],{"className":21864},[273],[238,21866],{"className":21867,"style":278},[277],[238,21869,260],{"className":21870,"style":284},[282,283]," 为有限集时，上确界就是最大值。",[238,21873,21875,21911],{"className":21874,"translate":242},[241],[238,21876,21878],{"className":21877},[246],[248,21879,21880],{"xmlns":250},[252,21881,21882,21908],{},[255,21883,21884,21886,21888,21890,21892,21894,21896,21898,21900,21906],{},[332,21885,335],{"stretchy":334},[258,21887,19971],{},[332,21889,335],{"stretchy":334},[258,21891,260],{},[332,21893,360],{"stretchy":334},[332,21895,354],{"separator":269},[258,21897,15588],{"mathvariant":344},[332,21899,9084],{},[812,21901,21902,21904],{},[258,21903,15588],{"mathvariant":344},[258,21905,1895],{"mathvariant":344},[332,21907,360],{"stretchy":334},[262,21909,21910],{"encoding":264},"(B(S),\\|\\cdot\\|_\\infty)",[238,21912,21914,21953],{"className":21913,"ariaHidden":269},[268],[238,21915,21917,21920,21923,21926,21929,21932,21935,21938,21941,21944,21947,21950],{"className":21916},[273],[238,21918],{"className":21919,"style":444},[277],[238,21921,335],{"className":21922},[381],[238,21924,19971],{"className":21925,"style":20011},[282,283],[238,21927,335],{"className":21928},[381],[238,21930,260],{"className":21931,"style":284},[282,283],[238,21933,360],{"className":21934},[462],[238,21936,354],{"className":21937},[451],[238,21939],{"className":21940,"style":455},[429],[238,21942,15588],{"className":21943},[282],[238,21945],{"className":21946,"style":660},[429],[238,21948,9084],{"className":21949},[664],[238,21951],{"className":21952,"style":660},[429],[238,21954,21956,21959,21999],{"className":21955},[273],[238,21957],{"className":21958,"style":444},[277],[238,21960,21962,21965],{"className":21961},[282],[238,21963,15588],{"className":21964},[282],[238,21966,21968],{"className":21967},[391],[238,21969,21971,21991],{"className":21970},[395,848],[238,21972,21974,21988],{"className":21973},[399],[238,21975,21977],{"className":21976,"style":1927},[403],[238,21978,21979,21982],{"style":1165},[238,21980],{"className":21981,"style":412},[411],[238,21983,21985],{"className":21984},[416,417,418,419],[238,21986,1895],{"className":21987},[282,419],[238,21989,872],{"className":21990},[871],[238,21992,21994],{"className":21993},[399],[238,21995,21997],{"className":21996,"style":879},[403],[238,21998],{},[238,22000,360],{"className":22001},[462]," 是完备的 Banach 空间。这个完备性是 Banach 不动点定理能够使用的关键。",[11,22004,22005,22006,22064,22065,22126,22127,22170,22171,22229,22230,22288],{},"贝尔曼期望算子 ",[238,22007,22009,22026],{"className":22008,"translate":242},[241],[238,22010,22012],{"className":22011},[246],[248,22013,22014],{"xmlns":250},[252,22015,22016,22024],{},[255,22017,22018],{},[337,22019,22020,22022],{},[258,22021,11971],{"mathvariant":816},[258,22023,2297],{},[262,22025,11976],{"encoding":264},[238,22027,22029],{"className":22028,"ariaHidden":269},[268],[238,22030,22032,22035],{"className":22031},[273],[238,22033],{"className":22034,"style":278},[277],[238,22036,22038,22041],{"className":22037},[282],[238,22039,11971],{"className":22040,"style":11992},[282,840],[238,22042,22044],{"className":22043},[391],[238,22045,22047],{"className":22046},[395],[238,22048,22050],{"className":22049},[399],[238,22051,22053],{"className":22052,"style":6609},[403],[238,22054,22055,22058],{"style":407},[238,22056],{"className":22057,"style":412},[411],[238,22059,22061],{"className":22060},[416,417,418,419],[238,22062,2297],{"className":22063,"style":2313},[282,283,419]," 与贝尔曼最优算子 ",[238,22066,22068,22087],{"className":22067,"translate":242},[241],[238,22069,22071],{"className":22070},[246],[248,22072,22073],{"xmlns":250},[252,22074,22075,22084],{},[255,22076,22077],{},[337,22078,22079,22081],{},[258,22080,11971],{"mathvariant":816},[332,22082,22083],{},"∗",[262,22085,22086],{"encoding":264},"\\mathcal{T}^*",[238,22088,22090],{"className":22089,"ariaHidden":269},[268],[238,22091,22093,22097],{"className":22092},[273],[238,22094],{"className":22095,"style":22096},[277],"height:0.6887em;",[238,22098,22100,22103],{"className":22099},[282],[238,22101,11971],{"className":22102,"style":11992},[282,840],[238,22104,22106],{"className":22105},[391],[238,22107,22109],{"className":22108},[395],[238,22110,22112],{"className":22111},[399],[238,22113,22115],{"className":22114,"style":22096},[403],[238,22116,22117,22120],{"style":407},[238,22118],{"className":22119,"style":412},[411],[238,22121,22123],{"className":22122},[416,417,418,419],[238,22124,22083],{"className":22125},[664,419]," 都将 ",[238,22128,22130,22149],{"className":22129,"translate":242},[241],[238,22131,22133],{"className":22132},[246],[248,22134,22135],{"xmlns":250},[252,22136,22137,22147],{},[255,22138,22139,22141,22143,22145],{},[258,22140,19971],{},[332,22142,335],{"stretchy":334},[258,22144,260],{},[332,22146,360],{"stretchy":334},[262,22148,21579],{"encoding":264},[238,22150,22152],{"className":22151,"ariaHidden":269},[268],[238,22153,22155,22158,22161,22164,22167],{"className":22154},[273],[238,22156],{"className":22157,"style":444},[277],[238,22159,19971],{"className":22160,"style":20011},[282,283],[238,22162,335],{"className":22163},[381],[238,22165,260],{"className":22166,"style":284},[282,283],[238,22168,360],{"className":22169},[462]," 映射到自身。",[238,22172,22174,22191],{"className":22173,"translate":242},[241],[238,22175,22177],{"className":22176},[246],[248,22178,22179],{"xmlns":250},[252,22180,22181,22189],{},[255,22182,22183],{},[337,22184,22185,22187],{},[258,22186,11971],{"mathvariant":816},[258,22188,2297],{},[262,22190,11976],{"encoding":264},[238,22192,22194],{"className":22193,"ariaHidden":269},[268],[238,22195,22197,22200],{"className":22196},[273],[238,22198],{"className":22199,"style":278},[277],[238,22201,22203,22206],{"className":22202},[282],[238,22204,11971],{"className":22205,"style":11992},[282,840],[238,22207,22209],{"className":22208},[391],[238,22210,22212],{"className":22211},[395],[238,22213,22215],{"className":22214},[399],[238,22216,22218],{"className":22217,"style":6609},[403],[238,22219,22220,22223],{"style":407},[238,22221],{"className":22222,"style":412},[411],[238,22224,22226],{"className":22225},[416,417,418,419],[238,22227,2297],{"className":22228,"style":2313},[282,283,419]," 的定义已在 2.2 节给出，",[238,22231,22233,22250],{"className":22232,"translate":242},[241],[238,22234,22236],{"className":22235},[246],[248,22237,22238],{"xmlns":250},[252,22239,22240,22248],{},[255,22241,22242],{},[337,22243,22244,22246],{},[258,22245,11971],{"mathvariant":816},[332,22247,22083],{},[262,22249,22086],{"encoding":264},[238,22251,22253],{"className":22252,"ariaHidden":269},[268],[238,22254,22256,22259],{"className":22255},[273],[238,22257],{"className":22258,"style":22096},[277],[238,22260,22262,22265],{"className":22261},[282],[238,22263,11971],{"className":22264,"style":11992},[282,840],[238,22266,22268],{"className":22267},[391],[238,22269,22271],{"className":22270},[395],[238,22272,22274],{"className":22273},[399],[238,22275,22277],{"className":22276,"style":22096},[403],[238,22278,22279,22282],{"style":407},[238,22280],{"className":22281,"style":412},[411],[238,22283,22285],{"className":22284},[416,417,418,419],[238,22286,22083],{"className":22287},[664,419]," 定义为",[238,22290,22292],{"className":22291,"translate":242},[945],[238,22293,22295,22411],{"className":22294,"translate":242},[241],[238,22296,22298],{"className":22297},[246],[248,22299,22300],{"xmlns":250,"display":954},[252,22301,22302,22408],{},[255,22303,22304,22306,22312,22314,22316,22318,22320,22322,22324,22334,22344,22346,22348,22354,22356,22358,22360,22362,22364,22406],{},[332,22305,335],{"stretchy":334},[337,22307,22308,22310],{},[258,22309,11971],{"mathvariant":816},[332,22311,22083],{},[258,22313,4429],{},[332,22315,360],{"stretchy":334},[332,22317,335],{"stretchy":334},[258,22319,341],{},[332,22321,360],{"stretchy":334},[332,22323,992],{},[6755,22325,22326,22332],{},[255,22327,22328,22330],{},[258,22329,1886],{},[332,22331,1889],{},[258,22333,357],{},[6755,22335,22336,22338],{},[332,22337,3390],{},[337,22339,22340,22342],{},[258,22341,341],{},[332,22343,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,22345,330],{},[332,22347,335],{"stretchy":334},[337,22349,22350,22352],{},[258,22351,341],{},[332,22353,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,22355,349],{},[258,22357,341],{},[332,22359,354],{"separator":269},[258,22361,357],{},[332,22363,360],{"stretchy":334},[255,22365,22366,22368,22370,22372,22374,22376,22378,22380,22386,22388,22390,22392,22394,22396,22402,22404],{},[332,22367,736],{"fence":269},[258,22369,600],{},[332,22371,335],{"stretchy":334},[258,22373,341],{},[332,22375,354],{"separator":269},[258,22377,357],{},[332,22379,354],{"separator":269},[337,22381,22382,22384],{},[258,22383,341],{},[332,22385,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,22387,360],{"stretchy":334},[332,22389,987],{},[258,22391,730],{},[258,22393,4429],{},[332,22395,335],{"stretchy":334},[337,22397,22398,22400],{},[258,22399,341],{},[332,22401,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,22403,360],{"stretchy":334},[332,22405,4479],{"fence":269},[258,22407,3102],{"mathvariant":344},[262,22409,22410],{"encoding":264},"(\\mathcal{T}^*V)(s)\n=\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')\\right].",[238,22412,22414,22477,22662],{"className":22413,"ariaHidden":269},[268],[238,22415,22417,22420,22423,22453,22456,22459,22462,22465,22468,22471,22474],{"className":22416},[273],[238,22418],{"className":22419,"style":444},[277],[238,22421,335],{"className":22422},[381],[238,22424,22426,22429],{"className":22425},[282],[238,22427,11971],{"className":22428,"style":11992},[282,840],[238,22430,22432],{"className":22431},[391],[238,22433,22435],{"className":22434},[395],[238,22436,22438],{"className":22437},[399],[238,22439,22442],{"className":22440,"style":22441},[403],"height:0.7387em;",[238,22443,22444,22447],{"style":1226},[238,22445],{"className":22446,"style":412},[411],[238,22448,22450],{"className":22449},[416,417,418,419],[238,22451,22083],{"className":22452},[664,419],[238,22454,4429],{"className":22455,"style":660},[282,283],[238,22457,360],{"className":22458},[462],[238,22460,335],{"className":22461},[381],[238,22463,341],{"className":22464},[282,283],[238,22466,360],{"className":22467},[462],[238,22469],{"className":22470,"style":430},[429],[238,22472,992],{"className":22473},[434],[238,22475],{"className":22476,"style":430},[429],[238,22478,22480,22484,22533,22536,22612,22615,22618,22621,22653,22656,22659],{"className":22479},[273],[238,22481],{"className":22482,"style":22483},[277],"height:2.344em;vertical-align:-1.294em;",[238,22485,22487],{"className":22486},[1943,3493],[238,22488,22490,22524],{"className":22489},[395,848],[238,22491,22493,22521],{"className":22492},[399],[238,22494,22496,22511],{"className":22495,"style":489},[403],[238,22497,22499,22502],{"style":22498},"top:-2.4em;margin-left:0em;",[238,22500],{"className":22501,"style":1134},[411],[238,22503,22505],{"className":22504},[416,417,418,419],[238,22506,22508],{"className":22507},[282,419],[238,22509,357],{"className":22510},[282,283,419],[238,22512,22513,22516],{"style":21798},[238,22514],{"className":22515,"style":1134},[411],[238,22517,22518],{},[238,22519,1886],{"className":22520},[1943],[238,22522,872],{"className":22523},[871],[238,22525,22527],{"className":22526},[399],[238,22528,22531],{"className":22529,"style":22530},[403],"height:0.7em;",[238,22532],{},[238,22534],{"className":22535,"style":455},[429],[238,22537,22539],{"className":22538},[1943,3493],[238,22540,22542,22604],{"className":22541},[395,848],[238,22543,22545,22601],{"className":22544},[399],[238,22546,22548,22591],{"className":22547,"style":6869},[403],[238,22549,22550,22553],{"style":7296},[238,22551],{"className":22552,"style":3510},[411],[238,22554,22556],{"className":22555},[416,417,418,419],[238,22557,22559],{"className":22558},[282,419],[238,22560,22562,22565],{"className":22561},[282,419],[238,22563,341],{"className":22564},[282,283,419],[238,22566,22568],{"className":22567},[391],[238,22569,22571],{"className":22570},[395],[238,22572,22574],{"className":22573},[399],[238,22575,22577],{"className":22576,"style":4196},[403],[238,22578,22579,22582],{"style":4199},[238,22580],{"className":22581,"style":4203},[411],[238,22583,22585],{"className":22584},[416,4207,4208,419],[238,22586,22588],{"className":22587},[282,419],[238,22589,346],{"className":22590},[282,419],[238,22592,22593,22596],{"style":3529},[238,22594],{"className":22595,"style":3510},[411],[238,22597,22598],{},[238,22599,3390],{"className":22600},[1943,3538,3539],[238,22602,872],{"className":22603},[871],[238,22605,22607],{"className":22606},[399],[238,22608,22610],{"className":22609,"style":7357},[403],[238,22611],{},[238,22613],{"className":22614,"style":455},[429],[238,22616,330],{"className":22617,"style":377},[282,283],[238,22619,335],{"className":22620},[381],[238,22622,22624,22627],{"className":22623},[282],[238,22625,341],{"className":22626},[282,283],[238,22628,22630],{"className":22629},[391],[238,22631,22633],{"className":22632},[395],[238,22634,22636],{"className":22635},[399],[238,22637,22639],{"className":22638,"style":1223},[403],[238,22640,22641,22644],{"style":1226},[238,22642],{"className":22643,"style":412},[411],[238,22645,22647],{"className":22646},[416,417,418,419],[238,22648,22650],{"className":22649},[282,419],[238,22651,346],{"className":22652},[282,419],[238,22654],{"className":22655,"style":430},[429],[238,22657,349],{"className":22658},[434],[238,22660],{"className":22661,"style":430},[429],[238,22663,22665,22669,22672,22675,22678,22681,22684,22687,22808,22811],{"className":22664},[273],[238,22666],{"className":22667,"style":22668},[277],"height:1.0519em;vertical-align:-0.25em;",[238,22670,341],{"className":22671},[282,283],[238,22673,354],{"className":22674},[451],[238,22676],{"className":22677,"style":455},[429],[238,22679,357],{"className":22680},[282,283],[238,22682,360],{"className":22683},[462],[238,22685],{"className":22686,"style":455},[429],[238,22688,22690,22693,22696,22699,22702,22705,22708,22711,22714,22717,22749,22752,22755,22758,22761,22764,22767,22770,22802,22805],{"className":22689},[3041],[238,22691,736],{"className":22692,"style":4791},[381,4790],[238,22694,600],{"className":22695,"style":637},[282,283],[238,22697,335],{"className":22698},[381],[238,22700,341],{"className":22701},[282,283],[238,22703,354],{"className":22704},[451],[238,22706],{"className":22707,"style":455},[429],[238,22709,357],{"className":22710},[282,283],[238,22712,354],{"className":22713},[451],[238,22715],{"className":22716,"style":455},[429],[238,22718,22720,22723],{"className":22719},[282],[238,22721,341],{"className":22722},[282,283],[238,22724,22726],{"className":22725},[391],[238,22727,22729],{"className":22728},[395],[238,22730,22732],{"className":22731},[399],[238,22733,22735],{"className":22734,"style":1223},[403],[238,22736,22737,22740],{"style":1226},[238,22738],{"className":22739,"style":412},[411],[238,22741,22743],{"className":22742},[416,417,418,419],[238,22744,22746],{"className":22745},[282,419],[238,22747,346],{"className":22748},[282,419],[238,22750,360],{"className":22751},[462],[238,22753],{"className":22754,"style":660},[429],[238,22756,987],{"className":22757},[664],[238,22759],{"className":22760,"style":660},[429],[238,22762,730],{"className":22763,"style":764},[282,283],[238,22765,4429],{"className":22766,"style":660},[282,283],[238,22768,335],{"className":22769},[381],[238,22771,22773,22776],{"className":22772},[282],[238,22774,341],{"className":22775},[282,283],[238,22777,22779],{"className":22778},[391],[238,22780,22782],{"className":22781},[395],[238,22783,22785],{"className":22784},[399],[238,22786,22788],{"className":22787,"style":1223},[403],[238,22789,22790,22793],{"style":1226},[238,22791],{"className":22792,"style":412},[411],[238,22794,22796],{"className":22795},[416,417,418,419],[238,22797,22799],{"className":22798},[282,419],[238,22800,346],{"className":22801},[282,419],[238,22803,360],{"className":22804},[462],[238,22806,4479],{"className":22807,"style":4791},[462,4790],[238,22809],{"className":22810,"style":455},[429],[238,22812,3102],{"className":22813},[282],[11,22815,22816,22817,22875],{},"以 ",[238,22818,22820,22837],{"className":22819,"translate":242},[241],[238,22821,22823],{"className":22822},[246],[248,22824,22825],{"xmlns":250},[252,22826,22827,22835],{},[255,22828,22829],{},[337,22830,22831,22833],{},[258,22832,11971],{"mathvariant":816},[258,22834,2297],{},[262,22836,11976],{"encoding":264},[238,22838,22840],{"className":22839,"ariaHidden":269},[268],[238,22841,22843,22846],{"className":22842},[273],[238,22844],{"className":22845,"style":278},[277],[238,22847,22849,22852],{"className":22848},[282],[238,22850,11971],{"className":22851,"style":11992},[282,840],[238,22853,22855],{"className":22854},[391],[238,22856,22858],{"className":22857},[395],[238,22859,22861],{"className":22860},[399],[238,22862,22864],{"className":22863,"style":6609},[403],[238,22865,22866,22869],{"style":407},[238,22867],{"className":22868,"style":412},[411],[238,22870,22872],{"className":22871},[416,417,418,419],[238,22873,2297],{"className":22874,"style":2313},[282,283,419]," 为例，",[238,22877,22879],{"className":22878,"translate":242},[945],[238,22880,22882,23052],{"className":22881,"translate":242},[241],[238,22883,22885],{"className":22884},[246],[248,22886,22887],{"xmlns":250,"display":954},[252,22888,22889,23049],{},[958,22890,22891,23007],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,22892,22893,22921],{},[966,22894,22895],{},[969,22896,22897],{"scriptlevel":740,"displaystyle":269},[255,22898,22899,22901,22903,22909,22911,22913,22915,22917,22919],{},[258,22900,349],{"mathvariant":344},[332,22902,335],{"stretchy":334},[337,22904,22905,22907],{},[258,22906,11971],{"mathvariant":816},[258,22908,2297],{},[258,22910,4429],{},[332,22912,360],{"stretchy":334},[332,22914,335],{"stretchy":334},[258,22916,341],{},[332,22918,360],{"stretchy":334},[258,22920,349],{"mathvariant":344},[966,22922,22923],{},[969,22924,22925],{"scriptlevel":740,"displaystyle":269},[255,22926,22927,22929,22931,22937,22939,22941,22943,22945,22947,22949,22959,22961,22963,22969,22971,22973,22975,22977,22979],{},[255,22928],{},[332,22930,2023],{},[6755,22932,22933,22935],{},[332,22934,3390],{},[258,22936,357],{},[258,22938,2297],{},[332,22940,335],{"stretchy":334},[258,22942,357],{},[332,22944,349],{},[258,22946,341],{},[332,22948,360],{"stretchy":334},[6755,22950,22951,22953],{},[332,22952,3390],{},[337,22954,22955,22957],{},[258,22956,341],{},[332,22958,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,22960,330],{},[332,22962,335],{"stretchy":334},[337,22964,22965,22967],{},[258,22966,341],{},[332,22968,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,22970,349],{},[258,22972,341],{},[332,22974,354],{"separator":269},[258,22976,357],{},[332,22978,360],{"stretchy":334},[255,22980,22981,22983,22991,22993,22995,22997,22999,23005],{},[332,22982,335],{"fence":269},[812,22984,22985,22987,22989],{},[258,22986,621],{},[258,22988,1886],{},[332,22990,1889],{},[332,22992,987],{},[258,22994,730],{},[258,22996,15588],{"mathvariant":344},[258,22998,4429],{},[812,23000,23001,23003],{},[258,23002,15588],{"mathvariant":344},[258,23004,1895],{"mathvariant":344},[332,23006,360],{"fence":269},[963,23008,23009,23015],{},[966,23010,23011],{},[969,23012,23013],{"scriptlevel":740,"displaystyle":269},[255,23014],{},[966,23016,23017],{},[969,23018,23019],{"scriptlevel":740,"displaystyle":269},[255,23020,23021,23023,23025,23033,23035,23037,23039,23041,23047],{},[255,23022],{},[332,23024,992],{},[812,23026,23027,23029,23031],{},[258,23028,621],{},[258,23030,1886],{},[332,23032,1889],{},[332,23034,987],{},[258,23036,730],{},[258,23038,15588],{"mathvariant":344},[258,23040,4429],{},[812,23042,23043,23045],{},[258,23044,15588],{"mathvariant":344},[258,23046,1895],{"mathvariant":344},[332,23048,354],{"separator":269},[262,23050,23051],{"encoding":264},"\\begin{aligned}\n|(\\mathcal{T}^\\pi V)(s)|\n&\\le\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left(R_{\\max}+\\gamma\\|V\\|_\\infty\\right)\\\\\n&=R_{\\max}+\\gamma\\|V\\|_\\infty,\n\\end{aligned}",[238,23053,23055],{"className":23054,"ariaHidden":269},[268],[238,23056,23058,23061],{"className":23057},[273],[238,23059],{"className":23060,"style":7153},[277],[238,23062,23064],{"className":23063},[282],[238,23065,23067,23159],{"className":23066},[958],[238,23068,23070],{"className":23069},[1117],[238,23071,23073,23151],{"className":23072},[395,848],[238,23074,23076,23148],{"className":23075},[399],[238,23077,23079,23140],{"className":23078,"style":7172},[403],[238,23080,23081,23084],{"style":7175},[238,23082],{"className":23083,"style":3510},[411],[238,23085,23087,23090,23093,23122,23125,23128,23131,23134,23137],{"className":23086},[282],[238,23088,349],{"className":23089},[282],[238,23091,335],{"className":23092},[381],[238,23094,23096,23099],{"className":23095},[282],[238,23097,11971],{"className":23098,"style":11992},[282,840],[238,23100,23102],{"className":23101},[391],[238,23103,23105],{"className":23104},[395],[238,23106,23108],{"className":23107},[399],[238,23109,23111],{"className":23110,"style":4620},[403],[238,23112,23113,23116],{"style":1226},[238,23114],{"className":23115,"style":412},[411],[238,23117,23119],{"className":23118},[416,417,418,419],[238,23120,2297],{"className":23121,"style":2313},[282,283,419],[238,23123,4429],{"className":23124,"style":660},[282,283],[238,23126,360],{"className":23127},[462],[238,23129,335],{"className":23130},[381],[238,23132,341],{"className":23133},[282,283],[238,23135,360],{"className":23136},[462],[238,23138,349],{"className":23139},[282],[238,23141,23142,23145],{"style":7231},[238,23143],{"className":23144,"style":3510},[411],[238,23146],{"className":23147},[282],[238,23149,872],{"className":23150},[871],[238,23152,23154],{"className":23153},[399],[238,23155,23157],{"className":23156,"style":7247},[403],[238,23158],{},[238,23160,23162],{"className":23161},[1422],[238,23163,23165,23667],{"className":23164},[395,848],[238,23166,23168,23664],{"className":23167},[399],[238,23169,23171,23531],{"className":23170,"style":7172},[403],[238,23172,23173,23176],{"style":7175},[238,23174],{"className":23175,"style":3510},[411],[238,23177,23179,23182,23185,23188,23191,23238,23241,23244,23247,23250,23253,23256,23259,23262,23265,23268,23344,23347,23350,23353,23385,23388,23391,23394,23397,23400,23403,23406,23409,23412],{"className":23178},[282],[238,23180],{"className":23181},[282],[238,23183],{"className":23184,"style":430},[429],[238,23186,2023],{"className":23187},[434],[238,23189],{"className":23190,"style":430},[429],[238,23192,23194],{"className":23193},[1943,3493],[238,23195,23197,23230],{"className":23196},[395,848],[238,23198,23200,23227],{"className":23199},[399],[238,23201,23203,23217],{"className":23202,"style":6869},[403],[238,23204,23205,23208],{"style":6872},[238,23206],{"className":23207,"style":3510},[411],[238,23209,23211],{"className":23210},[416,417,418,419],[238,23212,23214],{"className":23213},[282,419],[238,23215,357],{"className":23216},[282,283,419],[238,23218,23219,23222],{"style":3529},[238,23220],{"className":23221,"style":3510},[411],[238,23223,23224],{},[238,23225,3390],{"className":23226},[1943,3538,3539],[238,23228,872],{"className":23229},[871],[238,23231,23233],{"className":23232},[399],[238,23234,23236],{"className":23235,"style":6904},[403],[238,23237],{},[238,23239],{"className":23240,"style":455},[429],[238,23242,2297],{"className":23243,"style":2313},[282,283],[238,23245,335],{"className":23246},[381],[238,23248,357],{"className":23249},[282,283],[238,23251],{"className":23252,"style":430},[429],[238,23254,349],{"className":23255},[434],[238,23257],{"className":23258,"style":430},[429],[238,23260,341],{"className":23261},[282,283],[238,23263,360],{"className":23264},[462],[238,23266],{"className":23267,"style":455},[429],[238,23269,23271],{"className":23270},[1943,3493],[238,23272,23274,23336],{"className":23273},[395,848],[238,23275,23277,23333],{"className":23276},[399],[238,23278,23280,23323],{"className":23279,"style":6869},[403],[238,23281,23282,23285],{"style":7296},[238,23283],{"className":23284,"style":3510},[411],[238,23286,23288],{"className":23287},[416,417,418,419],[238,23289,23291],{"className":23290},[282,419],[238,23292,23294,23297],{"className":23293},[282,419],[238,23295,341],{"className":23296},[282,283,419],[238,23298,23300],{"className":23299},[391],[238,23301,23303],{"className":23302},[395],[238,23304,23306],{"className":23305},[399],[238,23307,23309],{"className":23308,"style":4196},[403],[238,23310,23311,23314],{"style":4199},[238,23312],{"className":23313,"style":4203},[411],[238,23315,23317],{"className":23316},[416,4207,4208,419],[238,23318,23320],{"className":23319},[282,419],[238,23321,346],{"className":23322},[282,419],[238,23324,23325,23328],{"style":3529},[238,23326],{"className":23327,"style":3510},[411],[238,23329,23330],{},[238,23331,3390],{"className":23332},[1943,3538,3539],[238,23334,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V\\|_\\infty\u003C\\infty",[238,23713,23715,23805],{"className":23714,"ariaHidden":269},[268],[238,23716,23718,23721,23724,23753,23756,23796,23799,23802],{"className":23717},[273],[238,23719],{"className":23720,"style":444},[277],[238,23722,15588],{"className":23723},[282],[238,23725,23727,23730],{"className":23726},[282],[238,23728,11971],{"className":23729,"style":11992},[282,840],[238,23731,23733],{"className":23732},[391],[238,23734,23736],{"className":23735},[395],[238,23737,23739],{"className":23738},[399],[238,23740,23742],{"className":23741,"style":6609},[403],[238,23743,23744,23747],{"style":407},[238,23745],{"className":23746,"style":412},[411],[238,23748,23750],{"className":23749},[416,417,418,419],[238,23751,2297],{"className":23752,"style":2313},[282,283,419],[238,23754,4429],{"className":23755,"style":660},[282,283],[238,23757,23759,23762],{"className":23758},[282],[238,23760,15588],{"className":23761},[282],[238,23763,23765],{"className":23764},[391],[238,23766,23768,23788],{"className":23767},[395,848],[238,23769,23771,23785],{"className":23770},[399],[238,23772,23774],{"className":23773,"style":1927},[403],[238,23775,23776,23779],{"style":1165},[238,23777],{"className":23778,"style":412},[411],[238,23780,23782],{"className":23781},[416,417,418,419],[238,23783,1895],{"className":23784},[282,419],[238,23786,872],{"className":23787},[871],[238,23789,23791],{"className":23790},[399],[238,23792,23794],{"className":23793,"style":879},[403],[238,23795],{},[238,23797],{"className":23798,"style":430},[429],[238,23800,1892],{"className":23801},[434],[238,23803],{"className":23804,"style":430},[429],[238,23806,23808,23811],{"className":23807},[273],[238,23809],{"className":23810,"style":489},[277],[238,23812,1895],{"className":23813},[282],"。对 ",[238,23816,23818,23835],{"className":23817,"translate":242},[241],[238,23819,23821],{"className":23820},[246],[248,23822,23823],{"xmlns":250},[252,23824,23825,23833],{},[255,23826,23827],{},[337,23828,23829,23831],{},[258,23830,11971],{"mathvariant":816},[332,23832,22083],{},[262,23834,22086],{"encoding":264},[238,23836,23838],{"className":23837,"ariaHidden":269},[268],[238,23839,23841,23844],{"className":23840},[273],[238,23842],{"className":23843,"style":22096},[277],[238,23845,23847,23850],{"className":23846},[282],[238,23848,11971],{"className":23849,"style":11992},[282,840],[238,23851,23853],{"className":23852},[391],[238,23854,23856],{"className":23855},[395],[238,23857,23859],{"className":23858},[399],[238,23860,23862],{"className":23861,"style":22096},[403],[238,23863,23864,23867],{"style":407},[238,23865],{"className":23866,"style":412},[411],[238,23868,23870],{"className":23869},[416,417,418,419],[238,23871,22083],{"className":23872},[664,419]," 有同样的界。因此算子迭代不会离开这个空间。",[225,23875,23877],{"id":23876},"_32-单调性与常数平移","3.2 单调性与常数平移",[230,23879,23882,24018,24259,24315,24772],{"id":23880,"title":23881},"proposition-monotonicity","命题 3.1（单调性与常数平移）",[11,23883,236,23884,23964,23965,24017],{},[238,23885,23887,23916],{"className":23886,"translate":242},[241],[238,23888,23890],{"className":23889},[246],[248,23891,23892],{"xmlns":250},[252,23893,23894,23913],{},[255,23895,23896,23898,23900,23903,23905,23907,23909,23911],{},[258,23897,4429],{},[332,23899,354],{"separator":269},[258,23901,23902],{},"W",[332,23904,733],{},[258,23906,19971],{},[332,23908,335],{"stretchy":334},[258,23910,260],{},[332,23912,360],{"stretchy":334},[262,23914,23915],{"encoding":264},"V,W\\in B(S)",[238,23917,23919,23946],{"className":23918,"ariaHidden":269},[268],[238,23920,23922,23925,23928,23931,23934,23937,23940,23943],{"className":23921},[273],[238,23923],{"className":23924,"style":6651},[277],[238,23926,4429],{"className":23927,"style":660},[282,283],[238,23929,354],{"className":23930},[451],[238,23932],{"className":23933,"style":455},[429],[238,23935,23902],{"className":23936,"style":377},[282,283],[238,23938],{"className":23939,"style":430},[429],[238,23941,733],{"className":23942},[434],[238,23944],{"className":23945,"style":430},[429],[238,23947,23949,23952,23955,23958,23961],{"className":23948},[273],[238,23950],{"className":23951,"style":444},[277],[238,23953,19971],{"className":23954,"style":20011},[282,283],[238,23956,335],{"className":23957},[381],[238,23959,260],{"className":23960,"style":284},[282,283],[238,23962,360],{"className":23963},[462]," 且 ",[238,23966,23968,23986],{"className":23967,"translate":242},[241],[238,23969,23971],{"className":23970},[246],[248,23972,23973],{"xmlns":250},[252,23974,23975,23983],{},[255,23976,23977,23979,23981],{},[258,23978,4429],{},[332,23980,2023],{},[258,23982,23902],{},[262,23984,23985],{"encoding":264},"V\\le W",[238,23987,23989,24008],{"className":23988,"ariaHidden":269},[268],[238,23990,23992,23996,23999,24002,24005],{"className":23991},[273],[238,23993],{"className":23994,"style":23995},[277],"height:0.8193em;vertical-align:-0.136em;",[238,23997,4429],{"className":23998,"style":660},[282,283],[238,24000],{"className":24001,"style":430},[429],[238,24003,2023],{"className":24004},[434],[238,24006],{"className":24007,"style":430},[429],[238,24009,24011,24014],{"className":24010},[273],[238,24012],{"className":24013,"style":278},[277],[238,24015,23902],{"className":24016,"style":377},[282,283]," 逐点成立。则",[238,24019,24021],{"className":24020,"translate":242},[945],[238,24022,24024,24078],{"className":24023,"translate":242},[241],[238,24025,24027],{"className":24026},[246],[248,24028,24029],{"xmlns":250,"display":954},[252,24030,24031,24075],{},[255,24032,24033,24039,24041,24043,24049,24051,24053,24055,24061,24063,24065,24071,24073],{},[337,24034,24035,24037],{},[258,24036,11971],{"mathvariant":816},[258,24038,2297],{},[258,24040,4429],{},[332,24042,2023],{},[337,24044,24045,24047],{},[258,24046,11971],{"mathvariant":816},[258,24048,2297],{},[258,24050,23902],{},[332,24052,354],{"separator":269},[429,24054],{"width":10450},[337,24056,24057,24059],{},[258,24058,11971],{"mathvariant":816},[332,24060,22083],{},[258,24062,4429],{},[332,24064,2023],{},[337,24066,24067,24069],{},[258,24068,11971],{"mathvariant":816},[332,24070,22083],{},[258,24072,23902],{},[258,24074,3102],{"mathvariant":344},[262,24076,24077],{"encoding":264},"\\mathcal{T}^\\pi V\\le\\mathcal{T}^\\pi W,\n\\qquad\n\\mathcal{T}^* V\\le\\mathcal{T}^* W.",[238,24079,24081,24129,24218],{"className":24080,"ariaHidden":269},[268],[238,24082,24084,24088,24117,24120,24123,24126],{"className":24083},[273],[238,24085],{"className":24086,"style":24087},[277],"height:0.8504em;vertical-align:-0.136em;",[238,24089,24091,24094],{"className":24090},[282],[238,24092,11971],{"className":24093,"style":11992},[282,840],[238,24095,24097],{"className":24096},[391],[238,24098,24100],{"className":24099},[395],[238,24101,24103],{"className":24102},[399],[238,24104,24106],{"className":24105,"style":4620},[403],[238,24107,24108,24111],{"style":1226},[238,24109],{"className":24110,"style":412},[411],[238,24112,24114],{"className":24113},[416,417,418,419],[238,24115,2297],{"className":24116,"style":2313},[282,283,419],[238,24118,4429],{"className":24119,"style":660},[282,283],[238,24121],{"className":24122,"style":430},[429],[238,24124,2023],{"className":24125},[434],[238,24127],{"className":24128,"style":430},[429],[238,24130,24132,24136,24165,24168,24171,24174,24177,24206,24209,24212,24215],{"className":24131},[273],[238,24133],{"className":24134,"style":24135},[277],"height:0.9331em;vertical-align:-0.1944em;",[238,24137,24139,24142],{"className":24138},[282],[238,24140,11971],{"className":24141,"style":11992},[282,840],[238,24143,24145],{"className":24144},[391],[238,24146,24148],{"className":24147},[395],[238,24149,24151],{"className":24150},[399],[238,24152,24154],{"className":24153,"style":4620},[403],[238,24155,24156,24159],{"style":1226},[238,24157],{"className":24158,"style":412},[411],[238,24160,24162],{"className":24161},[416,417,418,419],[238,24163,2297],{"className":24164,"style":2313},[282,283,419],[238,24166,23902],{"className":24167,"style":377},[282,283],[238,24169,354],{"className":24170},[451],[238,24172],{"className":24173,"style":10986},[429],[238,24175],{"className":24176,"style":455},[429],[238,24178,24180,24183],{"className":24179},[282],[238,24181,11971],{"className":24182,"style":11992},[282,840],[238,24184,24186],{"className":24185},[391],[238,24187,24189],{"className":24188},[395],[238,24190,24192],{"className":24191},[399],[238,24193,24195],{"className":24194,"style":22441},[403],[238,24196,24197,24200],{"style":1226},[238,24198],{"className":24199,"style":412},[411],[238,24201,24203],{"className":24202},[416,417,418,419],[238,24204,22083],{"className":24205},[664,419],[238,24207,4429],{"className":24208,"style":660},[282,283],[238,24210],{"className":24211,"style":430},[429],[238,24213,2023],{"className":24214},[434],[238,24216],{"className":24217,"style":430},[429],[238,24219,24221,24224,24253,24256],{"className":24220},[273],[238,24222],{"className":24223,"style":22441},[277],[238,24225,24227,24230],{"className":24226},[282],[238,24228,11971],{"className":24229,"style":11992},[282,840],[238,24231,24233],{"className":24232},[391],[238,24234,24236],{"className":24235},[395],[238,24237,24239],{"className":24238},[399],[238,24240,24242],{"className":24241,"style":22441},[403],[238,24243,24244,24247],{"style":1226},[238,24245],{"className":24246,"style":412},[411],[238,24248,24250],{"className":24249},[416,417,418,419],[238,24251,22083],{"className":24252},[664,419],[238,24254,23902],{"className":24255,"style":377},[282,283],[238,24257,3102],{"className":24258},[282],[11,24260,24261,24262,941],{},"此外，对任意常数 ",[238,24263,24265,24284],{"className":24264,"translate":242},[241],[238,24266,24268],{"className":24267},[246],[248,24269,24270],{"xmlns":250},[252,24271,24272,24281],{},[255,24273,24274,24277,24279],{},[258,24275,24276],{},"c",[332,24278,733],{},[258,24280,621],{"mathvariant":620},[262,24282,24283],{"encoding":264},"c\\in\\mathbb{R}",[238,24285,24287,24306],{"className":24286,"ariaHidden":269},[268],[238,24288,24290,24294,24297,24300,24303],{"className":24289},[273],[238,24291],{"className":24292,"style":24293},[277],"height:0.5782em;vertical-align:-0.0391em;",[238,24295,24276],{"className":24296},[282,283],[238,24298],{"className":24299,"style":430},[429],[238,24301,733],{"className":24302},[434],[238,24304],{"className":24305,"style":430},[429],[238,24307,24309,24312],{"className":24308},[273],[238,24310],{"className":24311,"style":710},[277],[238,24313,621],{"className":24314},[282,714],[238,24316,24318],{"className":24317,"translate":242},[945],[238,24319,24321,24442],{"className":24320,"translate":242},[241],[238,24322,24324],{"className":24323},[246],[248,24325,24326],{"xmlns":250,"display":954},[252,24327,24328,24439],{},[958,24329,24330,24385],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,24331,24332,24357],{},[966,24333,24334],{},[969,24335,24336],{"scriptlevel":740,"displaystyle":269},[255,24337,24338,24344,24346,24348,24350,24352,24355],{},[337,24339,24340,24342],{},[258,24341,11971],{"mathvariant":816},[258,24343,2297],{},[332,24345,335],{"stretchy":334},[258,24347,4429],{},[332,24349,987],{},[258,24351,24276],{},[738,24353,745],{"mathvariant":24354},"bold",[332,24356,360],{"stretchy":334},[966,24358,24359],{},[969,24360,24361],{"scriptlevel":740,"displaystyle":269},[255,24362,24363,24365,24367,24373,24375,24377,24379,24381,24383],{},[255,24364],{},[332,24366,992],{},[337,24368,24369,24371],{},[258,24370,11971],{"mathvariant":816},[258,24372,2297],{},[258,24374,4429],{},[332,24376,987],{},[258,24378,730],{},[258,24380,24276],{},[738,24382,745],{"mathvariant":24354},[332,24384,354],{"separator":269},[963,24386,24387,24411],{},[966,24388,24389],{},[969,24390,24391],{"scriptlevel":740,"displaystyle":269},[255,24392,24393,24399,24401,24403,24405,24407,24409],{},[337,24394,24395,24397],{},[258,24396,11971],{"mathvariant":816},[332,24398,22083],{},[332,24400,335],{"stretchy":334},[258,24402,4429],{},[332,24404,987],{},[258,24406,24276],{},[738,24408,745],{"mathvariant":24354},[332,24410,360],{"stretchy":334},[966,24412,24413],{},[969,24414,24415],{"scriptlevel":740,"displaystyle":269},[255,24416,24417,24419,24421,24427,24429,24431,24433,24435,24437],{},[255,24418],{},[332,24420,992],{},[337,24422,24423,24425],{},[258,24424,11971],{"mathvariant":816},[332,24426,22083],{},[258,24428,4429],{},[332,24430,987],{},[258,24432,730],{},[258,24434,24276],{},[738,24436,745],{"mathvariant":24354},[332,24438,354],{"separator":269},[262,24440,24441],{"encoding":264},"\\begin{aligned}\n\\mathcal{T}^\\pi(V+c\\mathbf{1})\n&=\\mathcal{T}^\\pi V+\\gamma c\\mathbf{1},\\\\\n\\mathcal{T}^*(V+c\\mathbf{1})\n&=\\mathcal{T}^* V+\\gamma c\\mathbf{1},\n\\end{aligned}",[238,24443,24445],{"className":24444,"ariaHidden":269},[268],[238,24446,24448,24451],{"className":24447},[273],[238,24449],{"className":24450,"style":4573},[277],[238,24452,24454],{"className":24453},[282],[238,24455,24457,24603],{"className":24456},[958],[238,24458,24460],{"className":24459},[1117],[238,24461,24463,24595],{"className":24462},[395,848],[238,24464,24466,24592],{"className":24465},[399],[238,24467,24469,24531],{"className":24468,"style":4592},[403],[238,24470,24471,24474],{"style":4595},[238,24472],{"className":24473,"style":1134},[411],[238,24475,24477,24506,24509,24512,24515,24518,24521,24524,24528],{"className":24476},[282],[238,24478,24480,24483],{"className":24479},[282],[238,24481,11971],{"className":24482,"style":11992},[282,840],[238,24484,24486],{"className":24485},[391],[238,24487,24489],{"className":24488},[395],[238,24490,24492],{"className":24491},[399],[238,24493,24495],{"className":24494,"style":4620},[403],[238,24496,24497,24500],{"style":1226},[238,24498],{"className":24499,"style":412},[411],[238,24501,24503],{"className":24502},[416,417,418,419],[238,24504,2297],{"className":24505,"style":2313},[282,283,419],[238,24507,335],{"className":24508},[381],[238,24510,4429],{"className":24511,"style":660},[282,283],[238,24513],{"className":24514,"style":660},[429],[238,24516,987],{"className":24517},[664],[238,24519],{"className":24520,"style":660},[429],[238,24522,24276],{"className":24523},[282,283],[238,24525,745],{"className":24526},[282,24527],"mathbf",[238,24529,360],{"className":24530},[462],[238,24532,24533,24536],{"style":4643},[238,24534],{"className":24535,"style":1134},[411],[238,24537,24539,24568,24571,24574,24577,24580,24583,24586,24589],{"className":24538},[282],[238,24540,24542,24545],{"className":24541},[282],[238,24543,11971],{"className":24544,"style":11992},[282,840],[238,24546,24548],{"className":24547},[391],[238,24549,24551],{"className":24550},[395],[238,24552,24554],{"className":24553},[399],[238,24555,24557],{"className":24556,"style":22441},[403],[238,24558,24559,24562],{"style":1226},[238,24560],{"className":24561,"style":412},[411],[238,24563,24565],{"className":24564},[416,417,418,419],[238,24566,22083],{"className":24567},[664,419],[238,24569,335],{"className":24570},[381],[238,24572,4429],{"className":24573,"style":660},[282,283],[238,24575],{"className":24576,"style":660},[429],[238,24578,987],{"className":24579},[664],[238,24581],{"className":24582,"style":660},[429],[238,24584,24276],{"className":24585},[282,283],[238,24587,745],{"className":24588},[282,24527],[238,24590,360],{"className":24591},[462],[238,24593,872],{"className":24594},[871],[238,24596,24598],{"className":24597},[399],[238,24599,24601],{"className":24600,"style":4706},[403],[238,24602],{},[238,24604,24606],{"className":24605},[1422],[238,24607,24609,24764],{"className":24608},[395,848],[238,24610,24612,24761],{"className":24611},[399],[238,24613,24615,24688],{"className":24614,"style":4592},[403],[238,24616,24617,24620],{"style":4595},[238,24618],{"className":24619,"style":1134},[411],[238,24621,24623,24626,24629,24632,24635,24664,24667,24670,24673,24676,24679,24682,24685],{"className":24622},[282],[238,24624],{"className":24625},[282],[238,24627],{"className":24628,"style":430},[429],[238,24630,992],{"className":24631},[434],[238,24633],{"className":24634,"style":430},[429],[238,24636,24638,24641],{"className":24637},[282],[238,24639,11971],{"className":24640,"style":11992},[282,840],[238,24642,24644],{"className":24643},[391],[238,24645,24647],{"className":24646},[395],[238,24648,24650],{"className":24649},[399],[238,24651,24653],{"className":24652,"style":4620},[403],[238,24654,24655,24658],{"style":1226},[238,24656],{"className":24657,"style":412},[411],[238,24659,24661],{"className":24660},[416,417,418,419],[238,24662,2297],{"className":24663,"style":2313},[282,283,419],[238,24665,4429],{"className":24666,"style":660},[282,283],[238,24668],{"className":24669,"style":660},[429],[238,24671,987],{"className":24672},[664],[238,24674],{"className":24675,"style":660},[429],[238,24677,730],{"className":24678,"style":764},[282,283],[238,24680,24276],{"className":24681},[282,283],[238,24683,745],{"className":24684},[282,24527],[238,24686,354],{"className":24687},[451],[238,24689,24690,24693],{"style":4643},[238,24691],{"className":24692,"style":1134},[411],[238,24694,24696,24699,24702,24705,24708,24737,24740,24743,24746,24749,24752,24755,24758],{"className":24695},[282],[238,24697],{"className":24698},[282],[238,24700],{"className":24701,"style":430},[429],[238,24703,992],{"className":24704},[434],[238,24706],{"className":24707,"style":430},[429],[238,24709,24711,24714],{"className":24710},[282],[238,24712,11971],{"className":24713,"style":11992},[282,840],[238,24715,24717],{"className":24716},[391],[238,24718,24720],{"className":24719},[395],[238,24721,24723],{"className":24722},[399],[238,24724,24726],{"className":24725,"style":22441},[403],[238,24727,24728,24731],{"style":1226},[238,24729],{"className":24730,"style":412},[411],[238,24732,24734],{"className":24733},[416,417,418,419],[238,24735,22083],{"className":24736},[664,419],[238,24738,4429],{"className":24739,"style":660},[282,283],[238,24741],{"className":24742,"style":660},[429],[238,24744,987],{"className":24745},[664],[238,24747],{"className":24748,"style":660},[429],[238,24750,730],{"className":24751,"style":764},[282,283],[238,24753,24276],{"className":24754},[282,283],[238,24756,745],{"className":24757},[282,24527],[238,24759,354],{"className":24760},[451],[238,24762,872],{"className":24763},[871],[238,24765,24767],{"className":24766},[399],[238,24768,24770],{"className":24769,"style":4706},[403],[238,24771],{},[11,24773,24774,24775,24804,24805,24833],{},"其中 ",[238,24776,24778,24792],{"className":24777,"translate":242},[241],[238,24779,24781],{"className":24780},[246],[248,24782,24783],{"xmlns":250},[252,24784,24785,24789],{},[255,24786,24787],{},[738,24788,745],{"mathvariant":24354},[262,24790,24791],{"encoding":264},"\\mathbf{1}",[238,24793,24795],{"className":24794,"ariaHidden":269},[268],[238,24796,24798,24801],{"className":24797},[273],[238,24799],{"className":24800,"style":2672},[277],[238,24802,745],{"className":24803},[282,24527]," 表示恒等于 ",[238,24806,24808,24821],{"className":24807,"translate":242},[241],[238,24809,24811],{"className":24810},[246],[248,24812,24813],{"xmlns":250},[252,24814,24815,24819],{},[255,24816,24817],{},[738,24818,745],{},[262,24820,745],{"encoding":264},[238,24822,24824],{"className":24823,"ariaHidden":269},[268],[238,24825,24827,24830],{"className":24826},[273],[238,24828],{"className":24829,"style":2672},[277],[238,24831,745],{"className":24832},[282]," 的函数。",[11,24835,24836,24838,24839,24889,24890,941],{},[1688,24837,5794],{}," 若 ",[238,24840,24842,24859],{"className":24841,"translate":242},[241],[238,24843,24845],{"className":24844},[246],[248,24846,24847],{"xmlns":250},[252,24848,24849,24857],{},[255,24850,24851,24853,24855],{},[258,24852,4429],{},[332,24854,2023],{},[258,24856,23902],{},[262,24858,23985],{"encoding":264},[238,24860,24862,24880],{"className":24861,"ariaHidden":269},[268],[238,24863,24865,24868,24871,24874,24877],{"className":24864},[273],[238,24866],{"className":24867,"style":23995},[277],[238,24869,4429],{"className":24870,"style":660},[282,283],[238,24872],{"className":24873,"style":430},[429],[238,24875,2023],{"className":24876},[434],[238,24878],{"className":24879,"style":430},[429],[238,24881,24883,24886],{"className":24882},[273],[238,24884],{"className":24885,"style":278},[277],[238,24887,23902],{"className":24888,"style":377},[282,283],"，则对每个 ",[238,24891,24893,24922],{"className":24892,"translate":242},[241],[238,24894,24896],{"className":24895},[246],[248,24897,24898],{"xmlns":250},[252,24899,24900,24920],{},[255,24901,24902,24904,24906,24908,24910,24912,24918],{},[332,24903,335],{"stretchy":334},[258,24905,341],{},[332,24907,354],{"separator":269},[258,24909,357],{},[332,24911,354],{"separator":269},[337,24913,24914,24916],{},[258,24915,341],{},[332,24917,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,24919,360],{"stretchy":334},[262,24921,2209],{"encoding":264},[238,24923,24925],{"className":24924,"ariaHidden":269},[268],[238,24926,24928,24931,24934,24937,24940,24943,24946,24949,24952,24984],{"className":24927},[273],[238,24929],{"className":24930,"style":373},[277],[238,24932,335],{"className":24933},[381],[238,24935,341],{"className":24936},[282,283],[238,24938,354],{"className":24939},[451],[238,24941],{"className":24942,"style":455},[429],[238,24944,357],{"className":24945},[282,283],[238,24947,354],{"className":24948},[451],[238,24950],{"className":24951,"style":455},[429],[238,24953,24955,24958],{"className":24954},[282],[238,24956,341],{"className":24957},[282,283],[238,24959,24961],{"className":24960},[391],[238,24962,24964],{"className":24963},[395],[238,24965,24967],{"className":24966},[399],[238,24968,24970],{"className":24969,"style":404},[403],[238,24971,24972,24975],{"style":407},[238,24973],{"className":24974,"style":412},[411],[238,24976,24978],{"className":24977},[416,417,418,419],[238,24979,24981],{"className":24980},[282,419],[238,24982,346],{"className":24983},[282,419],[238,24985,360],{"className":24986},[462],[238,24988,24990],{"className":24989,"translate":242},[945],[238,24991,24993,25081],{"className":24992,"translate":242},[241],[238,24994,24996],{"className":24995},[246],[248,24997,24998],{"xmlns":250,"display":954},[252,24999,25000,25078],{},[255,25001,25002,25004,25006,25008,25010,25012,25014,25020,25022,25024,25026,25028,25030,25036,25038,25040,25042,25044,25046,25048,25050,25052,25058,25060,25062,25064,25066,25068,25074,25076],{},[258,25003,600],{},[332,25005,335],{"stretchy":334},[258,25007,341],{},[332,25009,354],{"separator":269},[258,25011,357],{},[332,25013,354],{"separator":269},[337,25015,25016,25018],{},[258,25017,341],{},[332,25019,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,25021,360],{"stretchy":334},[332,25023,987],{},[258,25025,730],{},[258,25027,4429],{},[332,25029,335],{"stretchy":334},[337,25031,25032,25034],{},[258,25033,341],{},[332,25035,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,25037,360],{"stretchy":334},[332,25039,2023],{},[258,25041,600],{},[332,25043,335],{"stretchy":334},[258,25045,341],{},[332,25047,354],{"separator":269},[258,25049,357],{},[332,25051,354],{"separator":269},[337,25053,25054,25056],{},[258,25055,341],{},[332,25057,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,25059,360],{"stretchy":334},[332,25061,987],{},[258,25063,730],{},[258,25065,23902],{},[332,25067,335],{"stretchy":334},[337,25069,25070,25072],{},[258,25071,341],{},[332,25073,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,25075,360],{"stretchy":334},[258,25077,3102],{"mathvariant":344},[262,25079,25080],{"encoding":264},"r(s,a,s')+\\gamma V(s')\\le r(s,a,s')+\\gamma W(s').",[238,25082,25084,25158,25217,25291],{"className":25083,"ariaHidden":269},[268],[238,25085,25087,25090,25093,25096,25099,25102,25105,25108,25111,25114,25146,25149,25152,25155],{"className":25086},[273],[238,25088],{"className":25089,"style":22668},[277],[238,25091,600],{"className":25092,"style":637},[282,283],[238,25094,335],{"className":25095},[381],[238,25097,341],{"className":25098},[282,283],[238,25100,354],{"className":25101},[451],[238,25103],{"className":25104,"style":455},[429],[238,25106,357],{"className":25107},[282,283],[238,25109,354],{"className":25110},[451],[238,25112],{"className":25113,"style":455},[429],[238,25115,25117,25120],{"className":25116},[282],[238,25118,341],{"className":25119},[282,283],[238,25121,25123],{"className":25122},[391],[238,25124,25126],{"className":25125},[395],[238,25127,25129],{"className":25128},[399],[238,25130,25132],{"className":25131,"style":1223},[403],[238,25133,25134,25137],{"style":1226},[238,25135],{"className":25136,"style":412},[411],[238,25138,25140],{"className":25139},[416,417,418,419],[238,25141,25143],{"className":25142},[282,419],[238,25144,346],{"className":25145},[282,419],[238,25147,360],{"className":25148},[462],[238,25150],{"className":25151,"style":660},[429],[238,25153,987],{"className":25154},[664],[238,25156],{"className":25157,"style":660},[429],[238,25159,25161,25164,25167,25170,25173,25205,25208,25211,25214],{"className":25160},[273],[238,25162],{"className":25163,"style":22668},[277],[238,25165,730],{"className":25166,"style":764},[282,283],[238,25168,4429],{"className":25169,"style":660},[282,283],[238,25171,335],{"className":25172},[381],[238,25174,25176,25179],{"className":25175},[282],[238,25177,341],{"className":25178},[282,283],[238,25180,25182],{"className":25181},[391],[238,25183,25185],{"className":25184},[395],[238,25186,25188],{"className":25187},[399],[238,25189,25191],{"className":25190,"style":1223},[403],[238,25192,25193,25196],{"style":1226},[238,25194],{"className":25195,"style":412},[411],[238,25197,25199],{"className":25198},[416,417,418,419],[238,25200,25202],{"className":25201},[282,419],[238,25203,346],{"className":25204},[282,419],[238,25206,360],{"className":25207},[462],[238,25209],{"className":25210,"style":430},[429],[238,25212,2023],{"className":25213},[434],[238,25215],{"className":25216,"style":430},[429],[238,25218,25220,25223,25226,25229,25232,25235,25238,25241,25244,25247,25279,25282,25285,25288],{"className":25219},[273],[238,25221],{"className":25222,"style":22668},[277],[238,25224,600],{"className":25225,"style":637},[282,283],[238,25227,335],{"className":25228},[381],[238,25230,341],{"className":25231},[282,283],[238,25233,354],{"className":25234},[451],[238,25236],{"className":25237,"style":455},[429],[238,25239,357],{"className":25240},[282,283],[238,25242,354],{"className":25243},[451],[238,25245],{"className":25246,"style":455},[429],[238,25248,25250,25253],{"className":25249},[282],[238,25251,341],{"className":25252},[282,283],[238,25254,25256],{"className":25255},[391],[238,25257,25259],{"className":25258},[395],[238,25260,25262],{"className":25261},[399],[238,25263,25265],{"className":25264,"style":1223},[403],[238,25266,25267,25270],{"style":1226},[238,25268],{"className":25269,"style":412},[411],[238,25271,25273],{"className":25272},[416,417,418,419],[238,25274,25276],{"className":25275},[282,419],[238,25277,346],{"className":25278},[282,419],[238,25280,360],{"className":25281},[462],[238,25283],{"className":25284,"style":660},[429],[238,25286,987],{"className":25287},[664],[238,25289],{"className":25290,"style":660},[429],[238,25292,25294,25297,25300,25303,25306,25338,25341],{"className":25293},[273],[238,25295],{"className":25296,"style":22668},[277],[238,25298,730],{"className":25299,"style":764},[282,283],[238,25301,23902],{"className":25302,"style":377},[282,283],[238,25304,335],{"className":25305},[381],[238,25307,25309,25312],{"className":25308},[282],[238,25310,341],{"className":25311},[282,283],[238,25313,25315],{"className":25314},[391],[238,25316,25318],{"className":25317},[395],[238,25319,25321],{"className":25320},[399],[238,25322,25324],{"className":25323,"style":1223},[403],[238,25325,25326,25329],{"style":1226},[238,25327],{"className":25328,"style":412},[411],[238,25330,25332],{"className":25331},[416,417,418,419],[238,25333,25335],{"className":25334},[282,419],[238,25336,346],{"className":25337},[282,419],[238,25339,360],{"className":25340},[462],[238,25342,3102],{"className":25343},[282],[11,25345,20770,25346,23964,25435,25569,25570,23814,25691,25749,25750,1806],{},[238,25347,25349,25378],{"className":25348,"translate":242},[241],[238,25350,25352],{"className":25351},[246],[248,25353,25354],{"xmlns":250},[252,25355,25356,25375],{},[255,25357,25358,25360,25362,25364,25366,25368,25370,25373],{},[258,25359,2297],{},[332,25361,335],{"stretchy":334},[258,25363,357],{},[332,25365,349],{},[258,25367,341],{},[332,25369,360],{"stretchy":334},[332,25371,25372],{},"≥",[738,25374,740],{},[262,25376,25377],{"encoding":264},"\\pi(a\\mid s)\\ge0",[238,25379,25381,25405,25426],{"className":25380,"ariaHidden":269},[268],[238,25382,25384,25387,25390,25393,25396,25399,25402],{"className":25383},[273],[238,25385],{"className":25386,"style":444},[277],[238,25388,2297],{"className":25389,"style":2313},[282,283],[238,25391,335],{"className":25392},[381],[238,25394,357],{"className":25395},[282,283],[238,25397],{"className":25398,"style":430},[429],[238,25400,349],{"className":25401},[434],[238,25403],{"className":25404,"style":430},[429],[238,25406,25408,25411,25414,25417,25420,25423],{"className":25407},[273],[238,25409],{"className":25410,"style":444},[277],[238,25412,341],{"className":25413},[282,283],[238,25415,360],{"className":25416},[462],[238,25418],{"className":25419,"style":430},[429],[238,25421,25372],{"className":25422},[434],[238,25424],{"className":25425,"style":430},[429],[238,25427,25429,25432],{"className":25428},[273],[238,25430],{"className":25431,"style":2672},[277],[238,25433,740],{"className":25434},[282],[238,25436,25438,25474],{"className":25437,"translate":242},[241],[238,25439,25441],{"className":25440},[246],[248,25442,25443],{"xmlns":250},[252,25444,25445,25471],{},[255,25446,25447,25449,25451,25457,25459,25461,25463,25465,25467,25469],{},[258,25448,330],{},[332,25450,335],{"stretchy":334},[337,25452,25453,25455],{},[258,25454,341],{},[332,25456,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,25458,349],{},[258,25460,341],{},[332,25462,354],{"separator":269},[258,25464,357],{},[332,25466,360],{"stretchy":334},[332,25468,25372],{},[738,25470,740],{},[262,25472,25473],{"encoding":264},"P(s'\\mid s,a)\\ge0",[238,25475,25477,25530,25560],{"className":25476,"ariaHidden":269},[268],[238,25478,25480,25483,25486,25489,25521,25524,25527],{"className":25479},[273],[238,25481],{"className":25482,"style":373},[277],[238,25484,330],{"className":25485,"style":377},[282,283],[238,25487,335],{"className":25488},[381],[238,25490,25492,25495],{"className":25491},[282],[238,25493,341],{"className":25494},[282,283],[238,25496,25498],{"className":25497},[391],[238,25499,25501],{"className":25500},[395],[238,25502,25504],{"className":25503},[399],[238,25505,25507],{"className":25506,"style":404},[403],[238,25508,25509,25512],{"style":407},[238,25510],{"className":25511,"style":412},[411],[238,25513,25515],{"className":25514},[416,417,418,419],[238,25516,25518],{"className":25517},[282,419],[238,25519,346],{"className":25520},[282,419],[238,25522],{"className":25523,"style":430},[429],[238,25525,349],{"className":25526},[434],[238,25528],{"className":25529,"style":430},[429],[238,25531,25533,25536,25539,25542,25545,25548,25551,25554,25557],{"className":25532},[273],[238,25534],{"className":25535,"style":444},[277],[238,25537,341],{"className":25538},[282,283],[238,25540,354],{"className":25541},[451],[238,25543],{"className":25544,"style":455},[429],[238,25546,357],{"className":25547},[282,283],[238,25549,360],{"className":25550},[462],[238,25552],{"className":25553,"style":430},[429],[238,25555,25372],{"className":25556},[434],[238,25558],{"className":25559,"style":430},[429],[238,25561,25563,25566],{"className":25562},[273],[238,25564],{"className":25565,"style":2672},[277],[238,25567,740],{"className":25568},[282],"，加权求和保持不等号，故 ",[238,25571,25573,25603],{"className":25572,"translate":242},[241],[238,25574,25576],{"className":25575},[246],[248,25577,25578],{"xmlns":250},[252,25579,25580,25600],{},[255,25581,25582,25588,25590,25592,25598],{},[337,25583,25584,25586],{},[258,25585,11971],{"mathvariant":816},[258,25587,2297],{},[258,25589,4429],{},[332,25591,2023],{},[337,25593,25594,25596],{},[258,25595,11971],{"mathvariant":816},[258,25597,2297],{},[258,25599,23902],{},[262,25601,25602],{"encoding":264},"\\mathcal{T}^\\pi V\\le\\mathcal{T}^\\pi W",[238,25604,25606,25653],{"className":25605,"ariaHidden":269},[268],[238,25607,25609,25612,25641,25644,25647,25650],{"className":25608},[273],[238,25610],{"className":25611,"style":23995},[277],[238,25613,25615,25618],{"className":25614},[282],[238,25616,11971],{"className":25617,"style":11992},[282,840],[238,25619,25621],{"className":25620},[391],[238,25622,25624],{"className":25623},[395],[238,25625,25627],{"className":25626},[399],[238,25628,25630],{"className":25629,"style":6609},[403],[238,25631,25632,25635],{"style":407},[238,25633],{"className":25634,"style":412},[411],[238,25636,25638],{"className":25637},[416,417,418,419],[238,25639,2297],{"className":25640,"style":2313},[282,283,419],[238,25642,4429],{"className":25643,"style":660},[282,283],[238,25645],{"className":25646,"style":430},[429],[238,25648,2023],{"className":25649},[434],[238,25651],{"className":25652,"style":430},[429],[238,25654,25656,25659,25688],{"className":25655},[273],[238,25657],{"className":25658,"style":278},[277],[238,25660,25662,25665],{"className":25661},[282],[238,25663,11971],{"className":25664,"style":11992},[282,840],[238,25666,25668],{"className":25667},[391],[238,25669,25671],{"className":25670},[395],[238,25672,25674],{"className":25673},[399],[238,25675,25677],{"className":25676,"style":6609},[403],[238,25678,25679,25682],{"style":407},[238,25680],{"className":25681,"style":412},[411],[238,25683,25685],{"className":25684},[416,417,418,419],[238,25686,2297],{"className":25687,"style":2313},[282,283,419],[238,25689,23902],{"className":25690,"style":377},[282,283],[238,25692,25694,25711],{"className":25693,"translate":242},[241],[238,25695,25697],{"className":25696},[246],[248,25698,25699],{"xmlns":250},[252,25700,25701,25709],{},[255,25702,25703],{},[337,25704,25705,25707],{},[258,25706,11971],{"mathvariant":816},[332,25708,22083],{},[262,25710,22086],{"encoding":264},[238,25712,25714],{"className":25713,"ariaHidden":269},[268],[238,25715,25717,25720],{"className":25716},[273],[238,25718],{"className":25719,"style":22096},[277],[238,25721,25723,25726],{"className":25722},[282],[238,25724,11971],{"className":25725,"style":11992},[282,840],[238,25727,25729],{"className":25728},[391],[238,25730,25732],{"className":25731},[395],[238,25733,25735],{"className":25734},[399],[238,25736,25738],{"className":25737,"style":22096},[403],[238,25739,25740,25743],{"style":407},[238,25741],{"className":25742,"style":412},[411],[238,25744,25746],{"className":25745},[416,417,418,419],[238,25747,22083],{"className":25748},[664,419],"，逐点取最大值也保持逐点序，故 ",[238,25751,25753,25783],{"className":25752,"translate":242},[241],[238,25754,25756],{"className":25755},[246],[248,25757,25758],{"xmlns":250},[252,25759,25760,25780],{},[255,25761,25762,25768,25770,25772,25778],{},[337,25763,25764,25766],{},[258,25765,11971],{"mathvariant":816},[332,25767,22083],{},[258,25769,4429],{},[332,25771,2023],{},[337,25773,25774,25776],{},[258,25775,11971],{"mathvariant":816},[332,25777,22083],{},[258,25779,23902],{},[262,25781,25782],{"encoding":264},"\\mathcal{T}^*V\\le\\mathcal{T}^*W",[238,25784,25786,25834],{"className":25785,"ariaHidden":269},[268],[238,25787,25789,25793,25822,25825,25828,25831],{"className":25788},[273],[238,25790],{"className":25791,"style":25792},[277],"height:0.8247em;vertical-align:-0.136em;",[238,25794,25796,25799],{"className":25795},[282],[238,25797,11971],{"className":25798,"style":11992},[282,840],[238,25800,25802],{"className":25801},[391],[238,25803,25805],{"className":25804},[395],[238,25806,25808],{"className":25807},[399],[238,25809,25811],{"className":25810,"style":22096},[403],[238,25812,25813,25816],{"style":407},[238,25814],{"className":25815,"style":412},[411],[238,25817,25819],{"className":25818},[416,417,418,419],[238,25820,22083],{"className":25821},[664,419],[238,25823,4429],{"className":25824,"style":660},[282,283],[238,25826],{"className":25827,"style":430},[429],[238,25829,2023],{"className":25830},[434],[238,25832],{"className":25833,"style":430},[429],[238,25835,25837,25840,25869],{"className":25836},[273],[238,25838],{"className":25839,"style":22096},[277],[238,25841,25843,25846],{"className":25842},[282],[238,25844,11971],{"className":25845,"style":11992},[282,840],[238,25847,25849],{"className":25848},[391],[238,25850,25852],{"className":25851},[395],[238,25853,25855],{"className":25854},[399],[238,25856,25858],{"className":25857,"style":22096},[403],[238,25859,25860,25863],{"style":407},[238,25861],{"className":25862,"style":412},[411],[238,25864,25866],{"className":25865},[416,417,418,419],[238,25867,22083],{"className":25868},[664,419],[238,25870,23902],{"className":25871,"style":377},[282,283],[11,25873,25874],{},"常数平移式由直接计算得到：",[238,25876,25878],{"className":25877,"translate":242},[945],[238,25879,25881,26060],{"className":25880,"translate":242},[241],[238,25882,25884],{"className":25883},[246],[248,25885,25886],{"xmlns":250,"display":954},[252,25887,25888,26057],{},[958,25889,25890,26022],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,25891,25892,25916],{},[966,25893,25894],{},[969,25895,25896],{"scriptlevel":740,"displaystyle":269},[255,25897,25898,25904,25906,25908,25910,25912,25914],{},[337,25899,25900,25902],{},[258,25901,11971],{"mathvariant":816},[258,25903,2297],{},[332,25905,335],{"stretchy":334},[258,25907,4429],{},[332,25909,987],{},[258,25911,24276],{},[738,25913,745],{"mathvariant":24354},[332,25915,360],{"stretchy":334},[966,25917,25918],{},[969,25919,25920],{"scriptlevel":740,"displaystyle":269},[255,25921,25922,25924,25926,25932,25934,25936,25938,25940,25942,25944,25954,25956,25958,25964,25966,25968,25970,25972,25974],{},[255,25923],{},[332,25925,992],{},[6755,25927,25928,25930],{},[332,25929,3390],{},[258,25931,357],{},[258,25933,2297],{},[332,25935,335],{"stretchy":334},[258,25937,357],{},[332,25939,349],{},[258,25941,341],{},[332,25943,360],{"stretchy":334},[6755,25945,25946,25948],{},[332,25947,3390],{},[337,25949,25950,25952],{},[258,25951,341],{},[332,25953,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,25955,330],{},[332,25957,335],{"stretchy":334},[337,25959,25960,25962],{},[258,25961,341],{},[332,25963,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,25965,349],{},[258,25967,341],{},[332,25969,354],{"separator":269},[258,25971,357],{},[332,25973,360],{"stretchy":334},[255,25975,25976,25978,25980,25982,25984,25986,25988,25990,25996,25998,26000,26002,26004,26006,26012,26014,26016,26018,26020],{},[332,25977,736],{"fence":269},[258,25979,600],{},[332,25981,335],{"stretchy":334},[258,25983,341],{},[332,25985,354],{"separator":269},[258,25987,357],{},[332,25989,354],{"separator":269},[337,25991,25992,25994],{},[258,25993,341],{},[332,25995,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,25997,360],{"stretchy":334},[332,25999,987],{},[258,26001,730],{},[258,26003,4429],{},[332,26005,335],{"stretchy":334},[337,26007,26008,26010],{},[258,26009,341],{},[332,26011,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,26013,360],{"stretchy":334},[332,26015,987],{},[258,26017,730],{},[258,26019,24276],{},[332,26021,4479],{"fence":269},[963,26023,26024,26030],{},[966,26025,26026],{},[969,26027,26028],{"scriptlevel":740,"displaystyle":269},[255,26029],{},[966,26031,26032],{},[969,26033,26034],{"scriptlevel":740,"displaystyle":269},[255,26035,26036,26038,26040,26046,26048,26050,26052,26054],{},[255,26037],{},[332,26039,992],{},[337,26041,26042,26044],{},[258,26043,11971],{"mathvariant":816},[258,26045,2297],{},[258,26047,4429],{},[332,26049,987],{},[258,26051,730],{},[258,26053,24276],{},[738,26055,26056],{"mathvariant":24354},"1.",[262,26058,26059],{"encoding":264},"\\begin{aligned}\n\\mathcal{T}^\\pi(V+c\\mathbf{1})\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')+\\gamma c\\right]\\\\\n&=\\mathcal{T}^\\pi V+\\gamma c\\mathbf{1}.\n\\end{aligned}",[238,26061,26063],{"className":26062,"ariaHidden":269},[268],[238,26064,26066,26069],{"className":26065},[273],[238,26067],{"className":26068,"style":7153},[277],[238,26070,26072],{"className":26071},[282],[238,26073,26075,26167],{"className":26074},[958],[238,26076,26078],{"className":26077},[1117],[238,26079,26081,26159],{"className":26080},[395,848],[238,26082,26084,26156],{"className":26083},[399],[238,26085,26087,26148],{"className":26086,"style":7172},[403],[238,26088,26089,26092],{"style":7175},[238,26090],{"className":26091,"style":3510},[411],[238,26093,26095,26124,26127,26130,26133,26136,26139,26142,26145],{"className":26094},[282],[238,26096,26098,26101],{"className":26097},[282],[238,26099,11971],{"className":26100,"style":11992},[282,840],[238,26102,26104],{"className":26103},[391],[238,26105,26107],{"className":26106},[395],[238,26108,26110],{"className":26109},[399],[238,26111,26113],{"className":26112,"style":4620},[403],[238,26114,26115,26118],{"style":1226},[238,26116],{"className":26117,"style":412},[411],[238,26119,26121],{"className":26120},[416,417,418,419],[238,26122,2297],{"className":26123,"style":2313},[282,283,419],[238,26125,335],{"className":26126},[381],[238,26128,4429],{"className":26129,"style":660},[282,283],[238,26131],{"className":26132,"style":660},[429],[238,26134,987],{"className":26135},[664],[238,26137],{"className":26138,"style":660},[429],[238,26140,24276],{"className":26141},[282,283],[238,26143,745],{"className":26144},[282,24527],[238,26146,360],{"className":26147},[462],[238,26149,26150,26153],{"style":7231},[238,26151],{"className":26152,"style":3510},[411],[238,26154],{"className":26155},[282],[238,26157,872],{"className":26158},[871],[238,26160,26162],{"className":26161},[399],[238,26163,26165],{"className":26164,"style":7247},[403],[238,26166],{},[238,26168,26170],{"className":26169},[1422],[238,26171,26173,26632],{"className":26172},[395,848],[238,26174,26176,26629],{"className":26175},[399],[238,26177,26179,26556],{"className":26178,"style":7172},[403],[238,26180,26181,26184],{"style":7175},[238,26182],{"className":26183,"style":3510},[411],[238,26185,26187,26190,26193,26196,26199,26246,26249,26252,26255,26258,26261,26264,26267,26270,26273,26276,26352,26355,26358,26361,26393,26396,26399,26402,26405,26408,26411,26414,26417,26420],{"className":26186},[282],[238,26188],{"className":26189},[282],[238,26191],{"className":26192,"style":430},[429],[238,26194,992],{"className":26195},[434],[238,26197],{"className":26198,"style":430},[429],[238,26200,26202],{"className":26201},[1943,3493],[238,26203,26205,26238],{"className":26204},[395,848],[238,26206,26208,26235],{"className":26207},[399],[238,26209,26211,26225],{"className":26210,"style":6869},[403],[238,26212,26213,26216],{"style":6872},[238,26214],{"className":26215,"style":3510},[411],[238,26217,26219],{"className":26218},[416,417,418,419],[238,26220,26222],{"className":26221},[282,419],[238,26223,357],{"className":26224},[282,283,419],[238,26226,26227,26230],{"style":3529},[238,26228],{"className":26229,"style":3510},[411],[238,26231,26232],{},[238,26233,3390],{"className":26234},[1943,3538,3539],[238,26236,872],{"className":26237},[871],[238,26239,26241],{"className":26240},[399],[238,26242,26244],{"className":26243,"style":6904},[403],[238,26245],{},[238,26247],{"className":26248,"style":455},[429],[238,26250,2297],{"className":26251,"style":2313},[282,283],[238,26253,335],{"className":26254},[381],[238,26256,357],{"className":26257},[282,283],[238,26259],{"className":26260,"style":430},[429],[238,26262,349],{"className":26263},[434],[238,26265],{"className":26266,"style":430},[429],[238,26268,341],{"className":26269},[282,283],[238,26271,360],{"className":26272},[462],[238,26274],{"className":26275,"style":455},[429],[238,26277,26279],{"className":26278},[1943,3493],[238,26280,26282,26344],{"className":26281},[395,848],[238,26283,26285,26341],{"className":26284},[399],[238,26286,26288,26331],{"className":26287,"style":6869},[403],[238,26289,26290,26293],{"style":7296},[238,26291],{"className":26292,"style":3510},[411],[238,26294,26296],{"className":26295},[416,417,418,419],[238,26297,26299],{"className":26298},[282,419],[238,26300,26302,26305],{"className":26301},[282,419],[238,26303,341],{"className":26304},[282,283,419],[238,26306,26308],{"className":26307},[391],[238,26309,26311],{"className":26310},[395],[238,26312,26314],{"className":26313},[399],[238,26315,26317],{"className":26316,"style":4196},[403],[238,26318,26319,26322],{"style":4199},[238,26320],{"className":26321,"style":4203},[411],[238,26323,26325],{"className":26324},[416,4207,4208,419],[238,26326,26328],{"className":26327},[282,419],[238,26329,346],{"className":26330},[282,419],[238,26332,26333,26336],{"style":3529},[238,26334],{"className":26335,"style":3510},[411],[238,26337,26338],{},[238,26339,3390],{"className":26340},[1943,3538,3539],[238,26342,872],{"className":26343},[871],[238,26345,26347],{"className":26346},[399],[238,26348,26350],{"className":26349,"style":7357},[403],[238,26351],{},[238,26353],{"className":26354,"style":455},[429],[238,26356,330],{"className":26357,"style":377},[282,283],[238,26359,335],{"className":26360},[381],[238,26362,26364,26367],{"className":26363},[282],[238,26365,341],{"className":26366},[282,283],[238,26368,26370],{"className":26369},[391],[238,26371,26373],{"className":26372},[395],[238,26374,26376],{"className":26375},[399],[238,26377,26379],{"className":26378,"style":1223},[403],[238,26380,26381,26384],{"style":1226},[238,26382],{"className":26383,"style":412},[411],[238,26385,26387],{"className":26386},[416,417,418,419],[238,26388,26390],{"className":26389},[282,419],[238,26391,346],{"className":26392},[282,419],[238,26394],{"className":26395,"style":430},[429],[238,26397,349],{"className":26398},[434],[238,26400],{"className":26401,"style":430},[429],[238,26403,341],{"className":26404},[282,283],[238,26406,354],{"className":26407},[451],[238,26409],{"className":26410,"style":455},[429],[238,26412,357],{"className":26413},[282,283],[238,26415,360],{"className":26416},[462],[238,26418],{"className":26419,"style":455},[429],[238,26421,26423,26426,26429,26432,26435,26438,26441,26444,26447,26450,26482,26485,26488,26491,26494,26497,26500,26503,26535,26538,26541,26544,26547,26550,26553],{"className":26422},[3041],[238,26424,736],{"className":26425,"style":4791},[381,4790],[238,26427,600],{"className":26428,"style":637},[282,283],[238,26430,335],{"className":26431},[381],[238,26433,341],{"className":26434},[282,283],[238,26436,354],{"className":26437},[451],[238,26439],{"className":26440,"style":455},[429],[238,26442,357],{"className":26443},[282,283],[238,26445,354],{"className":26446},[451],[238,26448],{"className":26449,"style":455},[429],[238,26451,26453,26456],{"className":26452},[282],[238,26454,341],{"className":26455},[282,283],[238,26457,26459],{"className":26458},[391],[238,26460,26462],{"className":26461},[395],[238,26463,26465],{"className":26464},[399],[238,26466,26468],{"className":26467,"style":1223},[403],[238,26469,26470,26473],{"style":1226},[238,26471],{"className":26472,"style":412},[411],[238,26474,26476],{"className":26475},[416,417,418,419],[238,26477,26479],{"className":26478},[282,419],[238,26480,346],{"className":26481},[282,419],[238,26483,360],{"className":26484},[462],[238,26486],{"className":26487,"style":660},[429],[238,26489,987],{"className":26490},[664],[238,26492],{"className":26493,"style":660},[429],[238,26495,730],{"className":26496,"style":764},[282,283],[238,26498,4429],{"className":26499,"style":660},[282,283],[238,26501,335],{"className":26502},[381],[238,26504,26506,26509],{"className":26505},[282],[238,26507,341],{"className":26508},[282,283],[238,26510,26512],{"className":26511},[391],[238,26513,26515],{"className":26514},[395],[238,26516,26518],{"className":26517},[399],[238,26519,26521],{"className":26520,"style":1223},[403],[238,26522,26523,26526],{"style":1226},[238,26524],{"className":26525,"style":412},[411],[238,26527,26529],{"className":26528},[416,417,418,419],[238,26530,26532],{"className":26531},[282,419],[238,26533,346],{"className":26534},[282,419],[238,26536,360],{"className":26537},[462],[238,26539],{"className":26540,"style":660},[429],[238,26542,987],{"className":26543},[664],[238,26545],{"className":26546,"style":660},[429],[238,26548,730],{"className":26549,"style":764},[282,283],[238,26551,24276],{"className":26552},[282,283],[238,26554,4479],{"className":26555,"style":4791},[462,4790],[238,26557,26558,26561],{"style":7231},[238,26559],{"className":26560,"style":3510},[411],[238,26562,26564,26567,26570,26573,26576,26605,26608,26611,26614,26617,26620,26623,26626],{"className":26563},[282],[238,26565],{"className":26566},[282],[238,26568],{"className":26569,"style":430},[429],[238,26571,992],{"className":26572},[434],[238,26574],{"className":26575,"style":430},[429],[238,26577,26579,26582],{"className":26578},[282],[238,26580,11971],{"className":26581,"style":11992},[282,840],[238,26583,26585],{"className":26584},[391],[238,26586,26588],{"className":26587},[395],[238,26589,26591],{"className":26590},[399],[238,26592,26594],{"className":26593,"style":4620},[403],[238,26595,26596,26599],{"style":1226},[238,26597],{"className":26598,"style":412},[411],[238,26600,26602],{"className":26601},[416,417,418,419],[238,26603,2297],{"className":26604,"style":2313},[282,283,419],[238,26606,4429],{"className":26607,"style":660},[282,283],[238,26609],{"className":26610,"style":660},[429],[238,26612,987],{"className":26613},[664],[238,26615],{"className":26616,"style":660},[429],[238,26618,730],{"className":26619,"style":764},[282,283],[238,26621,24276],{"className":26622},[282,283],[238,26624,745],{"className":26625},[282,24527],[238,26627,3102],{"className":26628},[282],[238,26630,872],{"className":26631},[871],[238,26633,26635],{"className":26634},[399],[238,26636,26638],{"className":26637,"style":7247},[403],[238,26639],{},[11,26641,26642,26643,26701,26702,26736,26737,26765],{},"对 ",[238,26644,26646,26663],{"className":26645,"translate":242},[241],[238,26647,26649],{"className":26648},[246],[248,26650,26651],{"xmlns":250},[252,26652,26653,26661],{},[255,26654,26655],{},[337,26656,26657,26659],{},[258,26658,11971],{"mathvariant":816},[332,26660,22083],{},[262,26662,22086],{"encoding":264},[238,26664,26666],{"className":26665,"ariaHidden":269},[268],[238,26667,26669,26672],{"className":26668},[273],[238,26670],{"className":26671,"style":22096},[277],[238,26673,26675,26678],{"className":26674},[282],[238,26676,11971],{"className":26677,"style":11992},[282,840],[238,26679,26681],{"className":26680},[391],[238,26682,26684],{"className":26683},[395],[238,26685,26687],{"className":26686},[399],[238,26688,26690],{"className":26689,"style":22096},[403],[238,26691,26692,26695],{"style":407},[238,26693],{"className":26694,"style":412},[411],[238,26696,26698],{"className":26697},[416,417,418,419],[238,26699,22083],{"className":26700},[664,419],"，常数 ",[238,26703,26705,26721],{"className":26704,"translate":242},[241],[238,26706,26708],{"className":26707},[246],[248,26709,26710],{"xmlns":250},[252,26711,26712,26718],{},[255,26713,26714,26716],{},[258,26715,730],{},[258,26717,24276],{},[262,26719,26720],{"encoding":264},"\\gamma c",[238,26722,26724],{"className":26723,"ariaHidden":269},[268],[238,26725,26727,26730,26733],{"className":26726},[273],[238,26728],{"className":26729,"style":2853},[277],[238,26731,730],{"className":26732,"style":764},[282,283],[238,26734,24276],{"className":26735},[282,283]," 与动作 ",[238,26738,26740,26753],{"className":26739,"translate":242},[241],[238,26741,26743],{"className":26742},[246],[248,26744,26745],{"xmlns":250},[252,26746,26747,26751],{},[255,26748,26749],{},[258,26750,357],{},[262,26752,357],{"encoding":264},[238,26754,26756],{"className":26755,"ariaHidden":269},[268],[238,26757,26759,26762],{"className":26758},[273],[238,26760],{"className":26761,"style":489},[277],[238,26763,357],{"className":26764},[282,283]," 无关，可以从最大值中提出。证毕。",[11,26767,26768,26769,26798,26799,26827],{},"这两个性质看起来简单，却足以推出压缩性。单调性说明算子保持函数之间的序关系，常数平移式说明算子对整体平移的响应恰好按 ",[238,26770,26772,26786],{"className":26771,"translate":242},[241],[238,26773,26775],{"className":26774},[246],[248,26776,26777],{"xmlns":250},[252,26778,26779,26783],{},[255,26780,26781],{},[258,26782,730],{},[262,26784,26785],{"encoding":264},"\\gamma",[238,26787,26789],{"className":26788,"ariaHidden":269},[268],[238,26790,26792,26795],{"className":26791},[273],[238,26793],{"className":26794,"style":2853},[277],[238,26796,730],{"className":26797,"style":764},[282,283]," 缩小。二者结合，就得到「两个函数经过算子作用后距离至多变为原来的 ",[238,26800,26802,26815],{"className":26801,"translate":242},[241],[238,26803,26805],{"className":26804},[246],[248,26806,26807],{"xmlns":250},[252,26808,26809,26813],{},[255,26810,26811],{},[258,26812,730],{},[262,26814,26785],{"encoding":264},[238,26816,26818],{"className":26817,"ariaHidden":269},[268],[238,26819,26821,26824],{"className":26820},[273],[238,26822],{"className":26823,"style":2853},[277],[238,26825,730],{"className":26826,"style":764},[282,283]," 倍」这一结论。",[225,26829,26831],{"id":26830},"_33-压缩映射","3.3 压缩映射",[230,26833,26836,27261],{"id":26834,"title":26835},"theorem-contraction","定理 3.2（贝尔曼算子的压缩性）",[11,26837,236,26838,14826,26910,1838,26968,27026,27027,27070,27071,27099,27100,941],{},[238,26839,26841,26862],{"className":26840,"translate":242},[241],[238,26842,26844],{"className":26843},[246],[248,26845,26846],{"xmlns":250},[252,26847,26848,26860],{},[255,26849,26850,26852,26854,26856,26858],{},[738,26851,740],{},[332,26853,2023],{},[258,26855,730],{},[332,26857,1892],{},[738,26859,745],{},[262,26861,14776],{"encoding":264},[238,26863,26865,26883,26901],{"className":26864,"ariaHidden":269},[268],[238,26866,26868,26871,26874,26877,26880],{"className":26867},[273],[238,26869],{"className":26870,"style":14786},[277],[238,26872,740],{"className":26873},[282],[238,26875],{"className":26876,"style":430},[429],[238,26878,2023],{"className":26879},[434],[238,26881],{"className":26882,"style":430},[429],[238,26884,26886,26889,26892,26895,26898],{"className":26885},[273],[238,26887],{"className":26888,"style":760},[277],[238,26890,730],{"className":26891,"style":764},[282,283],[238,26893],{"className":26894,"style":430},[429],[238,26896,1892],{"className":26897},[434],[238,26899],{"className":26900,"style":430},[429],[238,26902,26904,26907],{"className":26903},[273],[238,26905],{"className":26906,"style":2672},[277],[238,26908,745],{"className":26909},[282],[238,26911,26913,26930],{"className":26912,"translate":242},[241],[238,26914,26916],{"className":26915},[246],[248,26917,26918],{"xmlns":250},[252,26919,26920,26928],{},[255,26921,26922],{},[337,26923,26924,26926],{},[258,26925,11971],{"mathvariant":816},[258,26927,2297],{},[262,26929,11976],{"encoding":264},[238,26931,26933],{"className":26932,"ariaHidden":269},[268],[238,26934,26936,26939],{"className":26935},[273],[238,26937],{"className":26938,"style":278},[277],[238,26940,26942,26945],{"className":26941},[282],[238,26943,11971],{"className":26944,"style":11992},[282,840],[238,26946,26948],{"className":26947},[391],[238,26949,26951],{"className":26950},[395],[238,26952,26954],{"className":26953},[399],[238,26955,26957],{"className":26956,"style":6609},[403],[238,26958,26959,26962],{"style":407},[238,26960],{"className":26961,"style":412},[411],[238,26963,26965],{"className":26964},[416,417,418,419],[238,26966,2297],{"className":26967,"style":2313},[282,283,419],[238,26969,26971,26988],{"className":26970,"translate":242},[241],[238,26972,26974],{"className":26973},[246],[248,26975,26976],{"xmlns":250},[252,26977,26978,26986],{},[255,26979,26980],{},[337,26981,26982,26984],{},[258,26983,11971],{"mathvariant":816},[332,26985,22083],{},[262,26987,22086],{"encoding":264},[238,26989,26991],{"className":26990,"ariaHidden":269},[268],[238,26992,26994,26997],{"className":26993},[273],[238,26995],{"className":26996,"style":22096},[277],[238,26998,27000,27003],{"className":26999},[282],[238,27001,11971],{"className":27002,"style":11992},[282,840],[238,27004,27006],{"className":27005},[391],[238,27007,27009],{"className":27008},[395],[238,27010,27012],{"className":27011},[399],[238,27013,27015],{"className":27014,"style":22096},[403],[238,27016,27017,27020],{"style":407},[238,27018],{"className":27019,"style":412},[411],[238,27021,27023],{"className":27022},[416,417,418,419],[238,27024,22083],{"className":27025},[664,419]," 都是 ",[238,27028,27030,27049],{"className":27029,"translate":242},[241],[238,27031,27033],{"className":27032},[246],[248,27034,27035],{"xmlns":250},[252,27036,27037,27047],{},[255,27038,27039,27041,27043,27045],{},[258,27040,19971],{},[332,27042,335],{"stretchy":334},[258,27044,260],{},[332,27046,360],{"stretchy":334},[262,27048,21579],{"encoding":264},[238,27050,27052],{"className":27051,"ariaHidden":269},[268],[238,27053,27055,27058,27061,27064,27067],{"className":27054},[273],[238,27056],{"className":27057,"style":444},[277],[238,27059,19971],{"className":27060,"style":20011},[282,283],[238,27062,335],{"className":27063},[381],[238,27065,260],{"className":27066,"style":284},[282,283],[238,27068,360],{"className":27069},[462]," 上的 ",[238,27072,27074,27087],{"className":27073,"translate":242},[241],[238,27075,27077],{"className":27076},[246],[248,27078,27079],{"xmlns":250},[252,27080,27081,27085],{},[255,27082,27083],{},[258,27084,730],{},[262,27086,26785],{"encoding":264},[238,27088,27090],{"className":27089,"ariaHidden":269},[268],[238,27091,27093,27096],{"className":27092},[273],[238,27094],{"className":27095,"style":2853},[277],[238,27097,730],{"className":27098,"style":764},[282,283],"-压缩映射，即对任意 ",[238,27101,27103,27139],{"className":27102,"translate":242},[241],[238,27104,27106],{"className":27105},[246],[248,27107,27108],{"xmlns":250},[252,27109,27110,27136],{},[255,27111,27112,27118,27120,27126,27128,27130,27132,27134],{},[812,27113,27114,27116],{},[258,27115,4429],{},[738,27117,745],{},[332,27119,354],{"separator":269},[812,27121,27122,27124],{},[258,27123,4429],{},[738,27125,17558],{},[332,27127,733],{},[258,27129,19971],{},[332,27131,335],{"stretchy":334},[258,27133,260],{},[332,27135,360],{"stretchy":334},[262,27137,27138],{"encoding":264},"V_1,V_2\\in B(S)",[238,27140,27142,27243],{"className":27141,"ariaHidden":269},[268],[238,27143,27145,27148,27188,27191,27194,27234,27237,27240],{"className":27144},[273],[238,27146],{"className":27147,"style":6651},[277],[238,27149,27151,27154],{"className":27150},[282],[238,27152,4429],{"className":27153,"style":660},[282,283],[238,27155,27157],{"className":27156},[391],[238,27158,27160,27180],{"className":27159},[395,848],[238,27161,27163,27177],{"className":27162},[399],[238,27164,27166],{"className":27165,"style":1162},[403],[238,27167,27168,27171],{"style":13806},[238,27169],{"className":27170,"style":412},[411],[238,27172,27174],{"className":27173},[416,417,418,419],[238,27175,745],{"className":27176},[282,419],[238,27178,872],{"className":27179},[871],[238,27181,27183],{"className":27182},[399],[238,27184,27186],{"className":27185,"style":879},[403],[238,27187],{},[238,27189,354],{"className":27190},[451],[238,27192],{"className":27193,"style":455},[429],[238,27195,27197,27200],{"className":27196},[282],[238,27198,4429],{"className":27199,"style":660},[282,283],[238,27201,27203],{"className":27202},[391],[238,27204,27206,27226],{"className":27205},[395,848],[238,27207,27209,27223],{"className":27208},[399],[238,27210,27212],{"className":27211,"style":1162},[403],[238,27213,27214,27217],{"style":13806},[238,27215],{"className":27216,"style":412},[411],[238,27218,27220],{"className":27219},[416,417,418,419],[238,27221,17558],{"className":27222},[282,419],[238,27224,872],{"className":27225},[871],[238,27227,27229],{"className":27228},[399],[238,27230,27232],{"className":27231,"style":879},[403],[238,27233],{},[238,27235],{"className":27236,"style":430},[429],[238,27238,733],{"className":27239},[434],[238,27241],{"className":27242,"style":430},[429],[238,27244,27246,27249,27252,27255,27258],{"className":27245},[273],[238,27247],{"className":27248,"style":444},[277],[238,27250,19971],{"className":27251,"style":20011},[282,283],[238,27253,335],{"className":27254},[381],[238,27256,260],{"className":27257,"style":284},[282,283],[238,27259,360],{"className":27260},[462],[238,27262,27264],{"className":27263,"translate":242},[945],[238,27265,27267,27333],{"className":27266,"translate":242},[241],[238,27268,27270],{"className":27269},[246],[248,27271,27272],{"xmlns":250,"display":954},[252,27273,27274,27330],{},[255,27275,27276,27278,27280,27286,27288,27290,27296,27302,27304,27306,27308,27314,27316,27322,27328],{},[258,27277,15588],{"mathvariant":344},[258,27279,11971],{"mathvariant":816},[812,27281,27282,27284],{},[258,27283,4429],{},[738,27285,745],{},[332,27287,5577],{},[258,27289,11971],{"mathvariant":816},[812,27291,27292,27294],{},[258,27293,4429],{},[738,27295,17558],{},[812,27297,27298,27300],{},[258,27299,15588],{"mathvariant":344},[258,27301,1895],{"mathvariant":344},[332,27303,2023],{},[258,27305,730],{},[258,27307,15588],{"mathvariant":344},[812,27309,27310,27312],{},[258,27311,4429],{},[738,27313,745],{},[332,27315,5577],{},[812,27317,27318,27320],{},[258,27319,4429],{},[738,27321,17558],{},[812,27323,27324,27326],{},[258,27325,15588],{"mathvariant":344},[258,27327,1895],{"mathvariant":344},[258,27329,3102],{"mathvariant":344},[262,27331,27332],{"encoding":264},"\\|\\mathcal{T}V_1-\\mathcal{T}V_2\\|_\\infty\n\\le\\gamma\\|V_1-V_2\\|_\\infty.",[238,27334,27336,27397,27495,27556],{"className":27335,"ariaHidden":269},[268],[238,27337,27339,27342,27345,27348,27388,27391,27394],{"className":27338},[273],[238,27340],{"className":27341,"style":444},[277],[238,27343,15588],{"className":27344},[282],[238,27346,11971],{"className":27347,"style":11992},[282,840],[238,27349,27351,27354],{"className":27350},[282],[238,27352,4429],{"className":27353,"style":660},[282,283],[238,27355,27357],{"className":27356},[391],[238,27358,27360,27380],{"className":27359},[395,848],[238,27361,27363,27377],{"className":27362},[399],[238,27364,27366],{"className":27365,"style":1162},[403],[238,27367,27368,27371],{"style":13806},[238,27369],{"className":27370,"style":412},[411],[238,27372,27374],{"className":27373},[416,417,418,419],[238,27375,745],{"className":27376},[282,419],[238,27378,872],{"className":27379},[871],[238,27381,27383],{"className":27382},[399],[238,27384,27386],{"className":27385,"style":879},[403],[238,27387],{},[238,27389],{"className":27390,"style":660},[429],[238,27392,5577],{"className":27393},[664],[238,27395],{"className":27396,"style":660},[429],[238,27398,27400,27403,27406,27446,27486,27489,27492],{"className":27399},[273],[238,27401],{"className":27402,"style":444},[277],[238,27404,11971],{"className":27405,"style":11992},[282,840],[238,27407,27409,27412],{"className":27408},[282],[238,27410,4429],{"className":27411,"style":660},[282,283],[238,27413,27415],{"className":27414},[391],[238,27416,27418,27438],{"className":27417},[395,848],[238,27419,27421,27435],{"className":27420},[399],[238,27422,27424],{"className":27423,"style":1162},[403],[238,27425,27426,27429],{"style":13806},[238,27427],{"className":27428,"s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令 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V_1\\le V_2+c\\mathbf{1}.",[238,27910,27912,27967,27988,28043,28098],{"className":27911,"ariaHidden":269},[268],[238,27913,27915,27918,27958,27961,27964],{"className":27914},[273],[238,27916],{"className":27917,"style":833},[277],[238,27919,27921,27924],{"className":27920},[282],[238,27922,4429],{"className":27923,"style":660},[282,283],[238,27925,27927],{"className":27926},[391],[238,27928,27930,27950],{"className":27929},[395,848],[238,27931,27933,27947],{"className":27932},[399],[238,27934,27936],{"className":27935,"style":1162},[403],[238,27937,27938,27941],{"style":13806},[238,27939],{"className":27940,"style":412},[411],[238,27942,27944],{"className":27943},[416,417,418,419],[238,27945,17558],{"className":27946},[282,419],[238,27948,872],{"className":27949},[871],[238,27951,27953],{"className":27952},[399],[238,27954,27956],{"className":27955,"style":879},[403],[238,27957],{},[238,27959],{"className":27960,"style":660},[429],[238,27962,5577],{"className":27963},[664],[238,27965],{"className":27966,"style":660},[429],[238,27968,27970,27973,27976,27979,27982,27985],{"className":27969},[273],[238,27971],{"className":27972,"style":14786},[277],[238,27974,24276],{"className":27975},[282,283],[238,27977,745],{"className":27978},[282,24527],[238,27980],{"className":27981,"style":430},[429],[238,27983,2023],{"className":27984},[434],[238,27986],{"className":27987,"style":430},[429],[238,27989,27991,27994,28034,28037,28040],{"className":27990},[273],[238,27992],{"className":27993,"style":833},[277],[238,27995,27997,28000],{"className":27996},[282],[238,27998,4429],{"className":27999,"style":660},[282,283],[238,28001,28003],{"className":28002},[391],[238,28004,28006,28026],{"className":28005},[395,848],[238,28007,28009,28023],{"className":28008},[399],[238,28010,28012],{"className":28011,"style":1162},[403],[238,28013,28014,28017],{"style":13806},[238,28015],{"className":28016,"style":412},[411],[238,28018,28020],{"className":28019},[416,417,418,419],[238,28021,745],{"className":28022},[282,419],[238,28024,872],{"className":28025},[871],[238,28027,28029],{"className":28028},[399],[238,28030,28032],{"className":28031,"style":879},[403],[238,28033],{},[238,28035],{"className":28036,"style":430},[429],[238,28038,2023],{"className":28039},[434],[238,28041],{"className":28042,"style":430},[429],[238,28044,28046,28049,28089,28092,28095],{"className":28045},[273],[238,28047],{"className":28048,"style":833},[277],[238,28050,28052,28055],{"className":28051},[282],[238,28053,4429],{"className":28054,"style":660},[282,283],[238,28056,28058],{"className":28057},[391],[238,28059,28061,28081],{"className":28060},[395,848],[238,28062,28064,28078],{"className":28063},[399],[238,28065,28067],{"className":28066,"style":1162},[403],[238,28068,28069,28072],{"style":13806},[238,28070],{"className":28071,"style":412},[411],[238,28073,28075],{"className":28074},[416,417,418,419],[238,28076,17558],{"className":28077},[282,419],[238,28079,872],{"className":28080},[871],[238,28082,28084],{"className":28083},[399],[238,28085,28087],{"className":28086,"style":879},[403],[238,28088],{},[238,28090],{"className":28091,"style":660},[429],[238,28093,987],{"className":28094},[664],[238,28096],{"className":28097,"style":660},[429],[238,28099,28101,28104,28107,28110],{"className":28100},[273],[238,28102],{"className":28103,"style":2672},[277],[238,28105,24276],{"className":28106},[282,283],[238,28108,745],{"className":28109},[282,24527],[238,28111,3102],{"className":28112},[282],[11,28114,16622,28115,28119],{},[357,28116,28118],{"href":28117},"#proposition-monotonicity","命题 3.1"," 的单调性与常数平移，",[238,28121,28123],{"className":28122,"translate":242},[945],[238,28124,28126,28182],{"className":28125,"translate":242},[241],[238,28127,28129],{"className":28128},[246],[248,28130,28131],{"xmlns":250,"display":954},[252,28132,28133,28179],{},[255,28134,28135,28137,28143,28145,28147,28149,28151,28153,28155,28161,28163,28165,28171,28173,28175,28177],{},[258,28136,11971],{"mathvariant":816},[812,28138,28139,28141],{},[258,28140,4429],{},[738,28142,17558],{},[332,28144,5577],{},[258,28146,730],{},[258,28148,24276],{},[738,28150,745],{"mathvariant":24354},[332,28152,2023],{},[258,28154,11971],{"mathvariant":816},[812,28156,28157,28159],{},[258,28158,4429],{},[738,28160,745],{},[332,28162,2023],{},[258,28164,11971],{"mathvariant":816},[812,28166,28167,28169],{},[258,28168,4429],{},[738,28170,17558],{},[332,28172,987],{},[258,28174,730],{},[258,28176,24276],{},[738,28178,26056],{"mathvariant":24354},[262,28180,28181],{"encoding":264},"\\mathcal{T}V_2-\\gamma c\\mathbf{1}\n\\le\\mathcal{T}V_1\n\\le\\mathcal{T}V_2+\\gamma c\\mathbf{1}.",[238,28183,28185,28243,28267,28325,28383],{"className":28184,"ariaHidden":269},[268],[238,28186,28188,28191,28194,28234,28237,28240],{"className":28187},[273],[238,28189],{"className":28190,"style":833},[277],[238,28192,11971],{"className":28193,"style":11992},[282,840],[238,28195,28197,28200],{"className":28196},[282],[238,28198,4429],{"className":28199,"style":660},[282,283],[238,28201,28203],{"className":28202},[391],[238,28204,28206,28226],{"className":28205},[395,848],[238,28207,28209,28223],{"className":28208},[399],[238,28210,28212],{"className":28211,"style":1162},[403],[238,28213,28214,28217],{"style":13806},[238,28215],{"className":28216,"style":412},[411],[238,28218,28220],{"className":28219},[416,417,418,419],[238,28221,17558],{"className":28222},[282,419],[238,28224,872],{"className":28225},[871],[238,28227,28229],{"className":28228},[399],[238,28230,28232],{"className":28231,"style":879},[403],[238,28233],{},[238,28235],{"className":28236,"style":660},[429],[238,28238,5577],{"className":28239},[664],[238,28241],{"className":28242,"style":660},[429],[238,28244,28246,28249,28252,28255,28258,28261,28264],{"className":28245},[273],[238,28247],{"className":28248,"style":17777},[277],[238,28250,730],{"className":28251,"style":764},[282,283],[238,28253,24276],{"className":28254},[282,283],[238,28256,745],{"className":28257},[282,24527],[238,28259],{"className":28260,"style":430},[429],[238,28262,2023],{"className":28263},[434],[238,28265],{"className":28266,"style":430},[429],[238,28268,28270,28273,28276,28316,28319,28322],{"className":28269},[273],[238,28271],{"className":28272,"style":833},[277],[238,28274,11971],{"className":28275,"style":11992},[282,840],[238,28277,28279,28282],{"className":28278},[282],[238,28280,4429],{"className":28281,"style":660},[282,283],[238,28283,28285],{"className":28284},[391],[238,28286,28288,28308],{"className":28287},[395,848],[238,28289,28291,28305],{"className":28290},[399],[238,28292,28294],{"className":28293,"style":1162},[403],[238,28295,28296,28299],{"style":13806},[238,28297],{"className":28298,"style":412},[411],[238,28300,28302],{"className":28301},[416,417,418,419],[238,28303,745],{"className":28304},[282,419],[238,28306,872],{"className":28307},[871],[238,28309,28311],{"className":28310},[399],[238,28312,28314],{"className":28313,"style":879},[403],[238,28315],{},[238,28317],{"className":28318,"style":430},[429],[238,28320,2023],{"className":28321},[434],[238,28323],{"className":28324,"style":430},[429],[238,28326,28328,28331,28334,28374,28377,28380],{"className":28327},[273],[238,28329],{"className":28330,"style":833},[277],[238,28332,11971],{"className":28333,"style":11992},[282,840],[238,28335,28337,28340],{"className":28336},[282],[238,28338,4429],{"className":28339,"style":660},[282,283],[238,28341,28343],{"className":28342},[391],[238,28344,28346,28366],{"className":28345},[395,848],[238,28347,28349,28363],{"className":28348},[399],[238,28350,28352],{"className":28351,"style":1162},[403],[238,28353,28354,28357],{"style":13806},[238,28355],{"className":28356,"style":412},[411],[238,28358,28360],{"className":28359},[416,417,418,419],[238,28361,17558],{"className":28362},[282,419],[238,28364,872],{"className":28365},[871],[238,28367,28369],{"className":28368},[399],[238,28370,28372],{"className":28371,"style":879},[403],[238,28373],{},[238,28375],{"className":28376,"style":660},[429],[238,28378,987],{"className":28379},[664],[238,28381],{"className":28382,"style":660},[429],[238,28384,28386,28389,28392,28395,28398],{"className":28385},[273],[238,28387],{"className":28388,"style":17777},[277],[238,28390,730],{"className":28391,"style":764},[282,283],[238,28393,24276],{"className":28394},[282,283],[238,28396,745],{"className":28397},[282,24527],[238,28399,3102],{"className":28400},[282],[11,28402,28403,28404,941],{},"因此对一切 ",[238,28405,28407,28420],{"className":28406,"translate":242},[241],[238,28408,28410],{"className":28409},[246],[248,28411,28412],{"xmlns":250},[252,28413,28414,28418],{},[255,28415,28416],{},[258,28417,341],{},[262,28419,341],{"encoding":264},[238,28421,28423],{"className":28422,"ariaHidden":269},[268],[238,28424,28426,28429],{"className":28425},[273],[238,28427],{"className":28428,"style":489},[277],[238,28430,341],{"className":28431},[282,283],[238,28433,28435],{"className":28434,"translate":242},[945],[238,28436,28438,28536],{"className":28437,"translate":242},[241],[238,28439,28441],{"className":28440},[246],[248,28442,28443],{"xmlns":250,"display":954},[252,28444,28445,28533],{},[958,28446,28447,28507],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,28448,28449,28455],{},[966,28450,28451],{},[969,28452,28453],{"scriptlevel":740,"displaystyle":269},[255,28454],{},[966,28456,28457],{},[969,28458,28459],{"scriptlevel":740,"displaystyle":269},[255,28460,28461,28463],{},[255,28462],{},[255,28464,28465,28467,28469,28471,28477,28479,28481,28483,28485,28487,28489,28491,28497,28499,28501,28503,28505],{},[332,28466,349],{"fence":269},[332,28468,335],{"stretchy":334},[258,28470,11971],{"mathvariant":816},[812,28472,28473,28475],{},[258,28474,4429],{},[738,28476,745],{},[332,28478,360],{"stretchy":334},[332,28480,335],{"stretchy":334},[258,28482,341],{},[332,28484,360],{"stretchy":334},[332,28486,5577],{},[332,28488,335],{"stretchy":334},[258,28490,11971],{"mathvariant":816},[812,28492,28493,28495],{},[258,28494,4429],{},[738,28496,17558],{},[332,28498,360],{"stretchy":334},[332,28500,335],{"stretchy":334},[258,28502,341],{},[332,28504,360],{"stretchy":334},[332,28506,349],{"fence":269},[963,28508,28509,28515],{},[966,28510,28511],{},[969,28512,28513],{"scriptlevel":740,"displaystyle":269},[255,28514],{},[966,28516,28517],{},[969,28518,28519],{"scriptlevel":740,"displaystyle":269},[255,28520,28521,28523,28525,28527,28529,28531],{},[255,28522],{},[429,28524],{"width":7090},[332,28526,2023],{},[258,28528,730],{},[258,28530,24276],{},[258,28532,3102],{"mathvariant":344},[262,28534,28535],{"encoding":264},"\\begin{aligned}\n&\\left|(\\mathcal{T}V_1)(s)-(\\mathcal{T}V_2)(s)\\right|\\\\\n&\\quad\\le\\gamma c.\n\\end{aligned}",[238,28537,28539],{"className":28538,"ariaHidden":269},[268],[238,28540,28542,28545],{"className":28541},[273],[238,28543],{"className":28544,"style":4573},[277],[238,28546,28548],{"className":28547},[282],[238,28549,28551,28592],{"className":28550},[958],[238,28552,28554],{"className":28553},[1117],[238,28555,28557,28584],{"className":28556},[395,848],[238,28558,28560,28581],{"className":28559},[399],[238,28561,28563,28573],{"className":28562,"style":4592},[403],[238,28564,28566,28570],{"style":28565},"top:-3.6em;",[238,28567],{"className":28568,"style":28569},[411],"height:2.84em;",[238,28571],{"className":28572},[282],[238,28574,28575,28578],{"style":6033},[238,28576],{"className":28577,"style":28569},[411],[238,28579],{"className":28580},[282],[238,28582,872],{"className":28583},[871],[238,28585,28587],{"className":28586},[399],[238,28588,28590],{"className":28589,"style":4706},[403],[238,28591],{},[238,28593,28595],{"className":28594},[1422],[238,28596,28598,28787],{"className":28597},[395,848],[238,28599,28601,28784],{"className":28600},[399],[238,28602,28604,28752],{"className":28603,"style":4592},[403],[238,28605,28606,28609],{"style":4595},[238,28607],{"className":28608,"style":1134},[411],[238,28610,28612,28615,28618],{"className":28611},[282],[238,28613],{"className":28614},[282],[238,28616],{"className":28617,"style":455},[429],[238,28619,28621,28624,28627,28630,28670,28673,28676,28679,28682,28685,28688,28691,28694,28697,28737,28740,28743,28746,28749],{"className":28620},[3041],[238,28622,349],{"className":28623,"style":4791},[381,4790],[238,28625,335],{"className":28626},[381],[238,28628,11971],{"className":28629,"style":11992},[282,840],[238,28631,28633,28636],{"className":28632},[282],[238,28634,4429],{"className":28635,"style":660},[282,283],[238,28637,28639],{"className":28638},[391],[238,28640,28642,28662],{"className":28641},[395,848],[238,28643,28645,28659],{"className":28644},[399],[238,28646,28648],{"className":28647,"style":1162},[403],[238,28649,28650,28653],{"style":13806},[238,28651],{"className":28652,"style":412},[411],[238,28654,28656],{"className":28655},[416,417,418,419],[238,28657,745],{"className":28658},[282,419],[238,28660,872],{"className":28661},[871],[238,28663,28665],{"className":28664},[399],[238,28666,28668],{"className":28667,"style":879},[403],[238,28669],{},[238,28671,360],{"className":28672},[462],[238,28674,335],{"className":28675},[381],[238,28677,341],{"className":28678},[282,283],[238,28680,360],{"className":28681},[462],[238,28683],{"className":28684,"style":660},[429],[238,28686,5577],{"className":28687},[664],[238,28689],{"className":28690,"style":660},[429],[238,28692,335],{"className":28693},[381],[238,28695,11971],{"className":28696,"style":11992},[282,840],[238,28698,28700,28703],{"className":28699},[282],[238,28701,4429],{"className":28702,"style":660},[282,283],[238,28704,28706],{"className":28705},[391],[238,28707,28709,28729],{"className":28708},[395,848],[238,28710,28712,28726],{"className":28711},[399],[238,28713,28715],{"className":28714,"style":1162},[403],[238,28716,28717,28720],{"style":13806},[238,28718],{"className":28719,"style":412},[411],[238,28721,28723],{"className":28722},[416,417,418,419],[238,28724,17558],{"className":28725},[282,419],[238,28727,872],{"className":28728},[871],[238,28730,28732],{"className":28731},[399],[238,28733,28735],{"className":28734,"style":879},[403],[238,28736],{},[238,28738,360],{"className":28739},[462],[238,28741,335],{"className":28742},[381],[238,28744,341],{"className":28745},[282,283],[238,28747,360],{"className":28748},[462],[238,28750,349],{"className":28751,"style":4791},[462,4790],[238,28753,28754,28757],{"style":4643},[238,28755],{"className":28756,"style":1134},[411],[238,28758,28760,28763,28766,28769,28772,28775,28778,28781],{"className":28759},[282],[238,28761],{"className":28762},[282],[238,28764],{"className":28765,"style":7439},[429],[238,28767],{"className":28768,"style":430},[429],[238,28770,2023],{"className":28771},[434],[238,28773],{"className":28774,"style":430},[429],[238,28776,730],{"className":28777,"style":764},[282,283],[238,28779,24276],{"className":28780},[282,283],[238,28782,3102],{"className":28783},[282],[238,28785,872],{"className":28786},[871],[238,28788,28790],{"className":28789},[399],[238,28791,28793],{"className":28792,"style":4706},[403],[238,28794],{},[11,28796,26642,28797,28825],{},[238,28798,28800,28813],{"className":28799,"translate":242},[241],[238,28801,28803],{"className":28802},[246],[248,28804,28805],{"xmlns":250},[252,28806,28807,28811],{},[255,28808,28809],{},[258,28810,341],{},[262,28812,341],{"encoding":264},[238,28814,28816],{"className":28815,"ariaHidden":269},[268],[238,28817,28819,28822],{"className":28818},[273],[238,28820],{"className":28821,"style":489},[277],[238,28823,341],{"className":28824},[282,283]," 取上确界即得结论。证毕。",[11,28827,28828,28829,28857,28858,28908,28909,28938,28939,28968,28969,941,28998,29027,29028,29114],{},"压缩性的直观含义是：无论两个价值函数相差多大，经过一次贝尔曼更新后，它们的最大差异最多变成原来的 ",[238,28830,28832,28845],{"className":28831,"translate":242},[241],[238,28833,28835],{"className":28834},[246],[248,28836,28837],{"xmlns":250},[252,28838,28839,28843],{},[255,28840,28841],{},[258,28842,730],{},[262,28844,26785],{"encoding":264},[238,28846,28848],{"className":28847,"ariaHidden":269},[268],[238,28849,28851,28854],{"className":28850},[273],[238,28852],{"className":28853,"style":2853},[277],[238,28855,730],{"className":28856,"style":764},[282,283]," 倍。若 ",[238,28859,28861,28878],{"className":28860,"translate":242},[241],[238,28862,28864],{"className":28863},[246],[248,28865,28866],{"xmlns":250},[252,28867,28868,28876],{},[255,28869,28870,28872,28874],{},[258,28871,730],{},[332,28873,992],{},[738,28875,17938],{},[262,28877,17941],{"encoding":264},[238,28879,28881,28899],{"className":28880,"ariaHidden":269},[268],[238,28882,28884,28887,28890,28893,28896],{"className":28883},[273],[238,28885],{"className":28886,"style":2853},[277],[238,28888,730],{"className":28889,"style":764},[282,283],[238,28891],{"className":28892,"style":430},[429],[238,28894,992],{"className":28895},[434],[238,28897],{"className":28898,"style":430},[429],[238,28900,28902,28905],{"className":28901},[273],[238,28903],{"className":28904,"style":2672},[277],[238,28906,17938],{"className":28907},[282],"，初始最大差异为 ",[238,28910,28912,28926],{"className":28911,"translate":242},[241],[238,28913,28915],{"className":28914},[246],[248,28916,28917],{"xmlns":250},[252,28918,28919,28924],{},[255,28920,28921],{},[738,28922,28923],{},"100",[262,28925,28923],{"encoding":264},[238,28927,28929],{"className":28928,"ariaHidden":269},[268],[238,28930,28932,28935],{"className":28931},[273],[238,28933],{"className":28934,"style":2672},[277],[238,28936,28923],{"className":28937},[282],"，则一次更新后差异不超过 ",[238,28940,28942,28956],{"className":28941,"translate":242},[241],[238,28943,28945],{"className":28944},[246],[248,28946,28947],{"xmlns":250},[252,28948,28949,28954],{},[255,28950,28951],{},[738,28952,28953],{},"90",[262,28955,28953],{"encoding":264},[238,28957,28959],{"className":28958,"ariaHidden":269},[268],[238,28960,28962,28965],{"className":28961},[273],[238,28963],{"className":28964,"style":2672},[277],[238,28966,28953],{"className":28967},[282],"，两次后不超过 ",[238,28970,28972,28986],{"className":28971,"translate":242},[241],[238,28973,28975],{"className":28974},[246],[248,28976,28977],{"xmlns":250},[252,28978,28979,28984],{},[255,28980,28981],{},[738,28982,28983],{},"81",[262,28985,28983],{"encoding":264},[238,28987,28989],{"className":28988,"ariaHidden":269},[268],[238,28990,28992,28995],{"className":28991},[273],[238,28993],{"className":28994,"style":2672},[277],[238,28996,28983],{"className":28997},[282],[238,28999,29001,29014],{"className":29000,"translate":242},[241],[238,29002,29004],{"className":29003},[246],[248,29005,29006],{"xmlns":250},[252,29007,29008,29012],{},[255,29009,29010],{},[258,29011,3395],{},[262,29013,3395],{"encoding":264},[238,29015,29017],{"className":29016,"ariaHidden":269},[268],[238,29018,29020,29024],{"className":29019},[273],[238,29021],{"className":29022,"style":29023},[277],"height:0.6944em;",[238,29025,3395],{"className":29026,"style":3520},[282,283]," 次后不超过 ",[238,29029,29031,29053],{"className":29030,"translate":242},[241],[238,29032,29034],{"className":29033},[246],[248,29035,29036],{"xmlns":250},[252,29037,29038,29050],{},[255,29039,29040,29046,29048],{},[337,29041,29042,29044],{},[738,29043,17938],{},[258,29045,3395],{},[332,29047,9084],{},[738,29049,28923],{},[262,29051,29052],{"encoding":264},"0.9^k\\cdot100",[238,29054,29056,29105],{"className":29055,"ariaHidden":269},[268],[238,29057,29059,29062,29066,29096,29099,29102],{"className":29058},[273],[238,29060],{"className":29061,"style":15776},[277],[238,29063,29065],{"className":29064},[282],"0.",[238,29067,29069,29073],{"className":29068},[282],[238,29070,29072],{"className":29071},[282],"9",[238,29074,29076],{"className":29075},[391],[238,29077,29079],{"className":29078},[395],[238,29080,29082],{"className":29081},[399],[238,29083,29085],{"className":29084,"style":15776},[403],[238,29086,29087,29090],{"style":407},[238,29088],{"className":29089,"style":412},[411],[238,29091,29093],{"className":29092},[416,417,418,419],[238,29094,3395],{"className":29095,"style":3520},[282,283,419],[238,29097],{"className":29098,"style":660},[429],[238,29100,9084],{"className":29101},[664],[238,29103],{"className":29104,"style":660},[429],[238,29106,29108,29111],{"className":29107},[273],[238,29109],{"className":29110,"style":2672},[277],[238,29112,28923],{"className":29113},[282],"。这正是后文几何收敛性的来源。",[11,29116,29117,29118,29168,29169,29220,29221,29592],{},"需要强调的是，压缩性依赖于 ",[238,29119,29121,29138],{"className":29120,"translate":242},[241],[238,29122,29124],{"className":29123},[246],[248,29125,29126],{"xmlns":250},[252,29127,29128,29136],{},[255,29129,29130,29132,29134],{},[258,29131,730],{},[332,29133,1892],{},[738,29135,745],{},[262,29137,17495],{"encoding":264},[238,29139,29141,29159],{"className":29140,"ariaHidden":269},[268],[238,29142,29144,29147,29150,29153,29156],{"className":29143},[273],[238,29145],{"className":29146,"style":760},[277],[238,29148,730],{"className":29149,"style":764},[282,283],[238,29151],{"className":29152,"style":430},[429],[238,29154,1892],{"className":29155},[434],[238,29157],{"className":29158,"style":430},[429],[238,29160,29162,29165],{"className":29161},[273],[238,29163],{"className":29164,"style":2672},[277],[238,29166,745],{"className":29167},[282],"。当 ",[238,29170,29172,29190],{"className":29171,"translate":242},[241],[238,29173,29175],{"className":29174},[246],[248,29176,29177],{"xmlns":250},[252,29178,29179,29187],{},[255,29180,29181,29183,29185],{},[258,29182,730],{},[332,29184,992],{},[738,29186,745],{},[262,29188,29189],{"encoding":264},"\\gamma=1",[238,29191,29193,29211],{"className":29192,"ariaHidden":269},[268],[238,29194,29196,29199,29202,29205,29208],{"className":29195},[273],[238,29197],{"className":29198,"style":2853},[277],[238,29200,730],{"className":29201,"style":764},[282,283],[238,29203],{"className":29204,"style":430},[429],[238,29206,992],{"className":29207},[434],[238,29209],{"className":29210,"style":430},[429],[238,29212,29214,29217],{"className":29213},[273],[238,29215],{"className":29216,"style":2672},[277],[238,29218,745],{"className":29219},[282]," 时，贝尔曼算子通常只是非扩张的，即只能得到 ",[238,29222,29224,29286],{"className":29223,"translate":242},[241],[238,29225,29227],{"className":29226},[246],[248,29228,29229],{"xmlns":250},[252,29230,29231,29283],{},[255,29232,29233,29235,29237,29243,29245,29247,29253,29259,29261,29263,29269,29271,29277],{},[258,29234,15588],{"mathvariant":344},[258,29236,11971],{"mathvariant":816},[812,29238,29239,29241],{},[258,29240,4429],{},[738,29242,745],{},[332,29244,5577],{},[258,29246,11971],{"mathvariant":816},[812,29248,29249,29251],{},[258,29250,4429],{},[738,29252,17558],{},[812,29254,29255,29257],{},[258,29256,15588],{"mathvariant":344},[258,29258,1895],{"mathvariant":344},[332,29260,2023],{},[258,29262,15588],{"mathvariant":344},[812,29264,29265,29267],{},[258,29266,4429],{},[738,29268,745],{},[332,29270,5577],{},[812,29272,29273,29275],{},[258,29274,4429],{},[738,29276,17558],{},[812,29278,29279,29281],{},[258,29280,15588],{"mathvariant":344},[258,29282,1895],{"mathvariant":344},[262,29284,29285],{"encoding":264},"\\|\\mathcal{T}V_1-\\mathcal{T}V_2\\|_\\infty\\le\\|V_1-V_2\\|_\\infty",[238,29287,29289,29350,29448,29506],{"className":29288,"ariaHidden":269},[268],[238,29290,29292,29295,29298,29301,29341,29344,29347],{"className":29291},[273],[238,29293],{"className":29294,"style":444},[277],[238,29296,15588],{"className":29297},[282],[238,29299,11971],{"className":29300,"style":11992},[282,840],[238,29302,29304,29307],{"className":29303},[282],[238,29305,4429],{"className":29306,"style":660},[282,283],[238,29308,29310],{"className":29309},[391],[238,29311,29313,29333],{"className":29312},[395,848],[238,29314,29316,29330],{"className":29315},[399],[238,29317,29319],{"className":29318,"style":1162},[403],[238,29320,29321,29324],{"style":13806},[238,29322],{"className":29323,"style":412},[411],[238,29325,29327],{"className":29326},[416,417,418,419],[238,29328,745],{"className":29329},[282,419],[238,29331,872],{"className":29332},[871],[238,29334,29336],{"className":29335},[399],[238,29337,29339],{"className":29338,"style":879},[403],[238,29340],{},[238,29342],{"className":29343,"style":660},[429],[238,29345,5577],{"className":29346},[664],[238,29348],{"className":29349,"style":660},[429],[238,29351,29353,29356,29359,29399,29439,29442,29445],{"className":29352},[273],[238,29354],{"className":29355,"style":444},[277],[238,29357,11971],{"className":29358,"style":11992},[282,840],[238,29360,29362,29365],{"className":29361},[282],[238,29363,4429],{"className":29364,"style":660},[282,283],[238,29366,29368],{"className":29367},[391],[238,29369,29371,29391],{"className":29370},[395,848],[238,29372,29374,29388],{"className":29373},[399],[238,29375,29377],{"className":29376,"style":1162},[403],[238,29378,29379,29382],{"style":13806},[238,29380],{"className":29381,"style":412},[411],[238,29383,29385],{"className":29384},[416,417,418,419],[238,29386,17558],{"className":29387},[282,419],[238,29389,872],{"className":29390},[871],[238,29392,29394],{"className":29393},[399],[238,29395,29397],{"className":29396,"style":879},[403],[238,29398],{},[238,29400,29402,29405],{"className":29401},[282],[238,29403,15588],{"className":29404},[282],[238,29406,29408],{"className":29407},[391],[238,29409,29411,29431],{"className":29410},[395,848],[238,29412,29414,29428],{"className":29413},[399],[238,29415,29417],{"className":29416,"style":1927},[403],[238,29418,29419,29422],{"style":1165},[238,29420],{"className":29421,"style":412},[411],[238,29423,29425],{"className":29424},[416,417,418,419],[238,29426,1895],{"className":29427},[282,419],[238,29429,872],{"className":29430},[871],[238,29432,29434],{"className":29433},[399],[238,29435,29437],{"className":29436,"style":879},[403],[238,29438],{},[238,29440],{"className":29441,"style":430},[429],[238,29443,2023],{"className":29444},[434],[238,29446],{"className":29447,"style":430},[429],[238,29449,29451,29454,29457,29497,29500,29503],{"className":29450},[273],[238,29452],{"className":29453,"style":444},[277],[238,29455,15588],{"className":29456},[282],[238,29458,29460,29463],{"className":29459},[282],[238,29461,4429],{"className":29462,"style":660},[282,283],[238,29464,29466],{"className":29465},[391],[238,29467,29469,29489],{"className":29468},[395,848],[238,29470,29472,29486],{"className":29471},[399],[238,29473,29475],{"className":29474,"style":1162},[403],[238,29476,29477,29480],{"style":13806},[238,29478],{"className":29479,"style":412},[411],[238,29481,29483],{"className":29482},[416,417,418,419],[238,29484,745],{"className":29485},[282,419],[238,29487,872],{"className":29488},[871],[238,29490,29492],{"className":29491},[399],[238,29493,29495],{"className":29494,"style":879},[403],[238,29496],{},[238,29498],{"className":29499,"style":660},[429],[238,29501,5577],{"className":29502},[664],[238,29504],{"className":29505,"style":660},[429],[238,29507,29509,29512,29552],{"className":29508},[273],[238,29510],{"className":29511,"style":444},[277],[238,29513,29515,29518],{"className":29514},[282],[238,29516,4429],{"className":29517,"style":660},[282,283],[238,29519,29521],{"className":29520},[391],[238,29522,29524,29544],{"className":29523},[395,848],[238,29525,29527,29541],{"className":29526},[399],[238,29528,29530],{"className":29529,"style":1162},[403],[238,29531,29532,29535],{"style":13806},[238,29533],{"className":29534,"style":412},[411],[238,29536,29538],{"className":29537},[416,417,418,419],[238,29539,17558],{"className":29540},[282,419],[238,29542,872],{"className":29543},[871],[238,29545,29547],{"className":29546},[399],[238,29548,29550],{"className":29549,"style":879},[403],[238,29551],{},[238,29553,29555,29558],{"className":29554},[282],[238,29556,15588],{"className":29557},[282],[238,29559,29561],{"className":29560},[391],[238,29562,29564,29584],{"className":29563},[395,848],[238,29565,29567,29581],{"className":29566},[399],[238,29568,29570],{"className":29569,"style":1927},[403],[238,29571,29572,29575],{"style":1165},[238,29573],{"className":29574,"style":412},[411],[238,29576,29578],{"className":29577},[416,417,418,419],[238,29579,1895],{"className":29580},[282,419],[238,29582,872],{"className":29583},[871],[238,29585,29587],{"className":29586},[399],[238,29588,29590],{"className":29589,"style":879},[403],[238,29591],{},"，不能保证距离严格缩小。第 5.4 节会给出反例。",[225,29594,29596],{"id":29595},"_34-banach-不动点定理","3.4 Banach 不动点定理",[230,29598,29601,29871,30001,30369],{"id":29599,"title":29600},"theorem-banach","定理 3.3（Banach 不动点定理）",[11,29602,236,29603,29657,29658,29731,29732,29805,29806,941],{},[238,29604,29606,29630],{"className":29605,"translate":242},[241],[238,29607,29609],{"className":29608},[246],[248,29610,29611],{"xmlns":250},[252,29612,29613,29627],{},[255,29614,29615,29617,29620,29622,29625],{},[332,29616,335],{"stretchy":334},[258,29618,29619],{},"X",[332,29621,354],{"separator":269},[258,29623,29624],{},"d",[332,29626,360],{"stretchy":334},[262,29628,29629],{"encoding":264},"(X,d)",[238,29631,29633],{"className":29632,"ariaHidden":269},[268],[238,29634,29636,29639,29642,29645,29648,29651,29654],{"className":29635},[273],[238,29637],{"className":29638,"style":444},[277],[238,29640,335],{"className":29641},[381],[238,29643,29619],{"className":29644,"style":14619},[282,283],[238,29646,354],{"className":29647},[451],[238,29649],{"className":29650,"style":455},[429],[238,29652,29624],{"className":29653},[282,283],[238,29655,360],{"className":29656},[462]," 是完备度量空间，",[238,29659,29661,29683],{"className":29660,"translate":242},[241],[238,29662,29664],{"className":29663},[246],[248,29665,29666],{"xmlns":250},[252,29667,29668,29680],{},[255,29669,29670,29672,29674,29676,29678],{},[258,29671,11971],{},[332,29673,603],{},[258,29675,29619],{},[332,29677,617],{},[258,29679,29619],{},[262,29681,29682],{"encoding":264},"T:X\\to X",[238,29684,29686,29704,29722],{"className":29685,"ariaHidden":269},[268],[238,29687,29689,29692,29695,29698,29701],{"className":29688},[273],[238,29690],{"className":29691,"style":278},[277],[238,29693,11971],{"className":29694,"style":377},[282,283],[238,29696],{"className":29697,"style":430},[429],[238,29699,603],{"className":29700},[434],[238,29702],{"className":29703,"style":430},[429],[238,29705,29707,29710,29713,29716,29719],{"className":29706},[273],[238,29708],{"className":29709,"style":278},[277],[238,29711,29619],{"className":29712,"style":14619},[282,283],[238,29714],{"className":29715,"style":430},[429],[238,29717,617],{"className":29718},[434],[238,29720],{"className":29721,"style":430},[429],[238,29723,29725,29728],{"className":29724},[273],[238,29726],{"className":29727,"style":278},[277],[238,29729,29619],{"className":29730,"style":14619},[282,283]," 满足：存在 ",[238,29733,29735,29760],{"className":29734,"translate":242},[241],[238,29736,29738],{"className":29737},[246],[248,29739,29740],{"xmlns":250},[252,29741,29742,29758],{},[255,29743,29744,29746,29748,29750,29752,29754,29756],{},[258,29745,730],{},[332,29747,733],{},[332,29749,736],{"stretchy":334},[738,29751,740],{},[332,29753,354],{"separator":269},[738,29755,745],{},[332,29757,360],{"stretchy":334},[262,29759,750],{"encoding":264},[238,29761,29763,29781],{"className":29762,"ariaHidden":269},[268],[238,29764,29766,29769,29772,29775,29778],{"className":29765},[273],[238,29767],{"className":29768,"style":760},[277],[238,29770,730],{"className":29771,"style":764},[282,283],[238,29773],{"className":29774,"style":430},[429],[238,29776,733],{"className":29777},[434],[238,29779],{"className":29780,"style":430},[429],[238,29782,29784,29787,29790,29793,29796,29799,29802],{"className":29783},[273],[238,29785],{"className":29786,"style":444},[277],[238,29788,736],{"className":29789},[381],[238,29791,740],{"className":29792},[282],[238,29794,354],{"className":29795},[451],[238,29797],{"className":29798,"style":455},[429],[238,29800,745],{"className":29801},[282],[238,29803,360],{"className":29804},[462],"，使得对任意 ",[238,29807,29809,29832],{"className":29808,"translate":242},[241],[238,29810,29812],{"className":29811},[246],[248,29813,29814],{"xmlns":250},[252,29815,29816,29829],{},[255,29817,29818,29820,29822,29825,29827],{},[258,29819,1954],{},[332,29821,354],{"separator":269},[258,29823,29824],{},"y",[332,29826,733],{},[258,29828,29619],{},[262,29830,29831],{"encoding":264},"x,y\\in X",[238,29833,29835,29862],{"className":29834,"ariaHidden":269},[268],[238,29836,29838,29841,29844,29847,29850,29853,29856,29859],{"className":29837},[273],[238,29839],{"className":29840,"style":760},[277],[238,29842,1954],{"className":29843},[282,283],[238,29845,354],{"className":29846},[451],[238,29848],{"className":29849,"style":455},[429],[238,29851,29824],{"className":29852,"style":2313},[282,283],[238,29854],{"className":29855,"style":430},[429],[238,29857,733],{"className":29858},[434],[238,29860],{"className":29861,"style":430},[429],[238,29863,29865,29868],{"className":29864},[273],[238,29866],{"className":29867,"style":278},[277],[238,29869,29619],{"className":29870,"style":14619},[282,283],[238,29872,29874],{"className":29873,"translate":242},[945],[238,29875,29877,29923],{"className":29876,"translate":242},[241],[238,29878,29880],{"className":29879},[246],[248,29881,29882],{"xmlns":250,"display":954},[252,29883,29884,29920],{},[255,29885,29886,29888,29890,29892,29894,29896,29898,29900,29902,29904,29906,29908,29910,29912,29914,29916,29918],{},[258,29887,29624],{},[332,29889,335],{"stretchy":334},[258,29891,11971],{},[258,29893,1954],{},[332,29895,354],{"separator":269},[258,29897,11971],{},[258,29899,29824],{},[332,29901,360],{"stretchy":334},[332,29903,2023],{},[258,29905,730],{},[258,29907,29624],{},[332,29909,335],{"stretchy":334},[258,29911,1954],{},[332,29913,354],{"separator":269},[258,29915,29824],{},[332,29917,360],{"stretchy":334},[258,29919,3102],{"mathvariant":344},[262,29921,29922],{"encoding":264},"d(Tx,Ty)\\le\\gamma d(x,y).",[238,29924,29926,29968],{"className":29925,"ariaHidden":269},[268],[238,29927,29929,29932,29935,29938,29941,29944,29947,29950,29953,29956,29959,29962,29965],{"className":29928},[273],[238,29930],{"className":29931,"style":444},[277],[238,29933,29624],{"className":29934},[282,283],[238,29936,335],{"className":29937},[381],[238,29939,11971],{"className":29940,"style":377},[282,283],[238,29942,1954],{"className":29943},[282,283],[238,29945,354],{"className":29946},[451],[238,29948],{"className":29949,"style":455},[429],[238,29951,11971],{"className":29952,"style":377},[282,283],[238,29954,29824],{"className":29955,"style":2313},[282,283],[238,29957,360],{"className":29958},[462],[238,29960],{"className":29961,"style":430},[429],[238,29963,2023],{"className":29964},[434],[238,29966],{"className":29967,"style":430},[429],[238,29969,29971,29974,29977,29980,29983,29986,29989,29992,29995,29998],{"className":29970},[273],[238,29972],{"className":29973,"style":444},[277],[238,29975,730],{"className":29976,"style":764},[282,283],[238,29978,29624],{"className":29979},[282,283],[238,29981,335],{"className":29982},[381],[238,29984,1954],{"className":29985},[282,283],[238,29987,354],{"className":29988},[451],[238,29990],{"className":29991,"style":455},[429],[238,29993,29824],{"className":29994,"style":2313},[282,283],[238,29996,360],{"className":29997},[462],[238,29999,3102],{"className":30000},[282],[11,30002,14372,30003,30031,30032,30060,30061,30120,30121,30214,30215,30368],{},[238,30004,30006,30019],{"className":30005,"translate":242},[241],[238,30007,30009],{"className":30008},[246],[248,30010,30011],{"xmlns":250},[252,30012,30013,30017],{},[255,30014,30015],{},[258,30016,11971],{},[262,30018,11971],{"encoding":264},[238,30020,30022],{"className":30021,"ariaHidden":269},[268],[238,30023,30025,30028],{"className":30024},[273],[238,30026],{"className":30027,"style":278},[277],[238,30029,11971],{"className":30030,"style":377},[282,283]," 在 ",[238,30033,30035,30048],{"className":30034,"translate":242},[241],[238,30036,30038],{"className":30037},[246],[248,30039,30040],{"xmlns":250},[252,30041,30042,30046],{},[255,30043,30044],{},[258,30045,29619],{},[262,30047,29619],{"encoding":264},[238,30049,30051],{"className":30050,"ariaHidden":269},[268],[238,30052,30054,30057],{"className":30053},[273],[238,30055],{"className":30056,"style":278},[277],[238,30058,29619],{"className":30059,"style":14619},[282,283]," 中存在唯一不动点 ",[238,30062,30064,30082],{"className":30063,"translate":242},[241],[238,30065,30067],{"className":30066},[246],[248,30068,30069],{"xmlns":250},[252,30070,30071,30079],{},[255,30072,30073],{},[337,30074,30075,30077],{},[258,30076,1954],{},[332,30078,22083],{},[262,30080,30081],{"encoding":264},"x^*",[238,30083,30085],{"className":30084,"ariaHidden":269},[268],[238,30086,30088,30091],{"className":30087},[273],[238,30089],{"className":30090,"style":22096},[277],[238,30092,30094,30097],{"className":30093},[282],[238,30095,1954],{"className":30096},[282,283],[238,30098,30100],{"className":30099},[391],[238,30101,30103],{"className":30102},[395],[238,30104,30106],{"className":30105},[399],[238,30107,30109],{"className":30108,"style":22096},[403],[238,30110,30111,30114],{"style":407},[238,30112],{"className":30113,"style":412},[411],[238,30115,30117],{"className":30116},[416,417,418,419],[238,30118,22083],{"className":30119},[664,419],"。此外，对任意初始点 ",[238,30122,30124,30146],{"className":30123,"translate":242},[241],[238,30125,30127],{"className":30126},[246],[248,30128,30129],{"xmlns":250},[252,30130,30131,30143],{},[255,30132,30133,30139,30141],{},[812,30134,30135,30137],{},[258,30136,1954],{},[738,30138,740],{},[332,30140,733],{},[258,30142,29619],{},[262,30144,30145],{"encoding":264},"x_0\\in X",[238,30147,30149,30205],{"className":30148,"ariaHidden":269},[268],[238,30150,30152,30156,30196,30199,30202],{"className":30151},[273],[238,30153],{"className":30154,"style":30155},[277],"height:0.6891em;vertical-align:-0.15em;",[238,30157,30159,30162],{"className":30158},[282],[238,30160,1954],{"className":30161},[282,283],[238,30163,30165],{"className":30164},[391],[238,30166,30168,30188],{"className":30167},[395,848],[238,30169,30171,30185],{"className":30170},[399],[238,30172,30174],{"className":30173,"style":1162},[403],[238,30175,30176,30179],{"style":1165},[238,30177],{"className":30178,"style":412},[411],[238,30180,30182],{"className":30181},[416,417,418,419],[238,30183,740],{"className":30184},[282,419],[238,30186,872],{"className":30187},[871],[238,30189,30191],{"className":30190},[399],[238,30192,30194],{"className":30193,"style":879},[403],[238,30195],{},[238,30197],{"className":30198,"style":430},[429],[238,30200,733],{"className":30201},[434],[238,30203],{"className":30204,"style":430},[429],[238,30206,30208,30211],{"className":30207},[273],[238,30209],{"className":30210,"style":278},[277],[238,30212,29619],{"className":30213,"style":14619},[282,283],"，迭代 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满足",[238,30370,30372],{"className":30371,"translate":242},[945],[238,30373,30375,30437],{"className":30374,"translate":242},[241],[238,30376,30378],{"className":30377},[246],[248,30379,30380],{"xmlns":250,"display":954},[252,30381,30382,30434],{},[255,30383,30384,30386,30388,30394,30396,30402,30404,30406,30412,30414,30416,30422,30424,30430,30432],{},[258,30385,29624],{},[332,30387,335],{"stretchy":334},[812,30389,30390,30392],{},[258,30391,1954],{},[258,30393,3395],{},[332,30395,354],{"separator":269},[337,30397,30398,30400],{},[258,30399,1954],{},[332,30401,22083],{},[332,30403,360],{"stretchy":334},[332,30405,2023],{},[337,30407,30408,30410],{},[258,30409,730],{},[258,30411,3395],{},[258,30413,29624],{},[332,30415,335],{"stretchy":334},[812,30417,30418,30420],{},[258,30419,1954],{},[738,30421,740],{},[332,30423,354],{"separator":269},[337,30425,30426,30428],{},[258,30427,1954],{},[332,30429,22083],{},[332,30431,360],{"stretchy":334},[258,30433,3102],{"mathvariant":344},[262,30435,30436],{"encoding":264},"d(x_k,x^*)\\le\\gamma^k 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先证唯一性。若 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都是不动点，则",[238,30787,30789],{"className":30788,"translate":242},[945],[238,30790,30792,30876],{"className":30791,"translate":242},[241],[238,30793,30795],{"className":30794},[246],[248,30796,30797],{"xmlns":250,"display":954},[252,30798,30799,30873],{},[255,30800,30801,30803,30805,30811,30813,30819,30821,30823,30825,30827,30829,30835,30837,30839,30845,30847,30849,30851,30853,30855,30861,30863,30869,30871],{},[258,30802,29624],{},[332,30804,335],{"stretchy":334},[337,30806,30807,30809],{},[258,30808,1954],{},[332,30810,22083],{},[332,30812,354],{"separator":269},[337,30814,30815,30817],{},[258,30816,29824],{},[332,30818,22083],{},[332,30820,360],{"stretchy":334},[332,30822,992],{},[258,30824,29624],{},[332,30826,335],{"stretchy":334},[258,30828,11971],{},[337,30830,30831,30833],{},[258,30832,1954],{},[332,30834,22083],{},[332,30836,354],{"separator":269},[258,30838,11971],{},[337,30840,30841,30843],{},[258,30842,29824],{},[332,30844,22083],{},[332,30846,360],{"stretchy":334},[332,30848,2023],{},[258,30850,730],{},[258,30852,29624],{},[332,30854,335],{"stretchy":334},[337,30856,30857,30859],{},[258,30858,1954],{},[332,30860,22083],{},[332,30862,354],{"separator":269},[337,30864,30865,30867],{},[258,30866,29824],{},[332,30868,22083],{},[332,30870,360],{"stretchy":334},[258,30872,3102],{"mathvariant":344},[262,30874,30875],{"encoding":264},"d(x^*,y^*)=d(Tx^*,Ty^*)\\le\\gamma d(x^*,y^*).",[238,30877,30879,30967,31061],{"className":30878,"ariaHidden":269},[268],[238,30880,30882,30885,30888,30891,30920,30923,30926,30955,30958,30961,30964],{"className":30881},[273],[238,30883],{"className":30884,"style":444},[277],[238,30886,29624],{"className":30887},[282,283],[238,30889,335],{"className":30890},[381],[238,30892,30894,30897],{"className":30893},[282],[238,30895,1954],{"className":30896},[282,283],[238,30898,30900],{"className":30899},[391],[238,30901,30903],{"className":30902},[395],[238,30904,30906],{"className":30905},[399],[238,30907,30909],{"className":30908,"style":22441},[403],[238,30910,30911,30914],{"style":1226},[238,30912],{"className":30913,"style":412},[411],[238,30915,30917],{"className":30916},[416,417,418,419],[238,30918,22083],{"className":30919},[664,419],[238,30921,354],{"className":30922},[451],[238,30924],{"className":30925,"style":455},[429],[238,30927,30929,30932],{"className":30928},[282],[238,30930,29824],{"className":30931,"style":2313},[282,283],[238,30933,30935],{"className":30934},[391],[238,30936,30938],{"className":30937},[395],[238,30939,30941],{"className":30940},[399],[238,30942,30944],{"className":30943,"style":22441},[403],[238,30945,30946,30949],{"style":1226},[238,30947],{"className":30948,"style":412},[411],[238,30950,30952],{"className":30951},[416,417,418,419],[238,30953,22083],{"className":30954},[664,419],[238,30956,360],{"className":30957},[462],[238,30959],{"className":30960,"style":430},[429],[238,30962,992],{"className":30963},[434],[238,30965],{"className":30966,"style":430},[429],[238,30968,30970,30973,30976,30979,30982,31011,31014,31017,31020,31049,31052,31055,31058],{"className":30969},[273],[238,30971],{"className":30972,"style":444},[277],[238,30974,29624],{"className":30975},[282,283],[238,30977,335],{"className":30978},[381],[238,30980,11971],{"className":30981,"style":377},[282,283],[238,30983,30985,30988],{"className":30984},[282],[238,30986,1954],{"className":30987},[282,283],[238,30989,30991],{"className":30990},[391],[238,30992,30994],{"className":30993},[395],[238,30995,30997],{"className":30996},[399],[238,30998,31000],{"className":30999,"style":22441},[403],[238,31001,31002,31005],{"style":1226},[238,31003],{"className":31004,"style":412},[411],[238,31006,31008],{"className":31007},[416,417,418,419],[238,31009,22083],{"className":31010},[664,419],[238,31012,354],{"className":31013},[451],[238,31015],{"className":31016,"style":455},[429],[238,31018,11971],{"className":31019,"style":377},[282,283],[238,31021,31023,31026],{"className":31022},[282],[238,31024,29824],{"className":31025,"style":2313},[282,283],[238,31027,31029],{"className":31028},[391],[238,31030,31032],{"className":31031},[395],[238,31033,31035],{"className":31034},[399],[238,31036,31038],{"className":31037,"style":22441},[403],[238,31039,31040,31043],{"style":1226},[238,31041],{"className":31042,"style":412},[411],[238,31044,31046],{"className":31045},[416,417,418,419],[238,31047,22083],{"className":31048},[664,419],[238,31050,360],{"className":31051},[462],[238,31053],{"className":31054,"style":430},[429],[238,31056,2023],{"className":31057},[434],[238,31059],{"className":31060,"style":430},[429],[238,31062,31064,31067,31070,31073,31076,31105,31108,31111,31140,31143],{"className":31063},[273],[238,31065],{"className":31066,"style":444},[277],[238,31068,730],{"className":31069,"style":764},[282,283],[238,31071,29624],{"className":31072},[282,283],[238,31074,335],{"className":31075},[381],[238,31077,31079,31082],{"className":31078},[282],[238,31080,1954],{"className":31081},[282,283],[238,31083,31085],{"className":31084},[391],[238,31086,31088],{"className":31087},[395],[238,31089,31091],{"className":31090},[399],[238,31092,31094],{"className":31093,"style":22441},[403],[238,31095,31096,31099],{"style":1226},[238,31097],{"className":31098,"style":412},[411],[238,31100,31102],{"className":31101},[416,417,418,419],[238,31103,22083],{"className":31104},[664,419],[238,31106,354],{"className":31107},[451],[238,31109],{"className":31110,"style":455},[429],[238,31112,31114,31117],{"className":31113},[282],[238,31115,29824],{"className":31116,"style":2313},[282,283],[238,31118,31120],{"className":31119},[391],[238,31121,31123],{"className":31122},[395],[238,31124,31126],{"className":31125},[399],[238,31127,31129],{"className":31128,"style":22441},[403],[238,31130,31131,31134],{"style":1226},[238,31132],{"className":31133,"style":412},[411],[238,31135,31137],{"className":31136},[416,417,418,419],[238,31138,22083],{"className":31139},[664,419],[238,31141,360],{"className":31142},[462],[238,31144,3102],{"className":31145},[282],[11,31147,20770,31148,31198,31199,31338,31339,1806],{},[238,31149,31151,31168],{"className":31150,"translate":242},[241],[238,31152,31154],{"className":31153},[246],[248,31155,31156],{"xmlns":250},[252,31157,31158,31166],{},[255,31159,31160,31162,31164],{},[258,31161,730],{},[332,31163,1892],{},[738,31165,745],{},[262,31167,17495],{"encoding":2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",[238,31200,31202,31238],{"className":31201,"translate":242},[241],[238,31203,31205],{"className":31204},[246],[248,31206,31207],{"xmlns":250},[252,31208,31209,31235],{},[255,31210,31211,31213,31215,31221,31223,31229,31231,31233],{},[258,31212,29624],{},[332,31214,335],{"stretchy":334},[337,31216,31217,31219],{},[258,31218,1954],{},[332,31220,22083],{},[332,31222,354],{"separator":269},[337,31224,31225,31227],{},[258,31226,29824],{},[332,31228,22083],{},[332,31230,360],{"stretchy":334},[332,31232,992],{},[738,31234,740],{},[262,31236,31237],{"encoding":264},"d(x^*,y^*)=0",[238,31239,31241,31329],{"className":31240,"ariaHidden":269},[268],[238,31242,31244,31247,31250,31253,31282,31285,31288,31317,31320,31323,31326],{"className":31243},[273],[238,31245],{"className":31246,"style":444},[277],[238,31248,29624],{"className":31249},[282,283],[238,31251,335],{"className":31252},[381],[238,31254,31256,31259],{"className":31255},[282],[238,31257,1954],{"className":31258},[282,283],[238,31260,31262],{"className":31261},[391],[238,31263,31265],{"className":31264},[395],[238,31266,31268],{"className":31267},[399],[238,31269,31271],{"className":31270,"style":22096},[403],[238,31272,31273,31276],{"style":407},[238,31274],{"className":31275,"style":412},[411],[238,31277,31279],{"className":31278},[416,417,418,419],[238,31280,22083],{"className":31281},[664,419],[238,31283,354],{"className":31284},[451],[238,31286],{"className":31287,"style":455},[429],[238,31289,31291,31294],{"className":31290},[282],[238,31292,29824],{"className":31293,"style":2313},[282,283],[238,31295,31297],{"className":31296},[391],[238,31298,31300],{"className":31299},[395],[238,31301,31303],{"className":31302},[399],[238,31304,31306],{"className":31305,"style":22096},[403],[238,31307,31308,31311],{"style":407},[238,31309],{"className":31310,"style":412},[411],[238,31312,31314],{"className":31313},[416,417,418,419],[238,31315,22083],{"className":31316},[664,419],[238,31318,360],{"className":31319},[462],[238,31321],{"className":31322,"style":430},[429],[238,31324,992],{"className":31325},[434],[238,31327],{"className":31328,"style":430},[429],[238,31330,31332,31335],{"className":31331},[273],[238,31333],{"className":31334,"style":2672},[277],[238,31336,740],{"className":31337},[282],"，即 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",[238,31454,31456,31477],{"className":31455,"translate":242},[241],[238,31457,31459],{"className":31458},[246],[248,31460,31461],{"xmlns":250},[252,31462,31463,31475],{},[255,31464,31465,31471,31473],{},[812,31466,31467,31469],{},[258,31468,1954],{},[738,31470,740],{},[332,31472,733],{},[258,31474,29619],{},[262,31476,30145],{"encoding":264},[238,31478,31480,31535],{"className":31479,"ariaHidden":269},[268],[238,31481,31483,31486,31526,31529,31532],{"className":31482},[273],[238,31484],{"className":31485,"style":30155},[277],[238,31487,31489,31492],{"className":31488},[282],[238,31490,1954],{"className":31491},[282,283],[238,31493,31495],{"className":31494},[391],[238,31496,31498,31518],{"className":31497},[395,848],[238,31499,31501,31515],{"className":31500},[399],[238,31502,31504],{"className":31503,"style":1162},[403],[238,31505,31506,31509],{"style":1165},[238,31507],{"className":31508,"style":412},[411],[238,31510,31512],{"className":31511},[416,417,418,419],[238,31513,740],{"className":31514},[282,419],[238,31516,872],{"className":31517},[871],[238,31519,31521],{"className":31520},[399],[238,31522,31524],{"className":31523,"style":879},[403],[238,31525],{},[238,31527],{"className":31528,"style":430},[429],[238,31530,733],{"className":31531},[434],[238,31533],{"className":31534,"style":430},[429],[238,31536,31538,31541],{"className":31537},[273],[238,31539],{"className":31540,"style":278},[277],[238,31542,29619],{"className":31543,"style":14619},[282,283],"，令 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是 Cauchy 列。由 ",[238,32845,32847,32860],{"className":32846,"translate":242},[241],[238,32848,32850],{"className":32849},[246],[248,32851,32852],{"xmlns":250},[252,32853,32854,32858],{},[255,32855,32856],{},[258,32857,29619],{},[262,32859,29619],{"encoding":264},[238,32861,32863],{"className":32862,"ariaHidden":269},[268],[238,32864,32866,32869],{"className":32865},[273],[238,32867],{"className":32868,"style":278},[277],[238,32870,29619],{"className":32871,"style":14619},[282,283]," 完备，存在 ",[238,32874,32876,32898],{"className":32875,"translate":242},[241],[238,32877,32879],{"className":32878},[246],[248,32880,32881],{"xmlns":250},[252,32882,32883,32895],{},[255,32884,32885,32891,32893],{},[337,32886,32887,32889],{},[258,32888,1954],{},[332,32890,22083],{},[332,32892,733],{},[258,32894,29619],{},[262,32896,32897],{"encoding":264},"x^*\\in X",[238,32899,32901,32946],{"className":32900,"ariaHidden":269},[268],[238,32902,32904,32908,32937,32940,32943],{"className":32903},[273],[238,32905],{"className":32906,"style":32907},[277],"height:0.7278em;vertical-align:-0.0391em;",[238,32909,32911,32914],{"className":32910},[282],[238,32912,1954],{"className":32913},[282,283],[238,32915,32917],{"className":32916},[391],[238,32918,32920],{"className":32919},[395],[238,32921,32923],{"className":32922},[399],[238,32924,32926],{"className":32925,"style":22096},[403],[238,32927,32928,32931],{"style":407},[238,32929],{"className":32930,"style":412},[411],[238,32932,32934],{"className":32933},[416,417,418,419],[238,32935,22083],{"className":32936},[664,419],[238,32938],{"className":32939,"style":430},[429],[238,32941,733],{"className":32942},[434],[238,32944],{"className":32945,"style":430},[429],[238,32947,32949,32952],{"className":32948},[273],[238,32950],{"className":32951,"style":278},[277],[238,32953,29619],{"className":32954,"style":14619},[282,283],[238,32956,32958,32984],{"className":32957,"translate":242},[241],[238,32959,32961],{"className":32960},[246],[248,32962,32963],{"xmlns":250},[252,32964,32965,32981],{},[255,32966,32967,32973,32975],{},[812,32968,32969,32971],{},[258,32970,1954],{},[258,32972,13161],{},[332,32974,617],{},[337,32976,32977,32979],{},[258,32978,1954],{},[332,32980,22083],{},[262,32982,32983],{"encoding":264},"x_n\\to x^*",[238,32985,32987,33042],{"className":32986,"ariaHidden":269},[268],[238,32988,32990,32993,33033,33036,33039],{"className":32989},[273],[238,32991],{"className":32992,"style":3115},[277],[238,32994,32996,32999],{"className":32995},[282],[238,32997,1954],{"className":32998},[282,283],[238,33000,33002],{"className":33001},[391],[238,33003,33005,33025],{"className":33004},[395,848],[238,33006,33008,33022],{"className":33007},[399],[238,33009,33011],{"className":33010,"style":1927},[403],[238,33012,33013,33016],{"style":1165},[238,33014],{"className":33015,"style":412},[411],[238,33017,33019],{"className":33018},[416,417,418,419],[238,33020,13161],{"className":33021},[282,283,419],[238,33023,872],{"className":33024},[871],[238,33026,33028],{"className":33027},[399],[238,33029,33031],{"className":33030,"style":879},[403],[238,33032],{},[238,33034],{"className":33035,"style":430},[429],[238,33037,617],{"className":33038},[434],[238,33040],{"className":33041,"style":430},[429],[238,33043,33045,33048],{"className":33044},[273],[238,33046],{"className":33047,"style":22096},[277],[238,33049,33051,33054],{"className":33050},[282],[238,33052,1954],{"className":33053},[282,283],[238,33055,33057],{"className":33056},[391],[238,33058,33060],{"className":33059},[395],[238,33061,33063],{"className":33062},[399],[238,33064,33066],{"className":33065,"style":22096},[403],[238,33067,33068,33071],{"style":407},[238,33069],{"className":33070,"style":412},[411],[238,33072,33074],{"className":33073},[416,417,418,419],[238,33075,22083],{"className":33076},[664,419],"。算子 ",[238,33079,33081,33094],{"className":33080,"translate":242},[241],[238,33082,33084],{"className":33083},[246],[248,33085,33086],{"xmlns":250},[252,33087,33088,33092],{},[255,33089,33090],{},[258,33091,11971],{},[262,33093,11971],{"encoding":264},[238,33095,33097],{"className":33096,"ariaHidden":269},[268],[238,33098,33100,33103],{"className":33099},[273],[238,33101],{"className":33102,"style":278},[277],[238,33104,11971],{"className":33105,"style":377},[282,283]," 是 Lipschitz 连续的，故",[238,33108,33110],{"className":33109,"translate":242},[945],[238,33111,33113,33264],{"className":33112,"translate":242},[241],[238,33114,33116],{"className":33115},[246],[248,33117,33118],{"xmlns":250,"display":954},[252,33119,33120,33261],{},[958,33121,33122,33205],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,33123,33124,33138],{},[966,33125,33126],{},[969,33127,33128],{"scriptlevel":740,"displaystyle":269},[255,33129,33130,33132],{},[258,33131,11971],{},[337,33133,33134,33136],{},[258,33135,1954],{},[332,33137,22083],{},[966,33139,33140],{},[969,33141,33142],{"scriptlevel":740,"displaystyle":269},[255,33143,33144,33146,33148,33150,33179,33181,33197,33199],{},[255,33145],{},[332,33147,992],{},[258,33149,11971],{},[255,33151,33152,33154,33171,33177],{},[332,33153,335],{"fence":269},[6755,33155,33156,33163],{},[255,33157,33158,33161],{},[258,33159,33160],{},"lim",[332,33162,1889],{},[255,33164,33165,33167,33169],{},[258,33166,13161],{},[332,33168,617],{},[258,33170,1895],{"mathvariant":344},[812,33172,33173,33175],{},[258,33174,1954],{},[258,33176,13161],{},[332,33178,360],{"fence":269},[332,33180,992],{},[6755,33182,33183,33189],{},[255,33184,33185,33187],{},[258,33186,33160],{},[332,33188,1889],{},[255,33190,33191,33193,33195],{},[258,33192,13161],{},[332,33194,617],{},[258,33196,1895],{"mathvariant":344},[258,33198,11971],{},[812,33200,33201,33203],{},[258,33202,1954],{},[258,33204,13161],{},[963,33206,33207,33213],{},[966,33208,33209],{},[969,33210,33211],{"scriptlevel":740,"displaystyle":269},[255,33212],{},[966,33214,33215],{},[969,33216,33217],{"scriptlevel":740,"displaystyle":269},[255,33218,33219,33221,33223,33239,33251,33253,33259],{},[255,33220],{},[332,33222,992],{},[6755,33224,33225,33231],{},[255,33226,33227,33229],{},[258,33228,33160],{},[332,33230,1889],{},[255,33232,33233,33235,33237],{},[258,33234,13161],{},[332,33236,617],{},[258,33238,1895],{"mathvariant":344},[812,33240,33241,33243],{},[258,33242,1954],{},[255,33244,33245,33247,33249],{},[258,33246,13161],{},[332,33248,987],{},[738,33250,745],{},[332,33252,992],{},[337,33254,33255,33257],{},[258,33256,1954],{},[332,33258,22083],{},[258,33260,3102],{"mathvariant":344},[262,33262,33263],{"encoding":264},"\\begin{aligned}\nTx^*&=T\\left(\\lim_{n\\to\\infty}x_n\\right)\n=\\lim_{n\\to\\infty}Tx_n\\\\\n&=\\lim_{n\\to\\infty}x_{n+1}=x^*.\n\\end{aligned}",[238,33265,33267],{"className":33266,"ariaHidden":269},[268],[238,33268,33270,33274],{"className":33269},[273],[238,33271],{"className":33272,"style":33273},[277],"height:3.69em;vertical-align:-1.595em;",[238,33275,33277],{"className":33276},[282],[238,33278,33280,33356],{"className":33279},[958],[238,33281,33283],{"className":33282},[1117],[238,33284,33286,33347],{"className":33285},[395,848],[238,33287,33289,33344],{"className":33288},[399],[238,33290,33293,33335],{"className":33291,"style":33292},[403],"height:2.095em;",[238,33294,33296,33300],{"style":33295},"top:-4.095em;",[238,33297],{"className":33298,"style":33299},[411],"height:3.15em;",[238,33301,33303,33306],{"className":33302},[282],[238,33304,11971],{"className":33305,"style":377},[282,283],[238,33307,33309,33312],{"className":33308},[282],[238,33310,1954],{"className":33311},[282,283],[238,33313,33315],{"className":33314},[391],[238,33316,33318],{"className":33317},[395],[238,33319,33321],{"className":33320},[399],[238,33322,33324],{"className":33323,"style":22441},[403],[238,33325,33326,33329],{"style":1226},[238,33327],{"className":33328,"style":412},[411],[238,33330,33332],{"className":33331},[416,417,418,419],[238,33333,22083],{"className":33334},[664,419],[238,33336,33338,33341],{"style":33337},"top:-2.255em;",[238,33339],{"className":33340,"style":33299},[411],[238,33342],{"className":33343},[282],[238,33345,872],{"className":33346},[871],[238,33348,33350],{"className":33349},[399],[238,33351,33354],{"className":33352,"style":33353},[403],"height:1.595em;",[238,33355],{},[238,33357,33359],{"className":33358},[1422],[238,33360,33362,33783],{"className":33361},[395,848],[238,33363,33365,33780],{"className":33364},[399],[238,33366,33368,33614],{"className":33367,"style":33292},[403],[238,33369,33370,33373],{"style":33295},[238,33371],{"className":33372,"style":33299},[411],[238,33374,33376,33379,33382,33385,33388,33391,33394,33506,33509,33512,33515,33568,33571,33574],{"className":33375},[282],[238,33377],{"className":33378},[282],[238,33380],{"className":33381,"style":430},[429],[238,33383,992],{"className":33384},[434],[238,33386],{"className":33387,"style":430},[429],[238,33389,11971],{"className":33390,"style":377},[282,283],[238,33392],{"className":33393,"style":455},[429],[238,33395,33397,33404,33457,33460,33500],{"className":33396},[3041],[238,33398,33400],{"className":33399,"style":4791},[381,4790],[238,33401,335],{"className":33402},[18719,33403],"size2",[238,33405,33407],{"className":33406},[1943,3493],[238,33408,33410,33449],{"className":33409},[395,848],[238,33411,33413,33446],{"className":33412},[399],[238,33414,33416,33436],{"className":33415,"style":29023},[403],[238,33417,33418,33421],{"style":22498},[238,33419],{"className":33420,"style":1134},[411],[238,33422,33424],{"className":33423},[416,417,418,419],[238,33425,33427,33430,33433],{"className":33426},[282,419],[238,33428,13161],{"className":33429},[282,283,419],[238,33431,617],{"className":33432},[434,419],[238,33434,1895],{"className":33435},[282,419],[238,33437,33438,33441],{"style":21798},[238,33439],{"className":33440,"style":1134},[411],[238,33442,33443],{},[238,33444,33160],{"className":33445},[1943],[238,33447,872],{"className":33448},[871],[238,33450,33452],{"className":33451},[399],[238,33453,33455],{"className":33454,"style":22530},[403],[238,33456],{},[238,33458],{"className":33459,"style":455},[429],[238,33461,33463,33466],{"className":33462},[282],[238,33464,1954],{"className":33465},[282,283],[238,33467,33469],{"className":33468},[391],[238,33470,33472,33492],{"className":33471},[395,848],[238,33473,33475,33489],{"className":33474},[399],[238,33476,33478],{"className":33477,"style":1927},[403],[238,33479,33480,33483],{"style":1165},[238,33481],{"className":33482,"style":412},[411],[238,33484,33486],{"className":33485},[416,417,418,419],[238,33487,13161],{"className":33488},[282,283,419],[238,33490,872],{"className":33491},[871],[238,33493,33495],{"className":33494},[399],[238,33496,33498],{"className":33497,"style":879},[403],[238,33499],{},[238,33501,33503],{"className":33502,"style":4791},[462,4790],[238,33504,360],{"className":33505},[18719,33403],[238,33507],{"className":33508,"style":430},[429],[238,33510,992],{"className":33511},[434],[238,33513],{"className":33514,"style":430},[429],[238,33516,33518],{"className":33517},[1943,3493],[238,33519,33521,33560],{"className":33520},[395,848],[238,33522,33524,33557],{"className":33523},[399],[238,33525,33527,33547],{"className":33526,"style":29023},[403],[238,33528,33529,33532],{"style":22498},[238,33530],{"className":33531,"style":1134},[411],[238,33533,33535],{"className":33534},[416,417,418,419],[238,33536,33538,33541,33544],{"className":33537},[282,419],[238,33539,13161],{"className":33540},[282,283,419],[238,33542,617],{"className":33543},[434,419],[238,33545,1895],{"className":33546},[282,419],[238,33548,33549,33552],{"style":21798},[238,33550],{"className":33551,"style":1134},[411],[238,33553,33554],{},[238,33555,33160],{"className":33556},[1943],[238,33558,872],{"className":33559},[871],[238,33561,33563],{"className":33562},[399],[238,33564,33566],{"className":33565,"style":22530},[403],[238,33567],{},[238,33569],{"className":33570,"style":455},[429],[238,33572,11971],{"className":33573,"style":377},[282,283],[238,33575,33577,33580],{"className":33576},[282],[238,33578,1954],{"className":33579},[282,283],[238,33581,33583],{"className":33582},[391],[238,33584,33586,33606],{"className":33585},[395,848],[238,33587,33589,33603],{"className":33588},[399],[238,33590,33592],{"className":33591,"style":1927},[403],[238,33593,33594,33597],{"style":1165},[238,33595],{"className":33596,"style":412},[411],[238,33598,33600],{"className":33599},[416,417,418,419],[238,33601,13161],{"className":33602},[282,283,419],[238,33604,872],{"className":33605},[871],[238,33607,33609],{"className":33608},[399],[238,33610,33612],{"className":33611,"style":879},[403],[238,33613],{},[238,33615,33616,33619],{"style":33337},[238,33617],{"className":33618,"style":33299},[411],[238,33620,33622,33625,33628,33631,33634,33687,33690,33739,33742,33745,33748,33777],{"className":33621},[282],[238,33623],{"className":33624},[282],[238,33626],{"className":33627,"style":430},[429],[238,33629,992],{"className":33630},[434],[238,33632],{"className":33633,"style":430},[429],[238,33635,33637],{"className":33636},[1943,3493],[238,33638,33640,33679],{"className":33639},[395,848],[238,33641,33643,33676],{"className":33642},[399],[238,33644,33646,33666],{"className":33645,"style":29023},[403],[238,33647,33648,33651],{"style":22498},[238,33649],{"className":33650,"style":1134},[411],[238,33652,33654],{"className":33653},[416,417,418,419],[238,33655,33657,33660,33663],{"className":33656},[282,419],[238,33658,13161],{"className":33659},[282,283,419],[238,33661,617],{"className":33662},[434,419],[238,33664,1895],{"className":33665},[282,419],[238,33667,33668,33671],{"style":21798},[238,33669],{"className":33670,"style":1134},[411],[238,33672,33673],{},[238,33674,33160],{"className":33675},[1943],[238,33677,872],{"className":33678},[871],[238,33680,33682],{"className":33681},[399],[238,33683,33685],{"className":33684,"style":22530},[403],[238,33686],{},[238,33688],{"className":33689,"style":455},[429],[238,33691,33693,33696],{"className":33692},[282],[238,33694,1954],{"className":33695},[282,283],[238,33697,33699],{"className":33698},[391],[238,33700,33702,33731],{"className":33701},[395,848],[238,33703,33705,33728],{"className":33704},[399],[238,33706,33708],{"className":33707,"style":1162},[403],[238,33709,33710,33713],{"style":1165},[238,33711],{"className":33712,"style":412},[411],[238,33714,33716],{"className":33715},[416,417,418,419],[238,33717,33719,33722,33725],{"className":33718},[282,419],[238,33720,13161],{"className":33721},[282,283,419],[238,33723,987],{"className":33724},[664,419],[238,33726,745],{"className":33727},[282,419],[238,33729,872],{"className":33730},[871],[238,33732,33734],{"className":33733},[399],[238,33735,33737],{"className":33736,"style":1193},[403],[238,33738],{},[238,33740],{"className":33741,"style":430},[429],[238,33743,992],{"className":33744},[434],[238,33746],{"className":33747,"style":430},[429],[238,33749,33751,33754],{"className":33750},[282],[238,33752,1954],{"className":33753},[282,283],[238,33755,33757],{"className":33756},[391],[238,33758,33760],{"className":33759},[395],[238,33761,33763],{"className":33762},[399],[238,33764,33766],{"className":33765,"style":22441},[403],[238,33767,33768,33771],{"style":1226},[238,33769],{"className":33770,"style":412},[411],[238,33772,33774],{"className":33773},[416,417,418,419],[238,33775,22083],{"className":33776},[664,419],[238,33778,3102],{"className":33779},[282],[238,33781,872],{"className":33782},[871],[238,33784,33786],{"className":33785},[399],[238,33787,33789],{"className":33788,"style":33353},[403],[238,33790],{},[11,33792,33793],{},"最后，由压缩性直接得到",[238,33795,33797],{"className":33796,"translate":242},[945],[238,33798,33800,33896],{"className":33799,"translate":242},[241],[238,33801,33803],{"className":33802},[246],[248,33804,33805],{"xmlns":250,"display":954},[252,33806,33807,33893],{},[255,33808,33809,33811,33813,33819,33821,33827,33829,33831,33833,33835,33841,33847,33849,33855,33861,33863,33865,33871,33873,33875,33881,33883,33889,33891],{},[258,33810,29624],{},[332,33812,335],{"stretchy":334},[812,33814,33815,33817],{},[258,33816,1954],{},[258,33818,3395],{},[332,33820,354],{"separator":269},[337,33822,33823,33825],{},[258,33824,1954],{},[332,33826,22083],{},[332,33828,360],{"stretchy":334},[332,33830,992],{},[258,33832,29624],{},[332,33834,335],{"stretchy":334},[337,33836,33837,33839],{},[258,33838,11971],{},[258,33840,3395],{},[812,33842,33843,33845],{},[258,33844,1954],{},[738,33846,740],{},[332,33848,354],{"separator":269},[337,33850,33851,33853],{},[258,33852,11971],{},[258,33854,3395],{},[337,33856,33857,33859],{},[258,33858,1954],{},[332,33860,22083],{},[332,33862,360],{"stretchy":334},[332,33864,2023],{},[337,33866,33867,33869],{},[258,33868,730],{},[258,33870,3395],{},[258,33872,29624],{},[332,33874,335],{"stretchy":334},[812,33876,33877,33879],{},[258,33878,1954],{},[738,33880,740],{},[332,33882,354],{"separator":269},[337,33884,33885,33887],{},[258,33886,1954],{},[332,33888,22083],{},[332,33890,360],{"stretchy":334},[258,33892,3102],{"mathvariant":344},[262,33894,33895],{"encoding":264},"d(x_k,x^*)=d(T^kx_0,T^kx^*)\\le\\gamma^k d(x_0,x^*).",[238,33897,33899,33998,34155],{"className":33898,"ariaHidden":269},[268],[238,33900,33902,33905,33908,33911,33951,33954,33957,33986,33989,33992,33995],{"className":33901},[273],[238,33903],{"className":33904,"style":444},[277],[238,33906,29624],{"className":33907},[282,283],[238,33909,335],{"className":33910},[381],[238,33912,33914,33917],{"className":33913},[282],[238,33915,1954],{"className":33916},[282,283],[238,33918,33920],{"className":33919},[391],[238,33921,33923,33943],{"className":33922},[395,848],[238,33924,33926,33940],{"className":33925},[399],[238,33927,33929],{"className":33928,"style":3618},[403],[238,33930,33931,33934],{"style":1165},[238,33932],{"className":33933,"style":412},[411],[238,33935,33937],{"className":33936},[416,417,418,419],[238,33938,3395],{"className":33939,"style":3520},[282,283,419],[238,33941,872],{"className":33942},[871],[238,33944,33946],{"className":33945},[399],[238,33947,33949],{"className":33948,"style":879},[403],[238,33950],{},[238,33952,354],{"className":33953},[451],[238,33955],{"className":33956,"style":455},[429],[238,33958,33960,33963],{"className":33959},[282],[238,33961,1954],{"className":33962},[282,283],[238,33964,33966],{"className":33965},[391],[238,33967,33969],{"className":33968},[395],[238,33970,33972],{"className":33971},[399],[238,33973,33975],{"className":33974,"style":22441},[403],[238,33976,33977,33980],{"style":1226},[238,33978],{"className":33979,"style":412},[411],[238,33981,33983],{"className":33982},[416,417,418,419],[238,33984,22083],{"className":33985},[664,419],[238,33987,360],{"className":33988},[462],[238,33990],{"className":33991,"style":430},[429],[238,33993,992],{"className":33994},[434],[238,33996],{"className":33997,"style":430},[429],[238,33999,34001,34004,34007,34010,34039,34079,34082,34085,34114,34143,34146,34149,34152],{"className":34000},[273],[238,34002],{"className":34003,"style":30545},[277],[238,34005,29624],{"className":34006},[282,283],[238,34008,335],{"className":34009},[381],[238,34011,34013,34016],{"className":34012},[282],[238,34014,11971],{"className":34015,"style":377},[282,283],[238,34017,34019],{"className":34018},[391],[238,34020,34022],{"className":34021},[395],[238,34023,34025],{"className":34024},[399],[238,34026,34028],{"className":34027,"style":3588},[403],[238,34029,34030,34033],{"style":1226},[238,34031],{"className":34032,"style":412},[411],[238,34034,34036],{"className":34035},[416,417,418,419],[238,34037,3395],{"className":34038,"style":3520},[282,283,419],[238,34040,34042,34045],{"className":34041},[282],[238,34043,1954],{"className":34044},[282,283],[238,34046,34048],{"className":34047},[391],[238,34049,34051,34071],{"className":34050},[395,848],[238,34052,34054,34068],{"className":34053},[399],[238,34055,34057],{"className":34056,"style":1162},[403],[238,34058,34059,34062],{"style":1165},[238,34060],{"className":34061,"style":412},[411],[238,34063,34065],{"className":34064},[416,417,418,419],[238,34066,740],{"className":34067},[282,419],[238,34069,872],{"className":34070},[871],[238,34072,34074],{"className":34073},[399],[238,34075,34077],{"className":34076,"style":879},[403],[238,34078],{},[238,34080,354],{"className":34081},[451],[238,34083],{"className":34084,"style":455},[429],[238,34086,34088,34091],{"className":34087},[282],[238,34089,11971],{"className":34090,"style":377},[282,283],[238,34092,34094],{"className":34093},[391],[238,34095,34097],{"className":34096},[395],[238,34098,34100],{"className":34099},[399],[238,34101,34103],{"className":34102,"style":3588},[403],[238,34104,34105,34108],{"style":1226},[238,34106],{"className":34107,"style":412},[411],[238,34109,34111],{"className":34110},[416,417,418,419],[238,34112,3395],{"className":34113,"style":3520},[282,283,419],[238,34115,34117,34120],{"className":34116},[282],[238,34118,1954],{"className":34119},[282,283],[238,34121,34123],{"className":34122},[391],[238,34124,34126],{"className":34125},[395],[238,34127,34129],{"className":34128},[399],[238,34130,34132],{"className":34131,"style":22441},[403],[238,34133,34134,34137],{"style":1226},[238,34135],{"className":34136,"style":412},[411],[238,34138,34140],{"className":34139},[416,417,418,419],[238,34141,22083],{"className":34142},[664,419],[238,34144,360],{"className":34145},[462],[238,34147],{"className":34148,"style":430},[429],[238,34150,2023],{"className":34151},[434],[238,34153],{"className":34154,"style":430},[429],[238,34156,34158,34161,34190,34193,34196,34236,34239,34242,34271,34274],{"className":34157},[273],[238,34159],{"className":34160,"style":30545},[277],[238,34162,34164,34167],{"className":34163},[282],[238,34165,730],{"className":34166,"style":764},[282,283],[238,34168,34170],{"className":34169},[391],[238,34171,34173],{"className":34172},[395],[238,34174,34176],{"className":34175},[399],[238,34177,34179],{"className":34178,"style":3588},[403],[238,34180,34181,34184],{"style":1226},[238,34182],{"className":34183,"style":412},[411],[238,34185,34187],{"className":34186},[416,417,418,419],[238,34188,3395],{"className":34189,"style":3520},[282,283,419],[238,34191,29624],{"className":34192},[282,283],[238,34194,335],{"className":34195},[381],[238,34197,34199,34202],{"className":34198},[282],[238,34200,1954],{"className":34201},[282,283],[238,34203,34205],{"className":34204},[391],[238,34206,34208,34228],{"className":34207},[395,848],[238,34209,34211,34225],{"className":34210},[399],[238,34212,34214],{"className":34213,"style":1162},[403],[238,34215,34216,34219],{"style":1165},[238,34217],{"className":34218,"style":412},[411],[238,34220,34222],{"className":34221},[416,417,418,419],[238,34223,740],{"className":34224},[282,419],[238,34226,872],{"className":34227},[871],[238,34229,34231],{"className":34230},[399],[238,34232,34234],{"className":34233,"style":879},[403],[238,34235],{},[238,34237,354],{"className":34238},[451],[238,34240],{"className":34241,"style":455},[429],[238,34243,34245,34248],{"className":34244},[282],[238,34246,1954],{"className":34247},[282,283],[238,34249,34251],{"className":34250},[391],[238,34252,34254],{"className":34253},[395],[238,34255,34257],{"className":34256},[399],[238,34258,34260],{"className":34259,"style":22441},[403],[238,34261,34262,34265],{"style":1226},[238,34263],{"className":34264,"style":412},[411],[238,34266,34268],{"className":34267},[416,417,418,419],[238,34269,22083],{"className":34270},[664,419],[238,34272,360],{"className":34273},[462],[238,34275,3102],{"className":34276},[282],[11,34278,11882],{},[225,34280,34282],{"id":34281},"_35-唯一性与上下界","3.5 唯一性与上下界",[230,34284,34287],{"id":34285,"title":34286},"corollary-unique-fixed-point","推论 3.4（不动点的存在唯一性）",[11,34288,236,34289,14826,34361,1838,34419,30031,34477,34520],{},[238,34290,34292,34313],{"className":34291,"translate":242},[241],[238,34293,34295],{"className":34294},[246],[248,34296,34297],{"xmlns":250},[252,34298,34299,34311],{},[255,34300,34301,34303,34305,34307,34309],{},[738,34302,740],{},[332,34304,2023],{},[258,34306,730],{},[332,34308,1892],{},[738,34310,745],{},[262,34312,14776],{"encoding":264},[238,34314,34316,34334,34352],{"className":34315,"ariaHidden":269},[268],[238,34317,34319,34322,34325,34328,34331],{"className":34318},[273],[238,34320],{"className":34321,"style":14786},[277],[238,34323,740],{"className":34324},[282],[238,34326],{"className":34327,"style":430},[429],[238,34329,2023],{"className":34330},[434],[238,34332],{"className":34333,"style":430},[429],[238,34335,34337,34340,34343,34346,34349],{"className":34336},[273],[238,34338],{"className":34339,"style":760},[277],[238,34341,730],{"className":34342,"style":764},[282,283],[238,34344],{"className":34345,"style":430},[429],[238,34347,1892],{"className":34348},[434],[238,34350],{"className":34351,"style":430},[429],[238,34353,34355,34358],{"className":34354},[273],[238,34356],{"className":34357,"style":2672},[277],[238,34359,745],{"className":34360},[282],[238,34362,34364,34381],{"className":34363,"translate":242},[241],[238,34365,34367],{"className":34366},[246],[248,34368,34369],{"xmlns":250},[252,34370,34371,34379],{},[255,34372,34373],{},[337,34374,34375,34377],{},[258,34376,11971],{"mathvariant":816},[258,34378,2297],{},[262,34380,11976],{"encoding":264},[238,34382,34384],{"className":34383,"ariaHidden":269},[268],[238,34385,34387,34390],{"className":34386},[273],[238,34388],{"className":34389,"style":278},[277],[238,34391,34393,34396],{"className":34392},[282],[238,34394,11971],{"className":34395,"style":11992},[282,840],[238,34397,34399],{"className":34398},[391],[238,34400,34402],{"className":34401},[395],[238,34403,34405],{"className":34404},[399],[238,34406,34408],{"className":34407,"style":6609},[403],[238,34409,34410,34413],{"style":407},[238,34411],{"className":34412,"style":412},[411],[238,34414,34416],{"className":34415},[416,417,418,419],[238,34417,2297],{"className":34418,"style":2313},[282,283,419],[238,34420,34422,34439],{"className":34421,"translate":242},[241],[238,34423,34425],{"className":34424},[246],[248,34426,34427],{"xmlns":250},[252,34428,34429,34437],{},[255,34430,34431],{},[337,34432,34433,34435],{},[258,34434,11971],{"mathvariant":816},[332,34436,22083],{},[262,34438,22086],{"encoding":264},[238,34440,34442],{"className":34441,"ariaHidden":269},[268],[238,34443,34445,34448],{"className":34444},[273],[238,34446],{"className":34447,"style":22096},[277],[238,34449,34451,34454],{"className":34450},[282],[238,34452,11971],{"className":34453,"style":11992},[282,840],[238,34455,34457],{"className":34456},[391],[238,34458,34460],{"className":34459},[395],[238,34461,34463],{"className":34462},[399],[238,34464,34466],{"className":34465,"style":22096},[403],[238,34467,34468,34471],{"style":407},[238,34469],{"className":34470,"style":412},[411],[238,34472,34474],{"className":34473},[416,417,418,419],[238,34475,22083],{"className":34476},[664,419],[238,34478,34480,34499],{"className":34479,"translate":242},[241],[238,34481,34483],{"className":34482},[246],[248,34484,34485],{"xmlns":250},[252,34486,34487,34497],{},[255,34488,34489,34491,34493,34495],{},[258,34490,19971],{},[332,34492,335],{"stretchy":334},[258,34494,260],{},[332,34496,360],{"stretchy":334},[262,34498,21579],{"encoding":264},[238,34500,34502],{"className":34501,"ariaHidden":269},[268],[238,34503,34505,34508,34511,34514,34517],{"className":34504},[273],[238,34506],{"className":34507,"style":444},[277],[238,34509,19971],{"className":34510,"style":20011},[282,283],[238,34512,335],{"className":34513},[381],[238,34515,260],{"className":34516,"style":284},[282,283],[238,34518,360],{"className":34519},[462]," 中各自存在唯一不动点。",[11,34522,34523,2282,34525,34568,34569,1838,34627,27026,34685,34713,34714,34718],{},[1688,34524,5794],{},[238,34526,34528,34547],{"className":34527,"translate":242},[241],[238,34529,34531],{"className":34530},[246],[248,34532,34533],{"xmlns":250},[252,34534,34535,34545],{},[255,34536,34537,34539,34541,34543],{},[258,34538,19971],{},[332,34540,335],{"stretchy":334},[258,34542,260],{},[332,34544,360],{"stretchy":334},[262,34546,21579],{"encoding":264},[238,34548,34550],{"className":34549,"ariaHidden":269},[268],[238,34551,34553,34556,34559,34562,34565],{"className":34552},[273],[238,34554],{"className":34555,"style":444},[277],[238,34557,19971],{"className":34558,"style":20011},[282,283],[238,34560,335],{"className":34561},[381],[238,34563,260],{"className":34564,"style":284},[282,283],[238,34566,360],{"className":34567},[462]," 在 sup 范数下完备，",[238,34570,34572,34589],{"className":34571,"translate":242},[241],[238,34573,34575],{"className":34574},[246],[248,34576,34577],{"xmlns":250},[252,34578,34579,34587],{},[255,34580,34581],{},[337,34582,34583,34585],{},[258,34584,11971],{"mathvariant":816},[258,34586,2297],{},[262,34588,11976],{"encoding":264},[238,34590,34592],{"className":34591,"ariaHidden":269},[268],[238,34593,34595,34598],{"className":34594},[273],[238,34596],{"className":34597,"style":278},[277],[238,34599,34601,34604],{"className":34600},[282],[238,34602,11971],{"className":34603,"style":11992},[282,840],[238,34605,34607],{"className":34606},[391],[238,34608,34610],{"className":34609},[395],[238,34611,34613],{"className":34612},[399],[238,34614,34616],{"className":34615,"style":6609},[403],[238,34617,34618,34621],{"style":407},[238,34619],{"className":34620,"style":412},[411],[238,34622,34624],{"className":34623},[416,417,418,419],[238,34625,2297],{"className":34626,"style":2313},[282,283,419],[238,34628,34630,34647],{"className":34629,"translate":242},[241],[238,34631,34633],{"className":34632},[246],[248,34634,34635],{"xmlns":250},[252,34636,34637,34645],{},[255,34638,34639],{},[337,34640,34641,34643],{},[258,34642,11971],{"mathvariant":816},[332,34644,22083],{},[262,34646,22086],{"encoding":264},[238,34648,34650],{"className":34649,"ariaHidden":269},[268],[238,34651,34653,34656],{"className":34652},[273],[238,34654],{"className":34655,"style":22096},[277],[238,34657,34659,34662],{"className":34658},[282],[238,34660,11971],{"className":34661,"style":11992},[282,840],[238,34663,34665],{"className":34664},[391],[238,34666,34668],{"className":34667},[395],[238,34669,34671],{"className":34670},[399],[238,34672,34674],{"className":34673,"style":22096},[403],[238,34675,34676,34679],{"style":407},[238,34677],{"className":34678,"style":412},[411],[238,34680,34682],{"className":34681},[416,417,418,419],[238,34683,22083],{"className":34684},[664,419],[238,34686,34688,34701],{"className":34687,"translate":242},[241],[238,34689,34691],{"className":34690},[246],[248,34692,34693],{"xmlns":250},[252,34694,34695,34699],{},[255,34696,34697],{},[258,34698,730],{},[262,34700,26785],{"encoding":264},[238,34702,34704],{"className":34703,"ariaHidden":269},[268],[238,34705,34707,34710],{"className":34706},[273],[238,34708],{"className":34709,"style":2853},[277],[238,34711,730],{"className":34712,"style":764},[282,283],"-压缩映射。由 ",[357,34715,34717],{"href":34716},"#theorem-banach","定理 3.3"," 即得。证毕。",[11,34720,26642,34721,941,34779,34781,34782,23814,34840,34898,34899,34958],{},[238,34722,34724,34741],{"className":34723,"translate":242},[241],[238,34725,34727],{"className":34726},[246],[248,34728,34729],{"xmlns":250},[252,34730,34731,34739],{},[255,34732,34733],{},[337,34734,34735,34737],{},[258,34736,11971],{"mathvariant":816},[258,34738,2297],{},[262,34740,11976],{"encoding":264},[238,34742,34744],{"className":34743,"ariaHidden":269},[268],[238,34745,34747,34750],{"className":34746},[273],[238,34748],{"className":34749,"style":278},[277],[238,34751,34753,34756],{"className":34752},[282],[238,34754,11971],{"className":34755,"style":11992},[282,840],[238,34757,34759],{"className":34758},[391],[238,34760,34762],{"className":34761},[395],[238,34763,34765],{"className":34764},[399],[238,34766,34768],{"className":34767,"style":6609},[403],[238,34769,34770,34773],{"style":407},[238,34771],{"className":34772,"style":412},[411],[238,34774,34776],{"className":34775},[416,417,418,419],[238,34777,2297],{"className":34778,"style":2313},[282,283,419],[357,34780,21493],{"href":21492}," 已经指出这个唯一不动点就是 ",[238,34783,34785,34802],{"className":34784,"translate":242},[241],[238,34786,34788],{"className":34787},[246],[248,34789,34790],{"xmlns":250},[252,34791,34792,34800],{},[255,34793,34794],{},[337,34795,34796,34798],{},[258,34797,4429],{},[258,34799,2297],{},[262,34801,6581],{"encoding":264},[238,34803,34805],{"className":34804,"ariaHidden":269},[268],[238,34806,34808,34811],{"className":34807},[273],[238,34809],{"className":34810,"style":278},[277],[238,34812,34814,34817],{"className":34813},[282],[238,34815,4429],{"className":34816,"style":660},[282,283],[238,34818,34820],{"className":34819},[391],[238,34821,34823],{"className":34822},[395],[238,34824,34826],{"className":34825},[399],[238,34827,34829],{"className":34828,"style":6609},[403],[238,34830,34831,34834],{"style":407},[238,34832],{"className":34833,"style":412},[411],[238,34835,34837],{"className":34836},[416,417,418,419],[238,34838,2297],{"className":34839,"style":2313},[282,283,419],[238,34841,34843,34860],{"className":34842,"translate":242},[241],[238,34844,34846],{"className":34845},[246],[248,34847,34848],{"xmlns":250},[252,34849,34850,34858],{},[255,34851,34852],{},[337,34853,34854,34856],{},[258,34855,11971],{"mathvariant":816},[332,34857,22083],{},[262,34859,22086],{"encoding":264},[238,34861,34863],{"className":34862,"ariaHidden":269},[268],[238,34864,34866,34869],{"className":34865},[273],[238,34867],{"className":34868,"style":22096},[277],[238,34870,34872,34875],{"className":34871},[282],[238,34873,11971],{"className":34874,"style":11992},[282,840],[238,34876,34878],{"className":34877},[391],[238,34879,34881],{"className":34880},[395],[238,34882,34884],{"className":34883},[399],[238,34885,34887],{"className":34886,"style":22096},[403],[238,34888,34889,34892],{"style":407},[238,34890],{"className":34891,"style":412},[411],[238,34893,34895],{"className":34894},[416,417,418,419],[238,34896,22083],{"className":34897},[664,419],"，第 4 节将证明它的唯一不动点就是最优值函数 ",[238,34900,34902,34920],{"className":34901,"translate":242},[241],[238,34903,34905],{"className":34904},[246],[248,34906,34907],{"xmlns":250},[252,34908,34909,34917],{},[255,34910,34911],{},[337,34912,34913,34915],{},[258,34914,4429],{},[332,34916,22083],{},[262,34918,34919],{"encoding":264},"V^*",[238,34921,34923],{"className":34922,"ariaHidden":269},[268],[238,34924,34926,34929],{"className":34925},[273],[238,34927],{"className":34928,"style":22096},[277],[238,34930,34932,34935],{"className":34931},[282],[238,34933,4429],{"className":34934,"style":660},[282,283],[238,34936,34938],{"className":34937},[391],[238,34939,34941],{"className":34940},[395],[238,34942,34944],{"className":34943},[399],[238,34945,34947],{"className":34946,"style":22096},[403],[238,34948,34949,34952],{"style":407},[238,34950],{"className":34951,"style":412},[411],[238,34953,34955],{"className":34954},[416,417,418,419],[238,34956,22083],{"className":34957},[664,419],"。因此，「贝尔曼方程有唯一解」并不是一个额外的假设，而是压缩性的直接推论。",[11,34960,34961],{},"单调性还给出了一个在理论上很有用的上下界判据。",[230,34963,34966],{"id":34964,"title":34965},"corollary-order-bounds","推论 3.5（上下界判据）",[11,34967,236,34968,21601,34997,35055,35056,35114,35115,35185,35186,35354,35355,35185,35500,35354,35662,1806],{},[238,34969,34971,34985],{"className":34970,"translate":242},[241],[238,34972,34974],{"className":34973},[246],[248,34975,34976],{"xmlns":250},[252,34977,34978,34982],{},[255,34979,34980],{},[258,34981,11971],{"mathvariant":816},[262,34983,34984],{"encoding":264},"\\mathcal{T}",[238,34986,34988],{"className":34987,"ariaHidden":269},[268],[238,34989,34991,34994],{"className":34990},[273],[238,34992],{"className":34993,"style":278},[277],[238,34995,11971],{"className":34996,"style":11992},[282,840],[238,34998,35000,35017],{"className":34999,"translate":242},[241],[238,35001,35003],{"className":35002},[246],[248,35004,35005],{"xmlns":250},[252,35006,35007,35015],{},[255,35008,35009],{},[337,35010,35011,35013],{},[258,35012,11971],{"mathvariant":816},[258,35014,2297],{},[262,35016,11976],{"encoding":264},[238,35018,35020],{"className":35019,"ariaHidden":269},[268],[238,35021,35023,35026],{"className":35022},[273],[238,35024],{"className":35025,"style":278},[277],[238,35027,35029,35032],{"className":35028},[282],[238,35030,11971],{"className":35031,"style":11992},[282,840],[238,35033,35035],{"className":35034},[391],[238,35036,35038],{"className":35037},[395],[238,35039,35041],{"className":35040},[399],[238,35042,35044],{"className":35043,"style":6609},[403],[238,35045,35046,35049],{"style":407},[238,35047],{"className":35048,"style":412},[411],[238,35050,35052],{"className":35051},[416,417,418,419],[238,35053,2297],{"className":35054,"style":2313},[282,283,419]," 或 ",[238,35057,35059,35076],{"className":35058,"translate":242},[241],[238,35060,35062],{"className":35061},[246],[248,35063,35064],{"xmlns":250},[252,35065,35066,35074],{},[255,35067,35068],{},[337,35069,35070,35072],{},[258,35071,11971],{"mathvariant":816},[332,35073,22083],{},[262,35075,22086],{"encoding":264},[238,35077,35079],{"className":35078,"ariaHidden":269},[268],[238,35080,35082,35085],{"className":35081},[273],[238,35083],{"className":35084,"style":22096},[277],[238,35086,35088,35091],{"className":35087},[282],[238,35089,11971],{"className":35090,"style":11992},[282,840],[238,35092,35094],{"className":35093},[391],[238,35095,35097],{"className":35096},[395],[238,35098,35100],{"className":35099},[399],[238,35101,35103],{"className":35102,"style":22096},[403],[238,35104,35105,35108],{"style":407},[238,35106],{"className":35107,"style":412},[411],[238,35109,35111],{"className":35110},[416,417,418,419],[238,35112,22083],{"className":35113},[664,419],"，其唯一不动点为 ",[238,35116,35118,35136],{"className":35117,"translate":242},[241],[238,35119,35121],{"className":35120},[246],[248,35122,35123],{"xmlns":250},[252,35124,35125,35133],{},[255,35126,35127],{},[812,35128,35129,35131],{},[258,35130,4429],{},[258,35132,1895],{"mathvariant":344},[262,35134,35135],{"encoding":264},"V_\\infty",[238,35137,35139],{"className":35138,"ariaHidden":269},[268],[238,35140,35142,35145],{"className":35141},[273],[238,35143],{"className":35144,"style":833},[277],[238,35146,35148,35151],{"className":35147},[282],[238,35149,4429],{"className":35150,"style":660},[282,283],[238,35152,35154],{"className":35153},[391],[238,35155,35157,35177],{"className":35156},[395,848],[238,35158,35160,35174],{"className":35159},[399],[238,35161,35163],{"className":35162,"style":1927},[403],[238,35164,35165,35168],{"style":13806},[238,35166],{"className":35167,"style":412},[411],[238,35169,35171],{"className":35170},[416,417,418,419],[238,35172,1895],{"className":35173},[282,419],[238,35175,872],{"className":35176},[871],[238,35178,35180],{"className":35179},[399],[238,35181,35183],{"className":35182,"style":879},[403],[238,35184],{},"。若 ",[238,35187,35189,35232],{"className":35188,"translate":242},[241],[238,35190,35192],{"className":35191},[246],[248,35193,35194],{"xmlns":250},[252,35195,35196,35229],{},[255,35197,35198,35213,35215,35217],{},[812,35199,35200,35202],{},[258,35201,4429],{},[255,35203,35204,35207,35210],{},[258,35205,35206],{"mathvariant":344},"l",[258,35208,35209],{"mathvariant":344},"o",[258,35211,35212],{"mathvariant":344},"w",[332,35214,2023],{},[258,35216,11971],{"mathvariant":816},[812,35218,35219,35221],{},[258,35220,4429],{},[255,35222,35223,35225,35227],{},[258,35224,35206],{"mathvariant":344},[258,35226,35209],{"mathvariant":344},[258,35228,35212],{"mathvariant":344},[262,35230,35231],{"encoding":264},"V_{\\mathrm{low}}\\le\\mathcal{T}V_{\\mathrm{low}}",[238,35233,35235,35299],{"className":35234,"ariaHidden":269},[268],[238,35236,35238,35241,35290,35293,35296],{"className":35237},[273],[238,35239],{"className":35240,"style":833},[277],[238,35242,35244,35247],{"className":35243},[282],[238,35245,4429],{"className":35246,"style":660},[282,283],[238,35248,35250],{"className":35249},[391],[238,35251,35253,35282],{"className":35252},[395,848],[238,35254,35256,35279],{"className":35255},[399],[238,35257,35259],{"className":35258,"style":3618},[403],[238,35260,35261,35264],{"style":13806},[238,35262],{"className":35263,"style":412},[411],[238,35265,35267],{"className":35266},[416,417,418,419],[238,35268,35270],{"className":35269},[282,419],[238,35271,35273],{"className":35272},[282,419],[238,35274,35278],{"className":35275,"style":35277},[282,35276,419],"mathrm","margin-right:0.0139em;","low",[238,35280,872],{"className":35281},[871],[238,35283,35285],{"className":35284},[399],[238,35286,35288],{"className":35287,"style":879},[403],[238,35289],{},[238,35291],{"className":35292,"style":430},[429],[238,35294,2023],{"className":35295},[434],[238,35297],{"className":35298,"style":430},[429],[238,35300,35302,35305,35308],{"className":35301},[273],[238,35303],{"className":35304,"style":833},[277],[238,35306,11971],{"className":35307,"style":11992},[282,840],[238,35309,35311,35314],{"className":35310},[282],[238,35312,4429],{"className":35313,"style":660},[282,283],[238,35315,35317],{"className":35316},[391],[238,35318,35320,35346],{"className":35319},[395,848],[238,35321,35323,35343],{"className":35322},[399],[238,35324,35326],{"className":35325,"style":3618},[403],[238,35327,35328,35331],{"style":13806},[238,35329],{"className":35330,"style":412},[411],[238,35332,35334],{"className":35333},[416,417,418,419],[238,35335,35337],{"className":35336},[282,419],[238,35338,35340],{"className":35339},[282,419],[238,35341,35278],{"className":35342,"style":35277},[282,35276,419],[238,35344,872],{"className":35345},[871],[238,35347,35349],{"className":35348},[399],[238,35350,35352],{"className":35351,"style":879},[403],[238,35353],{},"，则 ",[238,35356,35358,35390],{"className":35357,"translate":242},[241],[238,35359,35361],{"className":35360},[246],[248,35362,35363],{"xmlns":250},[252,35364,35365,35387],{},[255,35366,35367,35379,35381],{},[812,35368,35369,35371],{},[258,35370,4429],{},[255,35372,35373,35375,35377],{},[258,35374,35206],{"mathvariant":344},[258,35376,35209],{"mathvariant":344},[258,35378,35212],{"mathvariant":344},[332,35380,2023],{},[812,35382,35383,35385],{},[258,35384,4429],{},[258,35386,1895],{"mathvariant":344},[262,35388,35389],{"encoding":264},"V_{\\mathrm{low}}\\le V_\\infty",[238,35391,35393,35454],{"className":35392,"ariaHidden":269},[268],[238,35394,35396,35399,35445,35448,35451],{"className":35395},[273],[238,35397],{"className":35398,"style":833},[277],[238,35400,35402,35405],{"className":35401},[282],[238,35403,4429],{"className":35404,"style":660},[282,283],[238,35406,35408],{"className":35407},[391],[238,35409,35411,35437],{"className":35410},[395,848],[238,35412,35414,35434],{"className":35413},[399],[238,35415,35417],{"className":35416,"style":3618},[403],[238,35418,35419,35422],{"style":13806},[238,35420],{"className":35421,"style":412},[411],[238,35423,35425],{"className":35424},[416,417,418,419],[238,35426,35428],{"className":35427},[282,419],[238,35429,35431],{"className":35430},[282,419],[238,35432,35278],{"className":35433,"style":35277},[282,35276,419],[238,35435,872],{"className":35436},[871],[238,35438,35440],{"className":35439},[399],[238,35441,35443],{"className":35442,"style":879},[403],[238,35444],{},[238,35446],{"className":35447,"style":430},[429],[238,35449,2023],{"className":35450},[434],[238,35452],{"className":35453,"style":430},[429],[238,35455,35457,35460],{"className":35456},[273],[238,35458],{"className":35459,"style":833},[277],[238,35461,35463,35466],{"className":35462},[282],[238,35464,4429],{"className":35465,"style":660},[282,283],[238,35467,35469],{"className":35468},[391],[238,35470,35472,35492],{"className":35471},[395,848],[238,35473,35475,35489],{"className":35474},[399],[238,35476,35478],{"className":35477,"style":1927},[403],[238,35479,35480,35483],{"style":13806},[238,35481],{"className":35482,"style":412},[411],[238,35484,35486],{"className":35485},[416,417,418,419],[238,35487,1895],{"className":35488},[282,419],[238,35490,872],{"className":35491},[871],[238,35493,35495],{"className":35494},[399],[238,35496,35498],{"className":35497,"style":879},[403],[238,35499],{},[238,35501,35503,35540],{"className":35502,"translate":242},[241],[238,35504,35506],{"className":35505},[246],[248,35507,35508],{"xmlns":250},[252,35509,35510,35537],{},[255,35511,35512,35523,35525,35527],{},[812,35513,35514,35516],{},[258,35515,4429],{},[255,35517,35518,35521],{},[258,35519,35520],{"mathvariant":344},"u",[258,35522,11],{"mathvariant":344},[332,35524,25372],{},[258,35526,11971],{"mathvariant":816},[812,35528,35529,35531],{},[258,35530,4429],{},[255,35532,35533,35535],{},[258,35534,35520],{"mathvariant":344},[258,35536,11],{"mathvariant":344},[262,35538,35539],{"encoding":264},"V_{\\mathrm{up}}\\ge\\mathcal{T}V_{\\mathrm{up}}",[238,35541,35543,35607],{"className":35542,"ariaHidden":269},[268],[238,35544,35546,35550,35598,35601,35604],{"className":35545},[273],[238,35547],{"className":35548,"style":35549},[277],"height:0.9694em;vertical-align:-0.2861em;",[238,35551,35553,35556],{"className":35552},[282],[238,35554,4429],{"className":35555,"style":660},[282,283],[238,35557,35559],{"className":35558},[391],[238,35560,35562,35589],{"className":35561},[395,848],[238,35563,35565,35586],{"className":35564},[399],[238,35566,35568],{"className":35567,"style":1927},[403],[238,35569,35570,35573],{"style":13806},[238,35571],{"className":35572,"style":412},[411],[238,35574,35576],{"className":35575},[416,417,418,419],[238,35577,35579],{"className":35578},[282,419],[238,35580,35582],{"className":35581},[282,419],[238,35583,35585],{"className":35584},[282,35276,419],"up",[238,35587,872],{"className":35588},[871],[238,35590,35592],{"className":35591},[399],[238,35593,35596],{"className":35594,"style":35595},[403],"height:0.2861em;",[238,35597],{},[238,35599],{"className":35600,"style":430},[429],[238,35602,25372],{"className":35603},[434],[238,35605],{"className":35606,"style":430},[429],[238,35608,35610,35613,35616],{"className":35609},[273],[238,35611],{"className":35612,"style":35549},[277],[238,35614,11971],{"className":35615,"style":11992},[282,840],[238,35617,35619,35622],{"className":35618},[282],[238,35620,4429],{"className":35621,"style":660},[282,283],[238,35623,35625],{"className":35624},[391],[238,35626,35628,35654],{"className":35627},[395,848],[238,35629,35631,35651],{"className":35630},[399],[238,35632,35634],{"className":35633,"style":1927},[403],[238,35635,35636,35639],{"style":13806},[238,35637],{"className":35638,"style":412},[411],[238,35640,35642],{"className":35641},[416,417,418,419],[238,35643,35645],{"className":35644},[282,419],[238,35646,35648],{"className":35647},[282,419],[238,35649,35585],{"className":35650},[282,35276,419],[238,35652,872],{"className":35653},[871],[238,35655,35657],{"className":35656},[399],[238,35658,35660],{"className":35659,"style":35595},[403],[238,35661],{},[238,35663,35665,35695],{"className":35664,"translate":242},[241],[238,35666,35668],{"className":35667},[246],[248,35669,35670],{"xmlns":250},[252,35671,35672,35692],{},[255,35673,35674,35684,35686],{},[812,35675,35676,35678],{},[258,35677,4429],{},[255,35679,35680,35682],{},[258,35681,35520],{"mathvariant":344},[258,35683,11],{"mathvariant":344},[332,3568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",[238,36290,36292,36330],{"className":36291,"translate":242},[241],[238,36293,36295],{"className":36294},[246],[248,36296,36297],{"xmlns":250},[252,36298,36299,36327],{},[255,36300,36301,36307,36319,36321],{},[337,36302,36303,36305],{},[258,36304,11971],{"mathvariant":816},[258,36306,13161],{},[812,36308,36309,36311],{},[258,36310,4429],{},[255,36312,36313,36315,36317],{},[258,36314,35206],{"mathvariant":344},[258,36316,35209],{"mathvariant":344},[258,36318,35212],{"mathvariant":344},[332,36320,617],{},[812,36322,36323,36325],{},[258,36324,4429],{},[258,36326,1895],{"mathvariant":344},[262,36328,36329],{"encoding":264},"\\mathcal{T}^nV_{\\mathrm{low}}\\to V_\\infty",[238,36331,36333,36423],{"className":36332,"ariaHidden":269},[268],[238,36334,36336,36339,36368,36414,36417,36420],{"className":36335},[273],[238,36337],{"className":36338,"style":833},[277],[238,36340,36342,36345],{"className":36341},[282],[238,36343,11971],{"className":36344,"style":11992},[282,840],[238,36346,36348],{"className":36347},[391],[238,36349,36351],{"className":36350},[395],[238,36352,36354],{"className":36353},[399],[238,36355,36357],{"className":36356,"style":6609},[403],[238,36358,36359,36362],{"style":407},[238,36360],{"className":36361,"style":412},[411],[238,36363,36365],{"className":36364},[416,417,418,419],[238,36366,13161],{"className":36367},[282,283,419],[238,36369,36371,36374],{"className":36370},[282],[238,36372,4429],{"className":36373,"style":660},[282,283],[238,36375,36377],{"className":36376},[391],[238,36378,36380,36406],{"className":36379},[395,848],[238,36381,36383,36403],{"className":36382},[399],[238,36384,36386],{"className":36385,"style":3618},[403],[238,36387,36388,36391],{"style":13806},[238,36389],{"className":36390,"style":412},[411],[238,36392,36394],{"className":36393},[416,417,418,419],[238,36395,36397],{"className":36396},[282,419],[238,36398,36400],{"className":36399},[282,419],[238,36401,35278],{"className":36402,"style":35277},[282,35276,419],[238,36404,872],{"className":36405},[871],[238,36407,36409],{"className":36408},[399],[238,36410,36412],{"className":36411,"style":879},[403],[238,36413],{},[238,36415],{"className":36416,"style":430},[429],[238,36418,617],{"className":36419},[434],[238,36421],{"className":36422,"style":430},[429],[238,36424,36426,36429],{"className":36425},[273],[238,36427],{"className":36428,"style":833},[277],[238,36430,36432,36435],{"className":36431},[282],[238,36433,4429],{"className":36434,"style":660},[282,283],[238,36436,36438],{"className":36437},[391],[238,36439,36441,36461],{"className":36440},[395,848],[238,36442,36444,36458],{"className":36443},[399],[238,36445,36447],{"className":36446,"style":1927},[403],[238,36448,36449,36452],{"style":13806},[238,36450],{"className":36451,"style":412},[411],[238,36453,36455],{"className":36454},[416,417,418,419],[238,36456,1895],{"className":36457},[282,419],[238,36459,872],{"className":36460},[871],[238,36462,36464],{"className":36463},[399],[238,36465,36467],{"className":36466,"style":879},[403],[238,36468],{},"，故 ",[238,36471,36473,36504],{"className":36472,"translate":242},[241],[238,36474,36476],{"className":36475},[246],[248,36477,36478],{"xmlns":250},[252,36479,36480,36502],{},[255,36481,36482,36494,36496],{},[812,36483,36484,36486],{},[258,36485,4429],{},[255,36487,36488,36490,36492],{},[258,36489,35206],{"mathvariant":344},[258,36491,35209],{"mathvariant":344},[258,36493,35212],{"mathvariant":344},[332,36495,2023],{},[812,36497,36498,36500],{},[258,36499,4429],{},[258,36501,1895],{"mathvariant":344},[262,36503,35389],{"encoding":264},[238,36505,36507,36568],{"className":36506,"ariaHidden":269},[268],[238,36508,36510,36513,36559,36562,36565],{"className":36509},[273],[238,36511],{"className":36512,"style":833},[277],[238,36514,36516,36519],{"className":36515},[282],[238,36517,4429],{"className":36518,"style":660},[282,283],[238,36520,36522],{"className":36521},[391],[238,36523,36525,36551],{"className":36524},[395,848],[238,36526,36528,36548],{"className":36527},[399],[238,36529,36531],{"className":36530,"style":3618},[403],[238,36532,36533,36536],{"style":13806},[238,36534],{"className":36535,"style":412},[411],[238,36537,36539],{"className":36538},[416,417,418,419],[238,36540,36542],{"className":36541},[282,419],[238,36543,36545],{"className":36544},[282,419],[238,36546,35278],{"className":36547,"style":35277},[282,35276,419],[238,36549,872],{"className":36550},[871],[238,36552,36554],{"className":36553},[399],[238,36555,36557],{"className":36556,"style":879},[403],[238,36558],{},[238,36560],{"className":36561,"style":430},[429],[238,36563,2023],{"className":36564},[434],[238,36566],{"className":36567,"style":430},[429],[238,36569,36571,36574],{"className":36570},[273],[238,36572],{"className":36573,"style":833},[277],[238,36575,36577,36580],{"className":36576},[282],[238,36578,4429],{"className":36579,"style":660},[282,283],[238,36581,36583],{"className":36582},[391],[238,36584,36586,36606],{"className":36585},[395,848],[238,36587,36589,36603],{"className":36588},[399],[238,36590,36592],{"className":36591,"style":1927},[403],[238,36593,36594,36597],{"style":13806},[238,36595],{"className":36596,"style":412},[411],[238,36598,36600],{"className":36599},[416,417,418,419],[238,36601,1895],{"className":36602},[282,419],[238,36604,872],{"className":36605},[871],[238,36607,36609],{"className":36608},[399],[238,36610,36612],{"className":36611,"style":879},[403],[238,36613],{},"。上界情形同理，只是迭代序列单调下降。证毕。",[11,36616,36617,36618,36674,36675,36731],{},"这个判据说明，只要能够构造一个满足 ",[238,36619,36621,36641],{"className":36620,"translate":242},[241],[238,36622,36624],{"className":36623},[246],[248,36625,36626],{"xmlns":250},[252,36627,36628,36638],{},[255,36629,36630,36632,36634,36636],{},[258,36631,4429],{},[332,36633,2023],{},[258,36635,11971],{"mathvariant":816},[258,36637,4429],{},[262,36639,36640],{"encoding":264},"V\\le\\mathcal{T}V",[238,36642,36644,36662],{"className":36643,"ariaHidden":269},[268],[238,36645,36647,36650,36653,36656,36659],{"className":36646},[273],[238,36648],{"className":36649,"style":23995},[277],[238,36651,4429],{"className":36652,"style":660},[282,283],[238,36654],{"className":36655,"style":430},[429],[238,36657,2023],{"className":36658},[434],[238,36660],{"className":36661,"style":430},[429],[238,36663,36665,36668,36671],{"className":36664},[273],[238,36666],{"className":36667,"style":278},[277],[238,36669,11971],{"className":36670,"style":11992},[282,840],[238,36672,4429],{"className":36673,"style":660},[282,283]," 的下界函数，它就一直位于真实值函数之下。满足 ",[238,36676,36678,36698],{"className":36677,"translate":242},[241],[238,36679,36681],{"className":36680},[246],[248,36682,36683],{"xmlns":250},[252,36684,36685,36695],{},[255,36686,36687,36689,36691,36693],{},[258,36688,4429],{},[332,36690,25372],{},[258,36692,11971],{"mathvariant":816},[258,36694,4429],{},[262,36696,36697],{"encoding":264},"V\\ge\\mathcal{T}V",[238,36699,36701,36719],{"className":36700,"ariaHidden":269},[268],[238,36702,36704,36707,36710,36713,36716],{"className":36703},[273],[238,36705],{"className":36706,"style":23995},[277],[238,36708,4429],{"className":36709,"style":660},[282,283],[238,36711],{"className":36712,"style":430},[429],[238,36714,25372],{"className":36715},[434],[238,36717],{"className":36718,"style":430},[429],[238,36720,36722,36725,36728],{"className":36721},[273],[238,36723],{"className":36724,"style":278},[277],[238,36726,11971],{"className":36727,"style":11992},[282,840],[238,36729,4429],{"className":36730,"style":660},[282,283]," 的上界函数则一直位于真实值函数之上。这为值函数的区间估计提供了严格的数学基础。",[220,36733,36735],{"id":36734},"四贝尔曼最优方程","四、贝尔曼最优方程",[225,36737,36739],{"id":36738},"_41-最优值函数","4.1 最优值函数",[230,36741,36744,36747,37175],{"id":36742,"title":36743},"definition-optimal-value","定义 4.1（最优值函数）",[11,36745,36746],{},"最优状态值函数与最优动作值函数分别定义为",[238,36748,36750],{"className":36749,"translate":242},[945],[238,36751,36753,36851],{"className":36752,"translate":242},[241],[238,36754,36756],{"className":36755},[246],[248,36757,36758],{"xmlns":250,"display":954},[252,36759,36760,36848],{},[255,36761,36762,36768,36770,36772,36774,36776,36786,36792,36794,36796,36798,36800,36802,36808,36810,36812,36814,36816,36818,36820,36830,36836,36838,36840,36842,36844,36846],{},[337,36763,36764,36766],{},[258,36765,4429],{},[332,36767,22083],{},[332,36769,335],{"stretchy":334},[258,36771,341],{},[332,36773,360],{"stretchy":334},[332,36775,992],{},[6755,36777,36778,36784],{},[255,36779,36780,36782],{},[258,36781,21664],{},[332,36783,1889],{},[258,36785,2297],{},[337,36787,36788,36790],{},[258,36789,4429],{},[258,36791,2297],{},[332,36793,335],{"stretchy":334},[258,36795,341],{},[332,36797,360],{"stretchy":334},[332,36799,354],{"separator":269},[429,36801],{"width":10450},[337,36803,36804,36806],{},[258,36805,4494],{},[332,36807,22083],{},[332,36809,335],{"stretchy":334},[258,36811,341],{},[332,36813,354],{"separator":269},[258,36815,357],{},[332,36817,360],{"stretchy":334},[332,36819,992],{},[6755,36821,36822,36828],{},[255,36823,36824,36826],{},[258,36825,21664],{},[332,36827,1889],{},[258,36829,2297],{},[337,36831,36832,36834],{},[258,36833,4494],{},[258,36835,2297],{},[332,36837,335],{"stretchy":334},[258,36839,341],{},[332,36841,354],{"separator":269},[258,36843,357],{},[332,36845,360],{"stretchy":334},[258,36847,3102],{"mathvariant":344},[262,36849,36850],{"encoding":264},"V^*(s)=\\sup_{\\pi}V^\\pi(s),\n\\qquad\nQ^*(s,a)=\\sup_{\\pi}Q^\\pi(s,a).",[238,36852,36854,36907,37069],{"className":36853,"ariaHidden":269},[268],[238,36855,36857,36860,36889,36892,36895,36898,36901,36904],{"className":36856},[273],[238,36858],{"className":36859,"style":444},[277],[238,36861,36863,36866],{"className":36862},[282],[238,36864,4429],{"className":36865,"style":660},[282,283],[238,36867,36869],{"className":36868},[391],[238,36870,36872],{"className":36871},[395],[238,36873,36875],{"className":36874},[399],[238,36876,36878],{"className":36877,"style":22441},[403],[238,36879,36880,36883],{"style":1226},[238,36881],{"className":36882,"style":412},[411],[238,36884,36886],{"className":36885},[416,417,418,419],[238,36887,22083],{"className":36888},[664,419],[238,36890,335],{"className":36891},[381],[238,36893,341],{"className":36894},[282,283],[238,36896,360],{"className":36897},[462],[238,36899],{"className":36900,"style":430},[429],[238,36902,992],{"className":36903},[434],[238,36905],{"className":36906,"style":430},[429],[238,36908,36910,36914,36963,36966,36995,36998,37001,37004,37007,37010,37013,37042,37045,37048,37051,37054,37057,37060,37063,37066],{"className":36909},[273],[238,36911],{"className":36912,"style":36913},[277],"height:1.6444em;vertical-align:-0.8944em;",[238,36915,36917],{"className":36916},[1943,3493],[238,36918,36920,36954],{"className":36919},[395,848],[238,36921,36923,36951],{"className":36922},[399],[238,36924,36926,36941],{"className":36925,"style":489},[403],[238,36927,36929,36932],{"style":36928},"top:-2.2056em;margin-left:0em;",[238,36930],{"className":36931,"style":1134},[411],[238,36933,36935],{"className":36934},[416,417,418,419],[238,36936,36938],{"className":36937},[282,419],[238,36939,2297],{"className":36940,"style":2313},[282,283,419],[238,36942,36943,36946],{"style":21798},[238,36944],{"className":36945,"style":1134},[411],[238,36947,36948],{},[238,36949,21664],{"className":36950},[1943],[238,36952,872],{"className":36953},[871],[238,36955,36957],{"className":36956},[399],[238,36958,36961],{"className":36959,"style":36960},[403],"height:0.8944em;",[238,36962],{},[238,36964],{"className":36965,"style":455},[429],[238,36967,36969,36972],{"className":36968},[282],[238,36970,4429],{"className":36971,"style":660},[282,283],[238,36973,36975],{"className":36974},[391],[238,36976,36978],{"className":36977},[395],[238,36979,36981],{"className":36980},[399],[238,36982,36984],{"className":36983,"style":4620},[403],[238,36985,36986,36989],{"style":1226},[238,36987],{"className":36988,"style":412},[411],[238,36990,36992],{"className":36991},[416,417,418,419],[238,36993,2297],{"className":36994,"style":2313},[282,283,419],[238,36996,335],{"className":36997},[381],[238,36999,341],{"className":37000},[282,283],[238,37002,360],{"className":37003},[462],[238,37005,354],{"className":37006},[451],[238,37008],{"className":37009,"style":10986},[429],[238,37011],{"className":37012,"style":455},[429],[238,37014,37016,37019],{"className":37015},[282],[238,37017,4494],{"className":37018},[282,283],[238,37020,37022],{"className":37021},[391],[238,37023,37025],{"className":37024},[395],[238,37026,37028],{"className":37027},[399],[238,37029,37031],{"className":37030,"style":22441},[403],[238,37032,37033,37036],{"style":1226},[238,37034],{"className":37035,"style":412},[411],[238,37037,37039],{"className":37038},[416,417,418,419],[238,37040,22083],{"className":37041},[664,419],[238,37043,335],{"className":37044},[381],[238,37046,341],{"className":37047},[282,283],[238,37049,354],{"className":37050},[451],[238,37052],{"className":37053,"style":455},[429],[238,37055,357],{"className":37056},[282,283],[238,37058,360],{"className":37059},[462],[238,37061],{"className":37062,"style":430},[429],[238,37064,992],{"className":37065},[434],[238,37067],{"className":37068,"style":430},[429],[238,37070,37072,37075,37122,37125,37154,37157,37160,37163,37166,37169,37172],{"className":37071},[273],[238,37073],{"className":37074,"style":36913},[277],[238,37076,37078],{"className":37077},[1943,3493],[238,37079,37081,37114],{"className":37080},[395,848],[238,37082,37084,37111],{"className":37083},[399],[238,37085,37087,37101],{"className":37086,"style":489},[403],[238,37088,37089,37092],{"style":36928},[238,37090],{"className":37091,"style":1134},[411],[238,37093,37095],{"className":37094},[416,417,418,419],[238,37096,37098],{"className":37097},[282,419],[238,37099,2297],{"className":37100,"style":2313},[282,283,419],[238,37102,37103,37106],{"style":21798},[238,37104],{"className":37105,"style":1134},[411],[238,37107,37108],{},[238,37109,21664],{"className":37110},[1943],[238,37112,872],{"className":37113},[871],[238,37115,37117],{"className":37116},[399],[238,37118,37120],{"className":37119,"style":36960},[403],[238,37121],{},[238,37123],{"className":37124,"style":455},[429],[238,37126,37128,37131],{"className":37127},[282],[238,37129,4494],{"className":37130},[282,283],[238,37132,37134],{"className":37133},[391],[238,37135,37137],{"className":37136},[395],[238,37138,37140],{"className":37139},[399],[238,37141,37143],{"className":37142,"style":4620},[403],[238,37144,37145,37148],{"style":1226},[238,37146],{"className":37147,"style":412},[411],[238,37149,37151],{"className":37150},[416,417,418,419],[238,37152,2297],{"className":37153,"style":2313},[282,283,419],[238,37155,335],{"className":37156},[381],[238,37158,341],{"className":37159},[282,283],[238,37161,354],{"className":37162},[451],[238,37164],{"className":37165,"style":455},[429],[238,37167,357],{"className":37168},[282,283],[238,37170,360],{"className":37171},[462],[238,37173,3102],{"className":37174},[282],[11,37176,37177,37178,16685,37237,2676,37384,2705,37442,1806],{},"若策略 ",[238,37179,37181,37199],{"className":37180,"translate":242},[241],[238,37182,37184],{"className":37183},[246],[248,37185,37186],{"xmlns":250},[252,37187,37188,37196],{},[255,37189,37190],{},[337,37191,37192,37194],{},[258,37193,2297],{},[332,37195,22083],{},[262,37197,37198],{"encoding":264},"\\pi^*",[238,37200,37202],{"className":37201,"ariaHidden":269},[268],[238,37203,37205,37208],{"className":37204},[273],[238,37206],{"className":37207,"style":22096},[277],[238,37209,37211,37214],{"className":37210},[282],[238,37212,2297],{"className":37213,"style":2313},[282,283],[238,37215,37217],{"className":37216},[391],[238,37218,37220],{"className":37219},[395],[238,37221,37223],{"className":37222},[399],[238,37224,37226],{"className":37225,"style":22096},[403],[238,37227,37228,37231],{"style":407},[238,37229],{"className":37230,"style":412},[411],[238,37232,37234],{"className":37233},[416,417,418,419],[238,37235,22083],{"className":37236},[664,419],[238,37238,37240,37270],{"className":37239,"translate":242},[241],[238,37241,37243],{"className":37242},[246],[248,37244,37245],{"xmlns":250},[252,37246,37247,37267],{},[255,37248,37249,37259,37261],{},[337,37250,37251,37253],{},[258,37252,4429],{},[337,37254,37255,37257],{},[258,37256,2297],{},[332,37258,22083],{},[332,37260,992],{},[337,37262,37263,37265],{},[258,37264,4429],{},[332,37266,22083],{},[262,37268,37269],{"encoding":264},"V^{\\pi^*}=V^*",[238,37271,37273,37349],{"className":37272,"ariaHidden":269},[268],[238,37274,37276,37280,37340,37343,37346],{"className":37275},[273],[238,37277],{"className":37278,"style":37279},[277],"height:0.8973em;",[238,37281,37283,37286],{"className":37282},[282],[238,37284,4429],{"className":37285,"style":660},[282,283],[238,37287,37289],{"className":37288},[391],[238,37290,37292],{"className":37291},[395],[238,37293,37295],{"className":37294},[399],[238,37296,37298],{"className":37297,"style":37279},[403],[238,37299,37300,37303],{"style":407},[238,37301],{"className":37302,"style":412},[411],[238,37304,37306],{"className":37305},[416,417,418,419],[238,37307,37309],{"className":37308},[282,419],[238,37310,37312,37315],{"className":37311},[282,419],[238,37313,2297],{"className":37314,"style":2313},[282,283,419],[238,37316,37318],{"className":37317},[391],[238,37319,37321],{"className":37320},[395],[238,37322,37324],{"className":37323},[399],[238,37325,37328],{"className":37326,"style":37327},[403],"height:0.7633em;",[238,37329,37331,37334],{"style":37330},"top:-2.931em;margin-right:0.0714em;",[238,37332],{"className":37333,"style":4203},[411],[238,37335,37337],{"className":37336},[416,4207,4208,419],[238,37338,22083],{"className":37339},[664,419],[238,37341],{"className":37342,"style":430},[429],[238,37344,992],{"className":37345},[434],[238,37347],{"className":37348,"style":430},[429],[238,37350,37352,37355],{"className":37351},[273],[238,37353],{"className":37354,"style":22096},[277],[238,37356,37358,37361],{"className":37357},[282],[238,37359,4429],{"className":37360,"style":660},[282,283],[238,37362,37364],{"className":37363},[391],[238,37365,37367],{"className":37366},[395],[238,37368,37370],{"className":37369},[399],[238,37371,37373],{"className":37372,"style":22096},[403],[238,37374,37375,37378],{"style":407},[238,37376],{"className":37377,"style":412},[411],[238,37379,37381],{"className":37380},[416,417,418,419],[238,37382,22083],{"className":37383},[664,419],[238,37385,37387,37404],{"className":37386,"translate":242},[241],[238,37388,37390],{"className":37389},[246],[248,37391,37392],{"xmlns":250},[252,37393,37394,37402],{},[255,37395,37396],{},[337,37397,37398,37400],{},[258,37399,2297],{},[332,37401,22083],{},[262,37403,37198],{"encoding":264},[238,37405,37407],{"className":37406,"ariaHidden":269},[268],[238,37408,37410,37413],{"className":37409},[273],[238,37411],{"className":37412,"style":22096},[277],[238,37414,37416,37419],{"className":37415},[282],[238,37417,2297],{"className":37418,"style":2313},[282,283],[238,37420,37422],{"className":37421},[391],[238,37423,37425],{"className":37424},[395],[238,37426,37428],{"className":37427},[399],[238,37429,37431],{"className":37430,"style":22096},[403],[238,37432,37433,37436],{"style":407},[238,37434],{"className":37435,"style":412},[411],[238,37437,37439],{"className":37438},[416,417,418,419],[238,37440,22083],{"className":37441},[664,419],[1688,37443,37444],{},"最优策略",[11,37446,37447,37448,37452,37453,37484,37485,37516,37517,37545],{},"在有限状态与有限动作的折扣 MDP 中，上确界实际上可以取到，而且存在一个平稳确定性最优策略。这一点将由",[357,37449,37451],{"href":37450},"#theorem-bellman-optimality","定理 4.2"," 的证明一并给出。因此，本节的 ",[238,37454,37456,37472],{"className":37455,"translate":242},[241],[238,37457,37459],{"className":37458},[246],[248,37460,37461],{"xmlns":250},[252,37462,37463,37469],{},[255,37464,37465,37467],{},[258,37466,21664],{},[332,37468,1889],{},[262,37470,37471],{"encoding":264},"\\sup",[238,37473,37475],{"className":37474,"ariaHidden":269},[268],[238,37476,37478,37481],{"className":37477},[273],[238,37479],{"className":37480,"style":2853},[277],[238,37482,21664],{"className":37483},[1943]," 可以理解为 ",[238,37486,37488,37504],{"className":37487,"translate":242},[241],[238,37489,37491],{"className":37490},[246],[248,37492,37493],{"xmlns":250},[252,37494,37495,37501],{},[255,37496,37497,37499],{},[258,37498,1886],{},[332,37500,1889],{},[262,37502,37503],{"encoding":264},"\\max",[238,37505,37507],{"className":37506,"ariaHidden":269},[268],[238,37508,37510,37513],{"className":37509},[273],[238,37511],{"className":37512,"style":489},[277],[238,37514,1886],{"className":37515},[1943],"。本文的 ",[238,37518,37520,37533],{"className":37519,"translate":242},[241],[238,37521,37523],{"className":37522},[246],[248,37524,37525],{"xmlns":250},[252,37526,37527,37531],{},[255,37528,37529],{},[258,37530,2297],{},[262,37532,2300],{"encoding":264},[238,37534,37536],{"className":37535,"ariaHidden":269},[268],[238,37537,37539,37542],{"className":37538},[273],[238,37540],{"className":37541,"style":489},[277],[238,37543,2297],{"className":37544,"style":2313},[282,283]," 均指平稳策略。对于折扣有限 MDP，平稳策略类已经包含最优策略，因此这一限制不失一般性。",[225,37547,37549],{"id":37548},"_42-最优方程的证明","4.2 最优方程的证明",[11,37551,37552,37553,2282,37556,13548],{},"第 3.1 节已经定义了",[1688,37554,37555],{},"贝尔曼最优算子",[238,37557,37559,37576],{"className":37558,"translate":242},[241],[238,37560,37562],{"className":37561},[246],[248,37563,37564],{"xmlns":250},[252,37565,37566,37574],{},[255,37567,37568],{},[337,37569,37570,37572],{},[258,37571,11971],{"mathvariant":816},[332,37573,22083],{},[262,37575,22086],{"encoding":264},[238,37577,37579],{"className":37578,"ariaHidden":269},[268],[238,37580,37582,37585],{"className":37581},[273],[238,37583],{"className":37584,"style":22096},[277],[238,37586,37588,37591],{"className":37587},[282],[238,37589,11971],{"className":37590,"style":11992},[282,840],[238,37592,37594],{"className":37593},[391],[238,37595,37597],{"className":37596},[395],[238,37598,37600],{"className":37599},[399],[238,37601,37603],{"className":37602,"style":22096},[403],[238,37604,37605,37608],{"style":407},[238,37606],{"className":37607,"style":412},[411],[238,37609,37611],{"className":37610},[416,417,418,419],[238,37612,22083],{"className":37613},[664,419],[238,37615,37617],{"className":37616,"translate":242},[945],[238,37618,37620,37735],{"className":37619,"translate":242},[241],[238,37621,37623],{"className":37622},[246],[248,37624,37625],{"xmlns":250,"display":954},[252,37626,37627,37733],{},[255,37628,37629,37631,37637,37639,37641,37643,37645,37647,37649,37659,37669,37671,37673,37679,37681,37683,37685,37687,37689,37731],{},[332,37630,335],{"stretchy":334},[337,37632,37633,37635],{},[258,37634,11971],{"mathvariant":816},[332,37636,22083],{},[258,37638,4429],{},[332,37640,360],{"stretchy":334},[332,37642,335],{"stretchy":334},[258,37644,341],{},[332,37646,360],{"stretchy":334},[332,37648,992],{},[6755,37650,37651,37657],{},[255,37652,37653,37655],{},[258,37654,1886],{},[332,37656,1889],{},[258,37658,357],{},[6755,37660,37661,37663],{},[332,37662,3390],{},[337,37664,37665,37667],{},[258,37666,341],{},[332,37668,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,37670,330],{},[332,37672,335],{"stretchy":334},[337,37674,37675,37677],{},[258,37676,341],{},[332,37678,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,37680,349],{},[258,37682,341],{},[332,37684,354],{"separator":269},[258,37686,357],{},[332,37688,360],{"stretchy":334},[255,37690,37691,37693,37695,37697,37699,37701,37703,37705,37711,37713,37715,37717,37719,37721,37727,37729],{},[332,37692,736],{"fence":269},[258,37694,600],{},[332,37696,335],{"stretchy":334},[258,37698,341],{},[332,37700,354],{"separator":269},[258,37702,357],{},[332,37704,354],{"separator":269},[337,37706,37707,37709],{},[258,37708,341],{},[332,37710,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,37712,360],{"stretchy":334},[332,37714,987],{},[258,37716,730],{},[258,37718,4429],{},[332,37720,335],{"stretchy":334},[337,37722,37723,37725],{},[258,37724,341],{},[332,37726,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,37728,360],{"stretchy":334},[332,37730,4479],{"fence":269},[258,37732,3102],{"mathvariant":344},[262,37734,22410],{"encoding":264},[238,37736,37738,37800,37982],{"className":37737,"ariaHidden":269},[268],[238,37739,37741,37744,37747,37776,37779,37782,37785,37788,37791,37794,37797],{"className":37740},[273],[238,37742],{"className":37743,"style":444},[277],[238,37745,335],{"className":37746},[381],[238,37748,37750,37753],{"className":37749},[282],[238,37751,11971],{"className":37752,"style":11992},[282,840],[238,37754,37756],{"className":37755},[391],[238,37757,37759],{"className":37758},[395],[238,37760,37762],{"className":37761},[399],[238,37763,37765],{"className":37764,"style":22441},[403],[238,37766,37767,37770],{"style":1226},[238,37768],{"className":37769,"style":412},[411],[238,37771,37773],{"className":37772},[416,417,418,419],[238,37774,22083],{"className":37775},[664,419],[238,37777,4429],{"className":37778,"style":660},[282,283],[238,37780,360],{"className":37781},[462],[238,37783,335],{"className":37784},[381],[238,37786,341],{"className":37787},[282,283],[238,37789,360],{"className":37790},[462],[238,37792],{"className":37793,"style":430},[429],[238,37795,992],{"className":37796},[434],[238,37798],{"className":37799,"style":430},[429],[238,37801,37803,37806,37853,37856,37932,37935,37938,37941,37973,37976,37979],{"className":37802},[273],[238,37804],{"className":37805,"style":22483},[277],[238,37807,37809],{"className":37808},[1943,3493],[238,37810,37812,37845],{"className":37811},[395,848],[238,37813,37815,37842],{"className":37814},[399],[238,37816,37818,37832],{"className":37817,"style":489},[403],[238,37819,37820,37823],{"style":22498},[238,37821],{"className":37822,"style":1134},[411],[238,37824,37826],{"className":37825},[416,417,418,419],[238,37827,37829],{"className":37828},[282,419],[238,37830,357],{"className":37831},[282,283,419],[238,37833,37834,37837],{"style":21798},[238,37835],{"className":37836,"style":1134},[411],[238,37838,37839],{},[238,37840,1886],{"className":37841},[1943],[238,37843,872],{"className":37844},[871],[238,37846,37848],{"className":37847},[399],[238,37849,37851],{"className":37850,"style":22530},[403],[238,37852],{},[238,37854],{"className":37855,"style":455},[429],[238,37857,37859],{"className":37858},[1943,3493],[238,37860,37862,37924],{"className":37861},[395,848],[238,37863,37865,37921],{"className":37864},[399],[238,37866,37868,37911],{"className":37867,"style":6869},[403],[238,37869,37870,37873],{"style":7296},[238,37871],{"className":37872,"style":3510},[411],[238,37874,37876],{"className":37875},[416,417,418,419],[238,37877,37879],{"className":37878},[282,419],[238,37880,37882,37885],{"className":37881},[282,419],[238,37883,341],{"className":37884},[282,283,419],[238,37886,37888],{"className":37887},[391],[238,37889,37891],{"className":37890},[395],[238,37892,37894],{"className":37893},[399],[238,37895,37897],{"className":37896,"style":4196},[403],[238,37898,37899,37902],{"style":4199},[238,37900],{"className":37901,"style":4203},[411],[238,37903,37905],{"className":37904},[416,4207,4208,419],[238,37906,37908],{"className":37907},[282,419],[238,37909,346],{"className":37910},[282,419],[238,37912,37913,37916],{"style":3529},[238,37914],{"className":37915,"style":3510},[411],[238,37917,37918],{},[238,37919,3390],{"className":37920},[1943,3538,3539],[238,37922,872],{"className":37923},[871],[238,37925,37927],{"className":37926},[399],[238,37928,37930],{"className":37929,"style":7357},[403],[238,37931],{},[238,37933],{"className":37934,"style":455},[429],[238,37936,330],{"className":37937,"style":377},[282,283],[238,37939,335],{"className":37940},[381],[238,37942,37944,37947],{"className":37943},[282],[238,37945,341],{"className":37946},[282,283],[238,37948,37950],{"className":37949},[391],[238,37951,37953],{"className":37952},[395],[238,37954,37956],{"className":37955},[399],[238,37957,37959],{"className":37958,"style":1223},[403],[238,37960,37961,37964],{"style":1226},[238,37962],{"className":37963,"style":412},[411],[238,37965,37967],{"className":37966},[416,417,418,419],[238,37968,37970],{"className":37969},[282,419],[238,37971,346],{"className":37972},[282,419],[238,37974],{"className":37975,"style":430},[429],[238,37977,349],{"className":37978},[434],[238,37980],{"className":37981,"style":430},[429],[238,37983,37985,37988,37991,37994,37997,38000,38003,38006,38127,38130],{"className":37984},[273],[238,37986],{"className":37987,"style":22668},[277],[238,37989,341],{"className":37990},[282,283],[238,37992,354],{"className":37993},[451],[238,37995],{"className":37996,"style":455},[429],[238,37998,357],{"className":37999},[282,283],[238,38001,360],{"className":38002},[462],[238,38004],{"className":38005,"style":455},[429],[238,38007,38009,38012,38015,38018,38021,38024,38027,38030,38033,38036,38068,38071,38074,38077,38080,38083,38086,38089,38121,38124],{"className":38008},[3041],[238,38010,736],{"className":38011,"style":4791},[381,4790],[238,38013,600],{"className":38014,"style":637},[282,283],[238,38016,335],{"className":38017},[381],[238,38019,341],{"className":38020},[282,283],[238,38022,354],{"className":38023},[451],[238,38025],{"className":38026,"style":455},[429],[238,38028,357],{"className":38029},[282,283],[238,38031,354],{"className":38032},[451],[238,38034],{"className":38035,"style":455},[429],[238,38037,38039,38042],{"className":38038},[282],[238,38040,341],{"className":38041},[282,283],[238,38043,38045],{"className":38044},[391],[238,38046,38048],{"className":38047},[395],[238,38049,38051],{"className":38050},[399],[238,38052,38054],{"className":38053,"style":1223},[403],[238,38055,38056,38059],{"style":1226},[238,38057],{"className":38058,"style":412},[411],[238,38060,38062],{"className":38061},[416,417,418,419],[238,38063,38065],{"className":38064},[282,419],[238,38066,346],{"className":38067},[282,419],[238,38069,360],{"className":38070},[462],[238,38072],{"className":38073,"style":660},[429],[238,38075,987],{"className":38076},[664],[238,38078],{"className":38079,"style":660},[429],[238,38081,730],{"className":38082,"style":764},[282,283],[238,38084,4429],{"className":38085,"style":660},[282,283],[238,38087,335],{"className":38088},[381],[238,38090,38092,38095],{"className":38091},[282],[238,38093,341],{"className":38094},[282,283],[238,38096,38098],{"className":38097},[391],[238,38099,38101],{"className":38100},[395],[238,38102,38104],{"className":38103},[399],[238,38105,38107],{"className":38106,"style":1223},[403],[238,38108,38109,38112],{"style":1226},[238,38110],{"className":38111,"style":412},[411],[238,38113,38115],{"className":38114},[416,417,418,419],[238,38116,38118],{"className":38117},[282,419],[238,38119,346],{"className":38120},[282,419],[238,38122,360],{"className":38123},[462],[238,38125,4479],{"className":38126,"style":4791},[462,4790],[238,38128],{"className":38129,"style":455},[429],[238,38131,3102],{"className":38132},[282],[11,38134,38135,38136,38194,38195,38253],{},"与 ",[238,38137,38139,38156],{"className":38138,"translate":242},[241],[238,38140,38142],{"className":38141},[246],[248,38143,38144],{"xmlns":250},[252,38145,38146,38154],{},[255,38147,38148],{},[337,38149,38150,38152],{},[258,38151,11971],{"mathvariant":816},[258,38153,2297],{},[262,38155,11976],{"encoding":264},[238,38157,38159],{"className":38158,"ariaHidden":269},[268],[238,38160,38162,38165],{"className":38161},[273],[238,38163],{"className":38164,"style":278},[277],[238,38166,38168,38171],{"className":38167},[282],[238,38169,11971],{"className":38170,"style":11992},[282,840],[238,38172,38174],{"className":38173},[391],[238,38175,38177],{"className":38176},[395],[238,38178,38180],{"className":38179},[399],[238,38181,38183],{"className":38182,"style":6609},[403],[238,38184,38185,38188],{"style":407},[238,38186],{"className":38187,"style":412},[411],[238,38189,38191],{"className":38190},[416,417,418,419],[238,38192,2297],{"className":38193,"style":2313},[282,283,419]," 不同，",[238,38196,38198,38215],{"className":38197,"translate":242},[241],[238,38199,38201],{"className":38200},[246],[248,38202,38203],{"xmlns":250},[252,38204,38205,38213],{},[255,38206,38207],{},[337,38208,38209,38211],{},[258,38210,11971],{"mathvariant":816},[332,38212,22083],{},[262,38214,22086],{"encoding":264},[238,38216,38218],{"className":38217,"ariaHidden":269},[268],[238,38219,38221,38224],{"className":38220},[273],[238,38222],{"className":38223,"style":22096},[277],[238,38225,38227,38230],{"className":38226},[282],[238,38228,11971],{"className":38229,"style":11992},[282,840],[238,38231,38233],{"className":38232},[391],[238,38234,38236],{"className":38235},[395],[238,38237,38239],{"className":38238},[399],[238,38240,38242],{"className":38241,"style":22096},[403],[238,38243,38244,38247],{"style":407},[238,38245],{"className":38246,"style":412},[411],[238,38248,38250],{"className":38249},[416,417,418,419],[238,38251,22083],{"className":38252},[664,419]," 先对每个动作计算期望，再在动作上取最大值。这个顺序不能交换，原因见 4.3 节。",[230,38255,38258,38493,39125,39215],{"id":38256,"title":38257},"theorem-bellman-optimality","定理 4.2（贝尔曼最优方程）",[11,38259,236,38260,941,38333,1838,38361,38389,38390,38448,38449,38492],{},[238,38261,38263,38288],{"className":38262,"translate":242},[241],[238,38264,38266],{"className":38265},[246],[248,38267,38268],{"xmlns":250},[252,38269,38270,38286],{},[255,38271,38272,38274,38276,38278,38280,38282,38284],{},[258,38273,730],{},[332,38275,733],{},[332,38277,736],{"stretchy":334},[738,38279,740],{},[332,38281,354],{"separator":269},[738,38283,745],{},[332,38285,360],{"stretchy":334},[262,38287,750],{"encoding":264},[238,38289,38291,38309],{"className":38290,"ariaHidden":269},[268],[238,38292,38294,38297,38300,38303,38306],{"className":38293},[273],[238,38295],{"className":38296,"style":760},[277],[238,38298,730],{"className":38299,"style":764},[282,283],[238,38301],{"className":38302,"style":430},[429],[238,38304,733],{"className":38305},[434],[238,38307],{"className":38308,"style":430},[429],[238,38310,38312,38315,38318,38321,38324,38327,38330],{"className":38311},[273],[238,38313],{"className":38314,"style":444},[277],[238,38316,736],{"className":38317},[381],[238,38319,740],{"className":38320},[282],[238,38322,354],{"className":38323},[451],[238,38325],{"className":38326,"style":455},[429],[238,38328,745],{"className":38329},[282],[238,38331,360],{"className":38332},[462],[238,38334,38336,38349],{"className":38335,"translate":242},[241],[238,38337,38339],{"className":38338},[246],[248,38340,38341],{"xmlns":250},[252,38342,38343,38347],{},[255,38344,38345],{},[258,38346,260],{},[262,38348,260],{"encoding":264},[238,38350,38352],{"className":38351,"ariaHidden":269},[268],[238,38353,38355,38358],{"className":38354},[273],[238,38356],{"className":38357,"style":278},[277],[238,38359,260],{"className":38360,"style":284},[282,283],[238,38362,38364,38377],{"className":38363,"translate":242},[241],[238,38365,38367],{"className":38366},[246],[248,38368,38369],{"xmlns":250},[252,38370,38371,38375],{},[255,38372,38373],{},[258,38374,300],{},[262,38376,300],{"encoding":264},[238,38378,38380],{"className":38379,"ariaHidden":269},[268],[238,38381,38383,38386],{"className":38382},[273],[238,38384],{"className":38385,"style":278},[277],[238,38387,300],{"className":38388},[282,283]," 有限。则 ",[238,38391,38393,38410],{"className":38392,"translate":242},[241],[238,38394,38396],{"className":38395},[246],[248,38397,38398],{"xmlns":250},[252,38399,38400,38408],{},[255,38401,38402],{},[337,38403,38404,38406],{},[258,38405,4429],{},[332,38407,22083],{},[262,38409,34919],{"encoding":264},[238,38411,38413],{"className":38412,"ariaHidden":269},[268],[238,38414,38416,38419],{"className":38415},[273],[238,38417],{"className":38418,"style":22096},[277],[238,38420,38422,38425],{"className":38421},[282],[238,38423,4429],{"className":38424,"style":660},[282,283],[238,38426,38428],{"className":38427},[391],[238,38429,38431],{"className":38430},[395],[238,38432,38434],{"className":38433},[399],[238,38435,38437],{"className":38436,"style":22096},[403],[238,38438,38439,38442],{"style":407},[238,38440],{"className":38441,"style":412},[411],[238,38443,38445],{"className":38444},[416,417,418,419],[238,38446,22083],{"className":38447},[664,419]," 是 ",[238,38450,38452,38471],{"className":38451,"translate":242},[241],[238,38453,38455],{"className":38454},[246],[248,38456,38457],{"xmlns":250},[252,38458,38459,38469],{},[255,38460,38461,38463,38465,38467],{},[258,38462,19971],{},[332,38464,335],{"stretchy":334},[258,38466,260],{},[332,38468,360],{"stretchy":334},[262,38470,21579],{"encoding":264},[238,38472,38474],{"className":38473,"ariaHidden":269},[268],[238,38475,38477,38480,38483,38486,38489],{"className":38476},[273],[238,38478],{"className":38479,"style":444},[277],[238,38481,19971],{"className":38482,"style":20011},[282,283],[238,38484,335],{"className":38485},[381],[238,38487,260],{"className":38488,"style":284},[282,283],[238,38490,360],{"className":38491},[462]," 中满足",[238,38494,38496],{"className":38495,"translate":242},[945],[238,38497,38499,38651],{"className":38498,"translate":242},[241],[238,38500,38502],{"className":38501},[246],[248,38503,38504],{"xmlns":250,"display":954},[252,38505,38506,38648],{},[958,38507,38508,38574],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,38509,38510,38524],{},[966,38511,38512],{},[969,38513,38514],{"scriptlevel":740,"displaystyle":269},[255,38515,38516,38518,38520,38522],{},[258,38517,4429],{},[332,38519,335],{"stretchy":334},[258,38521,341],{},[332,38523,360],{"stretchy":334},[966,38525,38526],{},[969,38527,38528],{"scriptlevel":740,"displaystyle":269},[255,38529,38530,38532,38534,38544,38554,38556,38558,38564,38566,38568,38570,38572],{},[255,38531],{},[332,38533,992],{},[6755,38535,38536,38542],{},[255,38537,38538,38540],{},[258,38539,1886],{},[332,38541,1889],{},[258,38543,357],{},[6755,38545,38546,38548],{},[332,38547,3390],{},[337,38549,38550,38552],{},[258,38551,341],{},[332,38553,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,38555,330],{},[332,38557,335],{"stretchy":334},[337,38559,38560,38562],{},[258,38561,341],{},[332,38563,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,38565,349],{},[258,38567,341],{},[332,38569,354],{"separator":269},[258,38571,357],{},[332,38573,360],{"stretchy":334},[963,38575,38576,38582],{},[966,38577,38578],{},[969,38579,38580],{"scriptlevel":740,"displaystyle":269},[255,38581],{},[966,38583,38584],{},[969,38585,38586],{"scriptlevel":740,"displaystyle":269},[255,38587,38588,38590,38592,38594,38636,38638,38640,38642,38644,38646],{},[255,38589],{},[429,38591],{"width":7090},[332,38593,608],{},[255,38595,38596,38598,38600,38602,38604,38606,38608,38610,38616,38618,38620,38622,38624,38626,38632,38634],{},[332,38597,736],{"fence":269},[258,38599,600],{},[332,38601,335],{"stretchy":334},[258,38603,341],{},[332,38605,354],{"separator":269},[258,38607,357],{},[332,38609,354],{"separator":269},[337,38611,38612,38614],{},[258,38613,341],{},[332,38615,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,38617,360],{"stretchy":334},[332,38619,987],{},[258,38621,730],{},[258,38623,4429],{},[332,38625,335],{"stretchy":334},[337,38627,38628,38630],{},[258,38629,341],{},[332,38631,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,38633,360],{"stretchy":334},[332,38635,4479],{"fence":269},[332,38637,354],{"separator":269},[429,38639],{"width":10450},[258,38641,10453],{"mathvariant":344},[258,38643,341],{},[332,38645,733],{},[258,38647,260],{},[262,38649,38650],{"encoding":264},"\\begin{aligned}\nV(s)&=\\max_{a}\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V(s')\\right],\n\\qquad \\forall s\\in S\n\\end{aligned}",[238,38652,38654],{"className":38653,"ariaHidden":269},[268],[238,38655,38657,38660],{"className":38656},[273],[238,38658],{"className":38659,"style":7153},[277],[238,38661,38663],{"className":38662},[282],[238,38664,38666,38717],{"className":38665},[958],[238,38667,38669],{"className":38668},[1117],[238,38670,38672,38709],{"className":38671},[395,848],[238,38673,38675,38706],{"className":38674},[399],[238,38676,38678,38698],{"className":38677,"style":7172},[403],[238,38679,38680,38683],{"style":7175},[238,38681],{"className":38682,"style":3510},[411],[238,38684,38686,38689,38692,38695],{"className":38685},[282],[238,38687,4429],{"className":38688,"style":660},[282,283],[238,38690,335],{"className":38691},[381],[238,38693,341],{"className":38694},[282,283],[238,38696,360],{"className":38697},[462],[238,38699,38700,38703],{"style":7231},[238,38701],{"className":38702,"style":3510},[411],[238,38704],{"className":38705},[282],[238,38707,872],{"className":38708},[871],[238,38710,38712],{"className":38711},[399],[238,38713,38715],{"className":38714,"style":7247},[403],[238,38716],{},[238,38718,38720],{"className":38719},[1422],[238,38721,38723,39117],{"className":38722},[395,848],[238,38724,38726,39114],{"className":38725},[399],[238,38727,38729,38940],{"className":38728,"style":7172},[403],[238,38730,38731,38734],{"style":7175},[238,38732],{"className":38733,"style":3510},[411],[238,38735,38737,38740,38743,38746,38749,38796,38799,38875,38878,38881,38884,38916,38919,38922,38925,38928,38931,38934,38937],{"className":38736},[282],[238,38738],{"className":38739},[282],[238,38741],{"className":38742,"style":430},[429],[238,38744,992],{"className":38745},[434],[238,38747],{"className":38748,"style":430},[429],[238,38750,38752],{"className":38751},[1943,3493],[238,38753,38755,38788],{"className":38754},[395,848],[238,38756,38758,38785],{"className":38757},[399],[238,38759,38761,38775],{"className":38760,"style":489},[403],[238,38762,38763,38766],{"style":22498},[238,38764],{"className":38765,"style":1134},[411],[238,38767,38769],{"className":38768},[416,417,418,419],[238,38770,38772],{"className":38771},[282,419],[238,38773,357],{"className":38774},[282,283,419],[238,38776,38777,38780],{"style":21798},[238,38778],{"className":38779,"style":1134},[411],[238,38781,38782],{},[238,38783,1886],{"className":38784},[1943],[238,38786,872],{"className":38787},[871],[238,38789,38791],{"className":38790},[399],[238,38792,38794],{"className":38793,"style":22530},[403],[238,38795],{},[238,38797],{"className":38798,"style":455},[429],[238,38800,38802],{"className":38801},[1943,3493],[238,38803,38805,38867],{"className":38804},[395,848],[238,38806,38808,38864],{"className":38807},[399],[238,38809,38811,38854],{"className":38810,"style":6869},[403],[238,38812,38813,38816],{"style":7296},[238,38814],{"className":38815,"style":3510},[411],[238,38817,38819],{"className":38818},[416,417,418,419],[238,38820,38822],{"className":38821},[282,419],[238,38823,38825,38828],{"className":38824},[282,419],[238,38826,341],{"className":38827},[282,283,419],[238,38829,38831],{"className":38830},[391],[238,38832,38834],{"className":38833},[395],[238,38835,38837],{"className":38836},[399],[238,38838,38840],{"className":38839,"style":4196},[403],[238,38841,38842,38845],{"style":4199},[238,38843],{"className":38844,"style":4203},[411],[238,38846,38848],{"className":38847},[416,4207,4208,419],[238,38849,38851],{"className":38850},[282,419],[238,38852,346],{"className":38853},[282,419],[238,38855,38856,38859],{"style":3529},[238,38857],{"className":38858,"style":3510},[411],[238,38860,38861],{},[238,38862,3390],{"className":38863},[1943,3538,3539],[238,38865,872],{"className":38866},[871],[238,38868,38870],{"className":38869},[399],[238,38871,38873],{"className":38872,"style":7357},[403],[238,38874],{},[238,38876],{"className":38877,"style":455},[429],[238,38879,330],{"className":38880,"style":377},[282,283],[238,38882,335],{"className":38883},[381],[238,38885,38887,38890],{"className":38886},[282],[238,38888,341],{"className":38889},[282,283],[238,38891,38893],{"className":38892},[391],[238,38894,38896],{"className":38895},[395],[238,38897,38899],{"className":38898},[399],[238,38900,38902],{"className":38901,"style":1223},[403],[238,38903,38904,38907],{"style":1226},[238,38905],{"className":38906,"style":412},[411],[238,38908,38910],{"className":38909},[416,417,418,419],[238,38911,38913],{"className":38912},[282,419],[238,38914,346],{"className":38915},[282,419],[238,38917],{"className":38918,"style":430},[429],[238,38920,349],{"className":38921},[434],[238,38923],{"className":38924,"style":430},[429],[238,38926,341],{"className":38927},[282,283],[238,38929,354],{"className":38930},[451],[238,38932],{"className":38933,"style":455},[429],[238,38935,357],{"className":38936},[282,283],[238,38938,360],{"className":38939},[462],[238,38941,38942,38945],{"style":7231},[238,38943],{"className":38944,"style":3510},[411],[238,38946,38948,38951,38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",[238,39129,39131,39148],{"className":39130,"translate":242},[241],[238,39132,39134],{"className":39133},[246],[248,39135,39136],{"xmlns":250},[252,39137,39138,39146],{},[255,39139,39140],{},[337,39141,39142,39144],{},[258,39143,2297],{},[332,39145,22083],{},[262,39147,37198],{"encoding":264},[238,39149,39151],{"className":39150,"ariaHidden":269},[268],[238,39152,39154,39157],{"className":39153},[273],[238,39155],{"className":39156,"style":22096},[277],[238,39158,39160,39163],{"className":39159},[282],[238,39161,2297],{"className":39162,"style":2313},[282,283],[238,39164,39166],{"className":39165},[391],[238,39167,39169],{"className":39168},[395],[238,39170,39172],{"className":39171},[399],[238,39173,39175],{"className":39174,"style":22096},[403],[238,39176,39177,39180],{"style":407},[238,39178],{"className":39179,"style":412},[411],[238,39181,39183],{"className":39182},[416,417,418,419],[238,39184,22083],{"className":39185},[664,419]," 最优当且仅当对每个 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s,a)\n\\left[r(s,a,s')+\\gamma V^*(s')\\right].",[238,39341,39343,39396,39588],{"className":39342,"ariaHidden":269},[268],[238,39344,39346,39349,39378,39381,39384,39387,39390,39393],{"className":39345},[273],[238,39347],{"className":39348,"style":444},[277],[238,39350,39352,39355],{"className":39351},[282],[238,39353,2297],{"className":39354,"style":2313},[282,283],[238,39356,39358],{"className":39357},[391],[238,39359,39361],{"className":39360},[395],[238,39362,39364],{"className":39363},[399],[238,39365,39367],{"className":39366,"style":22441},[403],[238,39368,39369,39372],{"style":1226},[238,39370],{"className":39371,"style":412},[411],[238,39373,39375],{"className":39374},[416,417,418,419],[238,39376,22083],{"className":39377},[664,419],[238,39379,335],{"className":39380},[381],[238,39382,341],{"className":39383},[282,283],[238,39385,360],{"className":39386},[462],[238,39388],{"className":39389,"style":430},[429],[238,39391,733],{"className":39392},[434],[238,39394],{"className":39395,"style":430},[429],[238,39397,39399,39402,39409,39412,39459,39462,39538,39541,39544,39547,39579,39582,39585],{"className":39398},[273],[238,39400],{"className":39401,"style":22483},[277],[238,39403,39405,39406],{"className":39404},[1943],"ar",[238,39407,39408],{"style":35277},"g",[238,39410],{"className":39411,"style":455},[429],[238,39413,39415],{"className":39414},[1943,3493],[238,39416,39418,39451],{"className":39417},[395,848],[238,39419,39421,39448],{"className":39420},[399],[238,39422,39424,39438],{"className":39423,"style":489},[403],[238,39425,39426,39429],{"style":22498},[238,39427],{"className":39428,"style":1134},[411],[238,39430,39432],{"className":39431},[416,417,418,419],[238,39433,39435],{"className":39434},[282,419],[238,39436,357],{"className":39437},[282,283,419],[238,39439,39440,39443],{"style":21798},[238,39441],{"className":39442,"style":1134},[411],[238,39444,39445],{},[238,39446,1886],{"className":39447},[1943],[238,39449,872],{"className":39450},[871],[238,39452,39454],{"className":39453},[399],[238,39455,39457],{"className":39456,"style":22530},[403],[238,39458],{},[238,39460],{"className":39461,"style":455},[429],[238,39463,39465],{"className":39464},[1943,3493],[238,39466,39468,39530],{"className":39467},[395,848],[238,39469,39471,39527],{"className":39470},[399],[238,39472,39474,39517],{"className":39473,"style":6869},[403],[238,39475,39476,39479],{"style":7296},[238,39477],{"className":39478,"style":3510},[411],[238,39480,39482],{"className":39481},[416,417,418,419],[238,39483,39485],{"className":39484},[282,419],[238,39486,39488,39491],{"className":39487},[282,419],[238,39489,341],{"className":39490},[282,283,419],[238,39492,39494],{"className":39493},[391],[238,39495,39497],{"className":39496},[395],[238,39498,39500],{"className":39499},[399],[238,39501,39503],{"className":39502,"style":4196},[403],[238,39504,39505,39508],{"style":4199},[238,39506],{"className":39507,"style":4203},[411],[238,39509,39511],{"className":39510},[416,4207,4208,419],[238,39512,39514],{"className":39513},[282,419],[238,39515,346],{"className":39516},[282,419],[238,39518,39519,39522],{"style":3529},[238,39520],{"className":39521,"style":3510},[411],[238,39523,39524],{},[238,39525,3390],{"className":39526},[1943,3538,3539],[238,39528,872],{"className":39529},[871],[238,39531,39533],{"className":39532},[399],[238,39534,39536],{"className":39535,"style":7357},[403],[238,39537],{},[238,39539],{"className":39540,"style":455},[429],[238,39542,330],{"className":39543,"style":377},[282,283],[238,39545,335],{"className":39546},[381],[238,39548,39550,39553],{"className":39549},[282],[238,39551,341],{"className":39552},[282,283],[238,39554,39556],{"className":39555},[391],[238,39557,39559],{"className":39558},[395],[238,39560,39562],{"className":39561},[399],[238,39563,39565],{"className":39564,"style":1223},[403],[238,39566,39567,39570],{"style":1226},[238,39568],{"className":39569,"style":412},[411],[238,39571,39573],{"className":39572},[416,417,418,419],[238,39574,39576],{"className":39575},[282,419],[238,39577,346],{"className":39578},[282,419],[238,39580],{"className":39581,"style":430},[429],[238,39583,349],{"className":39584},[434],[238,39586],{"className":39587,"style":430},[429],[238,39589,39591,39594,39597,39600,39603,39606,39609,39612,39759,39762],{"className":39590},[273],[238,39592],{"className":39593,"style":22668},[277],[238,39595,341],{"className":39596},[282,283],[238,39598,354],{"className":39599},[451],[238,39601],{"className":39602,"style":455},[429],[238,39604,357],{"className":39605},[282,283],[238,39607,360],{"className":39608},[462],[238,39610],{"className":39611,"style":455},[429],[238,39613,39615,39618,39621,39624,39627,39630,39633,39636,39639,39642,39674,39677,39680,39683,39686,39689,39718,39721,39753,39756],{"className":39614},[3041],[238,39616,736],{"className":39617,"style":4791},[381,4790],[238,39619,600],{"className":39620,"style":637},[282,283],[238,39622,335],{"className":39623},[381],[238,39625,341],{"className":39626},[282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3.4",[238,39774,39776,39793],{"className":39775,"translate":242},[241],[238,39777,39779],{"className":39778},[246],[248,39780,39781],{"xmlns":250},[252,39782,39783,39791],{},[255,39784,39785],{},[337,39786,39787,39789],{},[258,39788,11971],{"mathvariant":816},[332,39790,22083],{},[262,39792,22086],{"encoding":264},[238,39794,39796],{"className":39795,"ariaHidden":269},[268],[238,39797,39799,39802],{"className":39798},[273],[238,39800],{"className":39801,"style":22096},[277],[238,39803,39805,39808],{"className":39804},[282],[238,39806,11971],{"className":39807,"style":11992},[282,840],[238,39809,39811],{"className":39810},[391],[238,39812,39814],{"className":39813},[395],[238,39815,39817],{"className":39816},[399],[238,39818,39820],{"className":39819,"style":22096},[403],[238,39821,39822,39825],{"style":407},[238,39823],{"className":39824,"style":412},[411],[238,39826,39828],{"className":39827},[416,417,418,419],[238,39829,22083],{"className":39830},[664,419],[238,39832,39834,39853],{"className":39833,"translate":242},[241],[238,39835,39837],{"className":39836},[246],[248,39838,39839],{"xmlns":250},[252,39840,39841,39851],{},[255,39842,39843,39845,39847,39849],{},[258,39844,19971],{},[332,39846,335],{"stretchy":334},[258,39848,260],{},[332,39850,360],{"stretchy":334},[262,39852,21579],{"encoding":264},[238,39854,39856],{"className":39855,"ariaHidden":269},[268],[238,39857,39859,39862,39865,39868,39871],{"className":39858},[273],[238,39860],{"className":39861,"style":444},[277],[238,39863,19971],{"className":39864,"style":20011},[282,283],[238,39866,335],{"className":39867},[381],[238,39869,260],{"className":39870,"style":284},[282,283],[238,39872,360],{"className":39873},[462]," 中存在唯一不动点，记作 ",[238,39876,39878,39898],{"className":39877,"translate":242},[241],[238,39879,39881],{"className":39880},[246],[248,39882,39883],{"xmlns":250},[252,39884,39885,39895],{},[255,39886,39887],{},[39888,39889,39890,39892],"mover",{"accent":269},[258,39891,4429],{},[332,39893,39894],{},"ˉ",[262,39896,39897],{"encoding":264},"\\bar V",[238,39899,39901],{"className":39900,"ariaHidden":269},[268],[238,39902,39904,39908],{"className":39903},[273],[238,39905],{"className":39906,"style":39907},[277],"height:0.8201em;",[238,39909,39912],{"className":39910},[282,39911],"accent",[238,39913,39915],{"className":39914},[395],[238,39916,39918],{"className":39917},[399],[238,39919,39921,39929],{"className":39920,"style":39907},[403],[238,39922,39923,39926],{"style":21798},[238,39924],{"className":39925,"style":1134},[411],[238,39927,4429],{"className":39928,"style":660},[282,283],[238,39930,39932,39935],{"style":39931},"top:-3.2523em;",[238,39933],{"className":39934,"style":1134},[411],[238,39936,39940],{"className":39937,"style":39939},[39938],"accent-body","left:-0.25em;",[238,39941,39894],{"className":39942},[282],"。下面证明 ",[238,39945,39947,39973],{"className":39946,"translate":242},[241],[238,39948,39950],{"className":39949},[246],[248,39951,39952],{"xmlns":250},[252,39953,39954,39970],{},[255,39955,39956,39962,39964],{},[39888,39957,39958,39960],{"accent":269},[258,39959,4429],{},[332,39961,39894],{},[332,39963,992],{},[337,39965,39966,39968],{},[258,39967,4429],{},[332,39969,22083],{},[262,39971,39972],{"encoding":264},"\\bar V=V^*",[238,39974,39976,40022],{"className":39975,"ariaHidden":269},[268],[238,39977,39979,39982,40013,40016,40019],{"className":39978},[273],[238,39980],{"className":39981,"style":39907},[277],[238,39983,39985],{"className":39984},[282,39911],[238,39986,39988],{"className":39987},[395],[238,39989,39991],{"className":39990},[399],[238,39992,39994,40002],{"className":39993,"style":39907},[403],[238,39995,39996,39999],{"style":21798},[238,39997],{"className":39998,"style":1134},[411],[238,40000,4429],{"className":40001,"style":660},[282,283],[238,40003,40004,40007],{"style":39931},[238,40005],{"className":40006,"style":1134},[411],[238,40008,40010],{"className":40009,"style":39939},[39938],[238,40011,39894],{"className":40012},[282],[238,40014],{"className":40015,"style":430},[429],[238,40017,992],{"className":40018},[434],[238,40020],{"className":40021,"style":430},[429],[238,40023,40025,40028],{"className":40024},[273],[238,40026],{"className":40027,"style":22096},[277],[238,40029,40031,40034],{"className":40030},[282],[238,40032,4429],{"className":40033,"style":660},[282,283],[238,40035,40037],{"className":40036},[391],[238,40038,40040],{"className":40039},[395],[238,40041,40043],{"className":40042},[399],[238,40044,40046],{"className":40045,"style":22096},[403],[238,40047,40048,40051],{"style":407},[238,40049],{"className":40050,"style":412},[411],[238,40052,40054],{"className":40053},[416,417,418,419],[238,40055,22083],{"className":40056},[664,419],[11,40058,40059,40060,40173,40174,40202,40203,40261,40262,40320],{},"先证 ",[238,40061,40063,40089],{"className":40062,"translate":242},[241],[238,40064,40066],{"className":40065},[246],[248,40067,40068],{"xmlns":250},[252,40069,40070,40086],{},[255,40071,40072,40078,40080],{},[337,40073,40074,40076],{},[258,40075,4429],{},[332,40077,22083],{},[332,40079,2023],{},[39888,40081,40082,40084],{"accent":269},[258,40083,4429],{},[332,40085,39894],{},[262,40087,40088],{"encoding":264},"V^*\\le\\bar 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",[238,40175,40177,40190],{"className":40176,"translate":242},[241],[238,40178,40180],{"className":40179},[246],[248,40181,40182],{"xmlns":250},[252,40183,40184,40188],{},[255,40185,40186],{},[258,40187,2297],{},[262,40189,2300],{"encoding":264},[238,40191,40193],{"className":40192,"ariaHidden":269},[268],[238,40194,40196,40199],{"className":40195},[273],[238,40197],{"className":40198,"style":489},[277],[238,40200,2297],{"className":40201,"style":2313},[282,283],"，由于 ",[238,40204,40206,40223],{"className":40205,"translate":242},[241],[238,40207,40209],{"className":40208},[246],[248,40210,40211],{"xmlns":250},[252,40212,40213,40221],{},[255,40214,40215],{},[337,40216,40217,40219],{},[258,40218,11971],{"mathvariant":816},[258,40220,2297],{},[262,40222,11976],{"encoding":264},[238,40224,40226],{"className":40225,"ariaHidden":269},[268],[238,40227,40229,40232],{"className":40228},[273],[238,40230],{"className":40231,"style":278},[277],[238,40233,40235,40238],{"className":40234},[282],[238,40236,11971],{"className":40237,"style":11992},[282,840],[238,40239,40241],{"className":40240},[391],[238,40242,40244],{"className":40243},[395],[238,40245,40247],{"className":40246},[399],[238,40248,40250],{"className":40249,"style":6609},[403],[238,40251,40252,40255],{"style":407},[238,40253],{"className":40254,"style":412},[411],[238,40256,40258],{"className":40257},[416,417,418,419],[238,40259,2297],{"className":40260,"style":2313},[282,283,419]," 是各动作值的加权平均，而 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在动作上取最大值，故逐点有",[238,40322,40324],{"className":40323,"translate":242},[945],[238,40325,40327,40375],{"className":40326,"translate":242},[241],[238,40328,40330],{"className":40329},[246],[248,40331,40332],{"xmlns":250,"display":954},[252,40333,40334,40372],{},[255,40335,40336,40342,40348,40350,40356,40362,40364,40370],{},[337,40337,40338,40340],{},[258,40339,11971],{"mathvariant":816},[258,40341,2297],{},[39888,40343,40344,40346],{"accent":269},[258,40345,4429],{},[332,40347,39894],{},[332,40349,2023],{},[337,40351,40352,40354],{},[258,40353,11971],{"mathvariant":816},[332,40355,22083],{},[39888,40357,40358,40360],{"accent":269},[258,40359,4429],{},[332,40361,39894],{},[332,40363,992],{},[39888,40365,40366,40368],{"accent":269},[258,40367,4429],{},[332,40369,39894],{},[258,40371,3102],{"mathvariant":344},[262,40373,40374],{"encoding":264},"\\mathcal{T}^\\pi\\bar V\\le\\mathcal{T}^*\\bar V=\\bar V.",[238,40376,40378,40454,40529],{"className":40377,"ariaHidden":269},[268],[238,40379,40381,40385,40414,40445,40448,40451],{"className":40380},[273],[238,40382],{"className":40383,"style":40384},[277],"height:0.9561em;vertical-align:-0.136em;",[238,40386,40388,40391],{"className":40387},[282],[238,40389,11971],{"className":40390,"style":11992},[282,840],[238,40392,40394],{"className":40393},[391],[238,40395,40397],{"className":40396},[395],[238,40398,40400],{"className":40399},[399],[238,40401,40403],{"className":40402,"style":4620},[403],[238,40404,40405,40408],{"style":1226},[238,40406],{"className":40407,"style":412},[411],[238,40409,40411],{"className":40410},[416,417,418,419],[238,40412,2297],{"className":40413,"style":2313},[282,283,419],[238,40415,40417],{"className":40416},[282,39911],[238,40418,40420],{"className":40419},[395],[238,40421,40423],{"className":40422},[399],[238,40424,40426,40434],{"className":40425,"style":39907},[403],[238,40427,40428,40431],{"style":21798},[238,40429],{"className":40430,"style":1134},[411],[238,40432,4429],{"className":40433,"style":660},[282,283],[238,40435,40436,40439],{"style":39931},[238,40437],{"className":40438,"style":1134},[411],[238,40440,40442],{"className":40441,"style":39939},[39938],[238,40443,39894],{"className":40444},[282],[238,40446],{"className":40447,"style":430},[429],[238,40449,2023],{"className":40450},[434],[238,40452],{"className":40453,"style":430},[429],[238,40455,40457,40460,40489,40520,40523,40526],{"className":40456},[273],[238,40458],{"className":40459,"style":39907},[277],[238,40461,40463,40466],{"className":40462},[282],[238,40464,11971],{"className":40465,"style":11992},[282,840],[238,40467,40469],{"className":40468},[391],[238,40470,40472],{"className":40471},[395],[238,40473,40475],{"className":40474},[399],[238,40476,40478],{"className":40477,"style":22441},[403],[238,40479,40480,40483],{"style":1226},[238,40481],{"className":40482,"style":412},[411],[238,40484,40486],{"className":40485},[416,417,418,419],[238,40487,22083],{"className":40488},[664,419],[238,40490,40492],{"className":40491},[282,39911],[238,40493,40495],{"className":40494},[395],[238,40496,40498],{"className":40497},[399],[238,40499,40501,40509],{"className":40500,"style":39907},[403],[238,40502,40503,40506],{"style":21798},[238,40504],{"className":40505,"style":1134},[411],[238,40507,4429],{"className":40508,"style":660},[282,283],[238,40510,40511,40514],{"style":39931},[238,40512],{"className":40513,"style":1134},[411],[238,40515,40517],{"className":40516,"style":39939},[39938],[238,40518,39894],{"className":40519},[282],[238,40521],{"className":40522,"style":430},[429],[238,40524,992],{"className":40525},[434],[238,40527],{"className":40528,"style":430},[429],[238,40530,40532,40535,40566],{"className":40531},[273],[238,40533],{"className":40534,"style":39907},[277],[238,40536,40538],{"className":40537},[282,39911],[238,40539,40541],{"className":40540},[395],[238,40542,40544],{"className":40543},[399],[238,40545,40547,40555],{"className":40546,"style":39907},[403],[238,40548,40549,40552],{"style":21798},[238,40550],{"className":40551,"style":1134},[411],[238,40553,4429],{"className":40554,"style":660},[282,283],[238,40556,40557,40560],{"style":39931},[238,40558],{"className":40559,"style":1134},[411],[238,40561,40563],{"className":40562,"style":39939},[39938],[238,40564,39894],{"className":40565},[282],[238,40567,3102],{"className":40568},[282],[11,40570,40571,40572,40600,40601,40792,40793,40844,40845,40903,40904,941,40906,36469,41094,23814,41207,41235,41236,1806],{},"由单调性，对一切 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V",[238,40645,40647,40755],{"className":40646,"ariaHidden":269},[268],[238,40648,40650,40654,40657,40686,40715,40746,40749,40752],{"className":40649},[273],[238,40651],{"className":40652,"style":40653},[277],"height:1.0701em;vertical-align:-0.25em;",[238,40655,335],{"className":40656},[381],[238,40658,40660,40663],{"className":40659},[282],[238,40661,11971],{"className":40662,"style":11992},[282,840],[238,40664,40666],{"className":40665},[391],[238,40667,40669],{"className":40668},[395],[238,40670,40672],{"className":40671},[399],[238,40673,40675],{"className":40674,"style":6609},[403],[238,40676,40677,40680],{"style":407},[238,40678],{"className":40679,"style":412},[411],[238,40681,40683],{"className":40682},[416,417,418,419],[238,40684,2297],{"className":40685,"style":2313},[282,283,419],[238,40687,40689,40692],{"className":40688},[462],[238,40690,360],{"className":40691},[462],[238,40693,40695],{"className":40694},[391],[238,40696,40698],{"className":40697},[395],[238,40699,40701],{"className":40700},[399],[238,40702,40704],{"className":40703,"style":6609},[403],[238,40705,40706,40709],{"style":407},[238,40707],{"className":40708,"style":412},[411],[238,40710,40712],{"className":40711},[416,417,418,419],[238,40713,13161],{"className":40714},[282,283,419],[238,40716,40718],{"className":40717},[282,39911],[238,40719,40721],{"className":40720},[395],[238,40722,40724],{"className":40723},[399],[238,40725,40727,40735],{"className":40726,"style":39907},[403],[238,40728,40729,40732],{"style":21798},[238,40730],{"className":40731,"style":1134},[411],[238,40733,4429],{"className":40734,"style":660},[282,283],[238,40736,40737,40740],{"style":39931},[238,40738],{"className":40739,"style":1134},[411],[238,40741,40743],{"className":40742,"style":39939},[39938],[238,40744,39894],{"className":40745},[282],[238,40747],{"className":40748,"style":430},[429],[238,40750,2023],{"className":40751},[434],[238,40753],{"className":40754,"style":430},[429],[238,40756,40758,40761],{"className":40757},[273],[238,40759],{"className":40760,"style":39907},[277],[238,40762,40764],{"className":40763},[282,39911],[238,40765,40767],{"className":40766},[395],[238,40768,40770],{"className":40769},[399],[238,40771,40773,40781],{"className":40772,"style":39907},[403],[238,40774,40775,40778],{"style":21798},[238,40776],{"className":40777,"style":1134},[411],[238,40779,4429],{"className":40780,"style":660},[282,283],[238,40782,40783,40786],{"style":39931},[238,40784],{"className":40785,"style":1134},[411],[238,40787,40789],{"className":40788,"style":39939},[39938],[238,40790,39894],{"className":40791},[282],"。令 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的压缩性与",[357,40905,21493],{"href":21492},[238,40907,40909,40949],{"className":40908,"translate":242},[241],[238,40910,40912],{"className":40911},[246],[248,40913,40914],{"xmlns":250},[252,40915,40916,40946],{},[255,40917,40918,40920,40926,40932,40938,40940],{},[332,40919,335],{"stretchy":334},[337,40921,40922,40924],{},[258,40923,11971],{"mathvariant":816},[258,40925,2297],{},[337,40927,40928,40930],{},[332,40929,360],{"stretchy":334},[258,40931,13161],{},[39888,40933,40934,40936],{"accent":269},[258,40935,4429],{},[332,40937,39894],{},[332,40939,617],{},[337,40941,40942,40944],{},[258,40943,4429],{},[258,40945,2297],{},[262,40947,40948],{"encoding":264},"(\\mathcal{T}^\\pi)^n\\bar V\\to 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取上确界得 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V=\\mathcal{T}^*\\bar V=\\bar V",[238,42249,42251,42369,42444],{"className":42250,"ariaHidden":269},[268],[238,42252,42254,42257,42329,42360,42363,42366],{"className":42253},[273],[238,42255],{"className":42256,"style":39907},[277],[238,42258,42260,42263],{"className":42259},[282],[238,42261,11971],{"className":42262,"style":11992},[282,840],[238,42264,42266],{"className":42265},[391],[238,42267,42269],{"className":42268},[395],[238,42270,42272],{"className":42271},[399],[238,42273,42275],{"className":42274,"style":6609},[403],[238,42276,42277,42280],{"style":407},[238,42278],{"className":42279,"style":412},[411],[238,42281,42283],{"className":42282},[416,417,418,419],[238,42284,42286],{"className":42285},[282,419],[238,42287,42289,42292],{"className":42288},[282,419],[238,42290,2297],{"className":42291,"style":2313},[282,283,419],[238,42293,42295],{"className":42294},[391],[238,42296,42298,42320],{"className":42297},[395,848],[238,42299,42301,42317],{"className":42300},[399],[238,42302,42305],{"className":42303,"style":42304},[403],"height:0.1645em;",[238,42306,42308,42311],{"style":42307},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[238,42309],{"className":42310,"style":4203},[411],[238,42312,42314],{"className":42313},[416,4207,4208,419],[238,42315,39408],{"className":42316,"style":2313},[282,283,419],[238,42318,872],{"className":42319},[871],[238,42321,42323],{"className":42322},[399],[238,42324,42327],{"className":42325,"style":42326},[403],"height:0.2819em;",[238,42328],{},[238,42330,42332],{"className":42331},[282,39911],[238,42333,42335],{"className":42334},[395],[238,42336,42338],{"className":42337},[399],[238,42339,42341,42349],{"className":42340,"style":39907},[403],[238,42342,42343,42346],{"style":21798},[238,42344],{"className":42345,"style":1134},[411],[238,42347,4429],{"className":42348,"style":660},[282,283],[238,42350,42351,42354],{"style":39931},[238,42352],{"className":42353,"style":1134},[411],[238,42355,42357],{"className":42356,"style":39939},[39938],[238,42358,39894],{"className":42359},[282],[238,42361],{"className":42362,"style":430},[429],[238,42364,992],{"className":42365},[434],[238,42367],{"className":42368,"style":430},[429],[238,42370,42372,42375,42404,42435,42438,42441],{"className":42371},[273],[238,42373],{"className":42374,"style":39907},[277],[238,42376,42378,42381],{"className":42377},[282],[238,42379,11971],{"className":42380,"style":11992},[282,840],[238,42382,42384],{"className":42383},[391],[238,42385,42387],{"className":42386},[395],[238,42388,42390],{"className":42389},[399],[238,42391,42393],{"className":42392,"style":22096},[403],[238,42394,42395,42398],{"style":407},[238,42396],{"className":42397,"style":412},[411],[238,42399,42401],{"className":42400},[416,417,418,419],[238,42402,22083],{"className":42403},[664,419],[238,42405,42407],{"className":42406},[282,39911],[238,42408,42410],{"className":42409},[395],[238,42411,42413],{"className":42412},[399],[238,42414,42416,42424],{"className":42415,"style":39907},[403],[238,42417,42418,42421],{"style":21798},[238,42419],{"className":42420,"style":1134},[411],[238,42422,4429],{"className":42423,"style":660},[282,283],[238,42425,42426,42429],{"style":39931},[238,42427],{"className":42428,"style":1134},[411],[238,42430,42432],{"className":42431,"style":39939},[39938],[238,42433,39894],{"className":42434},[282],[238,42436],{"className":42437,"style":430},[429],[238,42439,992],{"className":42440},[434],[238,42442],{"className":42443,"style":430},[429],[238,42445,42447,42450],{"className":42446},[273],[238,42448],{"className":42449,"style":39907},[277],[238,42451,42453],{"className":42452},[282,39911],[238,42454,42456],{"className":42455},[395],[238,42457,42459],{"className":42458},[399],[238,42460,42462,42470],{"className":42461,"style":39907},[403],[238,42463,42464,42467],{"style":21798},[238,42465],{"className":42466,"style":1134},[411],[238,42468,4429],{"className":42469,"style":660},[282,283],[238,42471,42472,42475],{"style":39931},[238,42473],{"className":42474,"style":1134},[411],[238,42476,42478],{"className":42477,"style":39939},[39938],[238,42479,39894],{"className":42480},[282],"。由",[357,42483,21493],{"href":21492},[238,42485,42487,42517],{"className":42486,"translate":242},[241],[238,42488,42490],{"className":42489},[246],[248,42491,42492],{"xmlns":250},[252,42493,42494,42514],{},[255,42495,42496,42502,42504],{},[39888,42497,42498,42500],{"accent":269},[258,42499,4429],{},[332,42501,39894],{},[332,42503,992],{},[337,42505,42506,42508],{},[258,42507,4429],{},[812,42509,42510,42512],{},[258,42511,2297],{},[258,42513,39408],{},[262,42515,42516],{"encoding":264},"\\bar V=V^{\\pi_g}",[238,42518,42520,42566],{"className":42519,"ariaHidden":269},[268],[238,42521,42523,42526,42557,42560,42563],{"className":42522},[273],[238,42524],{"className":42525,"style":39907},[277],[238,42527,42529],{"className":42528},[282,39911],[238,42530,42532],{"className":42531},[395],[238,42533,42535],{"className":42534},[399],[238,42536,42538,42546],{"className":42537,"style":39907},[403],[238,42539,42540,42543],{"style":21798},[238,42541],{"className":42542,"style":1134},[411],[238,42544,4429],{"className":42545,"style":660},[282,283],[238,42547,42548,42551],{"style":39931},[238,42549],{"className":42550,"style":1134},[411],[238,42552,42554],{"className":42553,"style":39939},[39938],[238,42555,39894],{"className":42556},[282],[238,42558],{"className":42559,"style":430},[429],[238,42561,992],{"className":42562},[434],[238,42564],{"className":42565,"style":430},[429],[238,42567,42569,42572],{"className":42568},[273],[238,42570],{"className":42571,"style":278},[277],[238,42573,42575,42578],{"className":42574},[282],[238,42576,4429],{"className":42577,"style":660},[282,283],[238,42579,42581],{"className":42580},[391],[238,42582,42584],{"className":42583},[395],[238,42585,42587],{"className":42586},[399],[238,42588,42590],{"className":42589,"style":6609},[403],[238,42591,42592,42595],{"style":407},[238,42593],{"className":42594,"style":412},[411],[238,42596,42598],{"className":42597},[416,417,418,419],[238,42599,42601],{"className":42600},[282,419],[238,42602,42604,42607],{"className":42603},[282,419],[238,42605,2297],{"className":42606,"style":2313},[282,283,419],[238,42608,42610],{"className":42609},[391],[238,42611,42613,42633],{"className":42612},[395,848],[238,42614,42616,42630],{"className":42615},[399],[238,42617,42619],{"className":42618,"style":42304},[403],[238,42620,42621,42624],{"style":42307},[238,42622],{"className":42623,"style":4203},[411],[238,42625,42627],{"className":42626},[416,4207,4208,419],[238,42628,39408],{"className":42629,"style":2313},[282,283,419],[238,42631,872],{"className":42632},[871],[238,42634,42636],{"className":42635},[399],[238,42637,42639],{"className":42638,"style":42326},[403],[238,42640],{},[238,42642,42644,42669],{"className":42643,"translate":242},[241],[238,42645,42647],{"className":42646},[246],[248,42648,42649],{"xmlns":250},[252,42650,42651,42667],{},[255,42652,42653,42659,42661],{},[39888,42654,42655,42657],{"accent":269},[258,42656,4429],{},[332,42658,39894],{},[332,42660,2023],{},[337,42662,42663,42665],{},[258,42664,4429],{},[332,42666,22083],{},[262,42668,41379],{"encoding":264},[238,42670,42672,42718],{"className":42671,"ariaHidden":269},[268],[238,42673,42675,42678,42709,42712,42715],{"className":42674},[273],[238,42676],{"className":42677,"style":40384},[277],[238,42679,42681],{"className":42680},[282,39911],[238,42682,42684],{"className":42683},[395],[238,42685,42687],{"className":42686},[399],[238,42688,42690,42698],{"className":42689,"style":39907},[403],[238,42691,42692,42695],{"style":21798},[238,42693],{"className":42694,"style":1134},[411],[238,42696,4429],{"className":42697,"style":660},[282,283],[238,42699,42700,42703],{"style":39931},[238,42701],{"className":42702,"style":1134},[411],[238,42704,42706],{"className":42705,"style":39939},[39938],[238,42707,39894],{"className":42708},[282],[238,42710],{"className":42711,"style":430},[429],[238,42713,2023],{"className":42714},[434],[238,42716],{"className":42717,"style":430},[429],[238,42719,42721,42724],{"className":42720},[273],[238,42722],{"className":42723,"style":22096},[277],[238,42725,42727,42730],{"className":42726},[282],[238,42728,4429],{"className":42729,"style":660},[282,283],[238,42731,42733],{"className":42732},[391],[238,42734,42736],{"className":42735},[395],[238,42737,42739],{"className":42738},[399],[238,42740,42742],{"className":42741,"style":22096},[403],[238,42743,42744,42747],{"style":407},[238,42745],{"className":42746,"style":412},[411],[238,42748,42750],{"className":42749},[416,417,418,419],[238,42751,22083],{"className":42752},[664,419],"。结合两个不等式得 ",[238,42755,42757,42782],{"className":42756,"translate":242},[241],[238,42758,42760],{"className":42759},[246],[248,42761,42762],{"xmlns":250},[252,42763,42764,42780],{},[255,42765,42766,42772,42774],{},[39888,42767,42768,42770],{"accent":269},[258,42769,4429],{},[332,42771,39894],{},[332,42773,992],{},[337,42775,42776,42778],{},[258,42777,4429],{},[332,42779,22083],{},[262,42781,39972],{"encoding":264},[238,42783,42785,42831],{"className":42784,"ariaHidden":269},[268],[238,42786,42788,42791,42822,42825,42828],{"className":42787},[273],[238,42789],{"className":42790,"style":39907},[277],[238,42792,42794],{"className":42793},[282,39911],[238,42795,42797],{"className":42796},[395],[238,42798,42800],{"className":42799},[399],[238,42801,42803,42811],{"className":42802,"style":39907},[403],[238,42804,42805,42808],{"style":21798},[238,42806],{"className":42807,"style":1134},[411],[238,42809,4429],{"className":42810,"style":660},[282,283],[238,42812,42813,42816],{"style":39931},[238,42814],{"className":42815,"style":1134},[411],[238,42817,42819],{"className":42818,"style":39939},[39938],[238,42820,39894],{"className":42821},[282],[238,42823],{"className":42824,"style":430},[429],[238,42826,992],{"className":42827},[434],[238,42829],{"className":42830,"style":430},[429],[238,42832,42834,42837],{"className":42833},[273],[238,42835],{"className":42836,"style":22096},[277],[238,42838,42840,42843],{"className":42839},[282],[238,42841,4429],{"className":42842,"style":660},[282,283],[238,42844,42846],{"className":42845},[391],[238,42847,42849],{"className":42848},[395],[238,42850,42852],{"className":42851},[399],[238,42853,42855],{"className":42854,"style":22096},[403],[238,42856,42857,42860],{"style":407},[238,42858],{"className":42859,"style":412},[411],[238,42861,42863],{"className":42862},[416,417,418,419],[238,42864,22083],{"className":42865},[664,419],[11,42867,42868,42869,42897,42898,43131,43132,43134,43135,43246,43247,43357],{},"最后证明策略的最优性判据。若确定性策略 ",[238,42870,42872,42885],{"className":42871,"translate":242},[241],[238,42873,42875],{"className":42874},[246],[248,42876,42877],{"xmlns":250},[252,42878,42879,42883],{},[255,42880,42881],{},[258,42882,2297],{},[262,42884,2300],{"encoding":264},[238,42886,42888],{"className":42887,"ariaHidden":269},[268],[238,42889,42891,42894],{"className":42890},[273],[238,42892],{"className":42893,"style":489},[277],[238,42895,2297],{"className":42896,"style":2313},[282,283]," 在每个状态都取到最大值，则 ",[238,42899,42901,42947],{"className":42900,"translate":242},[241],[238,42902,42904],{"className":42903},[246],[248,42905,42906],{"xmlns":250},[252,42907,42908,42944],{},[255,42909,42910,42916,42922,42924,42930,42936,42938],{},[337,42911,42912,42914],{},[258,42913,11971],{"mathvariant":816},[258,42915,2297],{},[337,42917,42918,42920],{},[258,42919,4429],{},[332,42921,22083],{},[332,42923,992],{},[337,42925,42926,42928],{},[258,42927,11971],{"mathvariant":816},[332,42929,22083],{},[337,42931,42932,42934],{},[258,42933,4429],{},[332,42935,22083],{},[332,42937,992],{},[337,42939,42940,42942],{},[258,42941,4429],{},[332,42943,22083],{},[262,42945,42946],{"encoding":264},"\\mathcal{T}^\\pi V^*=\\mathcal{T}^*V^*=V^*",[238,42948,42950,43023,43096],{"className":42949,"ariaHidden":269},[268],[238,42951,42953,42956,42985,43014,43017,43020],{"className":42952},[273],[238,42954],{"className":42955,"style":22096},[277],[238,42957,42959,42962],{"className":42958},[282],[238,42960,11971],{"className":42961,"style":11992},[282,840],[238,42963,42965],{"className":42964},[391],[238,42966,42968],{"className":42967},[395],[238,42969,42971],{"className":42970},[399],[238,42972,42974],{"className":42973,"style":6609},[403],[238,42975,42976,42979],{"style":407},[238,42977],{"className":42978,"style":412},[411],[238,42980,42982],{"className":42981},[416,417,418,419],[238,42983,2297],{"className":42984,"style":2313},[282,283,419],[238,42986,42988,42991],{"className":42987},[282],[238,42989,4429],{"className":42990,"style":660},[282,283],[238,42992,42994],{"className":42993},[391],[238,42995,42997],{"className":42996},[395],[238,42998,43000],{"className":42999},[399],[238,43001,43003],{"className":43002,"style":22096},[403],[238,43004,43005,43008],{"style":407},[238,43006],{"className":43007,"style":412},[411],[238,43009,43011],{"className":43010},[416,417,418,419],[238,43012,22083],{"className":43013},[664,419],[238,43015],{"className":43016,"style":430},[429],[238,43018,992],{"className":43019},[434],[238,43021],{"className":43022,"style":430},[429],[238,43024,43026,43029,43058,43087,43090,43093],{"className":43025},[273],[238,43027],{"className":43028,"style":22096},[277],[238,43030,43032,43035],{"className":43031},[282],[238,43033,11971],{"className":43034,"style":11992},[282,840],[238,43036,43038],{"className":43037},[391],[238,43039,43041],{"className":43040},[395],[238,43042,43044],{"className":43043},[399],[238,43045,43047],{"className":43046,"style":22096},[403],[238,43048,43049,43052],{"style":407},[238,43050],{"className":43051,"style":412},[411],[238,43053,43055],{"className":43054},[416,417,418,419],[238,43056,22083],{"className":43057},[664,419],[238,43059,43061,43064],{"className":43060},[282],[238,43062,4429],{"className":43063,"style":660},[282,283],[238,43065,43067],{"className":43066},[391],[238,43068,43070],{"className":43069},[395],[238,43071,43073],{"className":43072},[399],[238,43074,43076],{"className":43075,"style":22096},[403],[238,43077,43078,43081],{"style":407},[238,43079],{"className":43080,"style":412},[411],[238,43082,43084],{"className":43083},[416,417,418,419],[238,43085,22083],{"className":43086},[664,419],[238,43088],{"className":43089,"style":430},[429],[238,43091,992],{"className":43092},[434],[238,43094],{"className":43095,"style":430},[429],[238,43097,43099,43102],{"className":43098},[273],[238,43100],{"className":43101,"style":22096},[277],[238,43103,43105,43108],{"className":43104},[282],[238,43106,4429],{"className":43107,"style":660},[282,283],[238,43109,43111],{"className":43110},[391],[238,43112,43114],{"className":43113},[395],[238,43115,43117],{"className":43116},[399],[238,43118,43120],{"className":43119,"style":22096},[403],[238,43121,43122,43125],{"style":407},[238,43123],{"className":43124,"style":412},[411],[238,43126,43128],{"className":43127},[416,417,418,419],[238,43129,22083],{"className":43130},[664,419],"，由",[357,43133,21493],{"href":21492}," 得 ",[238,43136,43138,43164],{"className":43137,"translate":242},[241],[238,43139,43141],{"className":43140},[246],[248,43142,43143],{"xmlns":250},[252,43144,43145,43161],{},[255,43146,43147,43153,43155],{},[337,43148,43149,43151],{},[258,43150,4429],{},[258,43152,2297],{},[332,43154,992],{},[337,43156,43157,43159],{},[258,43158,4429],{},[332,43160,22083],{},[262,43162,43163],{"encoding":264},"V^\\pi=V^*",[238,43165,43167,43211],{"className":43166,"ariaHidden":269},[268],[238,43168,43170,43173,43202,43205,43208],{"className":43169},[273],[238,43171],{"className":43172,"style":278},[277],[238,43174,43176,43179],{"className":43175},[282],[238,43177,4429],{"className":43178,"style":660},[282,283],[238,43180,43182],{"className":43181},[391],[238,43183,43185],{"className":43184},[395],[238,43186,43188],{"className":43187},[399],[238,43189,43191],{"className":43190,"style":6609},[403],[238,43192,43193,43196],{"style":407},[238,43194],{"className":43195,"style":412},[411],[238,43197,43199],{"className":43198},[416,417,418,419],[238,43200,2297],{"className":43201,"style":2313},[282,283,419],[238,43203],{"className":43204,"style":430},[429],[238,43206,992],{"className":43207},[434],[238,43209],{"className":43210,"style":430},[429],[238,43212,43214,43217],{"className":43213},[273],[238,43215],{"className":43216,"style":22096},[277],[238,43218,43220,43223],{"className":43219},[282],[238,43221,4429],{"className":43222,"style":660},[282,283],[238,43224,43226],{"className":43225},[391],[238,43227,43229],{"className":43228},[395],[238,43230,43232],{"className":43231},[399],[238,43233,43235],{"className":43234,"style":22096},[403],[238,43236,43237,43240],{"style":407},[238,43238],{"className":43239,"style":412},[411],[238,43241,43243],{"className":43242},[416,417,418,419],[238,43244,22083],{"className":43245},[664,419],"。反之，若 ",[238,43248,43250,43275],{"className":43249,"translate":242},[241],[238,43251,43253],{"className":43252},[246],[248,43254,43255],{"xmlns":250},[252,43256,43257,43273],{},[255,43258,43259,43265,43267],{},[337,43260,43261,43263],{},[258,43262,4429],{},[258,43264,2297],{},[332,43266,992],{},[337,43268,43269,43271],{},[258,43270,4429],{},[332,43272,22083],{},[262,43274,43163],{"encoding":264},[238,43276,43278,43322],{"className":43277,"ariaHidden":269},[268],[238,43279,43281,43284,43313,43316,43319],{"className":43280},[273],[238,43282],{"className":43283,"style":278},[277],[238,43285,43287,43290],{"className":43286},[282],[238,43288,4429],{"className":43289,"style":660},[282,283],[238,43291,43293],{"className":43292},[391],[238,43294,43296],{"className":43295},[395],[238,43297,43299],{"className":43298},[399],[238,43300,43302],{"className":43301,"style":6609},[403],[238,43303,43304,43307],{"style":407},[238,43305],{"className":43306,"style":412},[411],[238,43308,43310],{"className":43309},[416,417,418,419],[238,43311,2297],{"className":43312,"style":2313},[282,283,419],[238,43314],{"className":43315,"style":430},[429],[238,43317,992],{"className":43318},[434],[238,43320],{"className":43321,"style":430},[429],[238,43323,43325,43328],{"className":43324},[273],[238,43326],{"className":43327,"style":22096},[277],[238,43329,43331,43334],{"className":43330},[282],[238,43332,4429],{"className":43333,"style":660},[282,283],[238,43335,43337],{"className":43336},[391],[238,43338,43340],{"className":43339},[395],[238,43341,43343],{"className":43342},[399],[238,43344,43346],{"className":43345,"style":22096},[403],[238,43347,43348,43351],{"style":407},[238,43349],{"className":43350,"style":412},[411],[238,43352,43354],{"className":43353},[416,417,418,419],[238,43355,22083],{"className":43356},[664,419],"，则",[238,43359,43361],{"className":43360,"translate":242},[945],[238,43362,43364,43466],{"className":43363,"translate":242},[241],[238,43365,43367],{"className":43366},[246],[248,43368,43369],{"xmlns":250,"display":954},[252,43370,43371,43463],{},[958,43372,43373,43423],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,43374,43375,43393],{},[966,43376,43377],{},[969,43378,43379],{"scriptlevel":740,"displaystyle":269},[255,43380,43381,43387],{},[337,43382,43383,43385],{},[258,43384,11971],{"mathvariant":816},[258,43386,2297],{},[337,43388,43389,43391],{},[258,43390,4429],{},[332,43392,22083],{},[966,43394,43395],{},[969,43396,43397],{"scriptlevel":740,"displaystyle":269},[255,43398,43399,43401,43403,43409,43415,43417],{},[255,43400],{},[332,43402,992],{},[337,43404,43405,43407],{},[258,43406,11971],{"mathvariant":816},[258,43408,2297],{},[337,43410,43411,43413],{},[258,43412,4429],{},[258,43414,2297],{},[332,43416,992],{},[337,43418,43419,43421],{},[258,43420,4429],{},[258,43422,2297],{},[963,43424,43425,43431],{},[966,43426,43427],{},[969,43428,43429],{"scriptlevel":740,"displaystyle":269},[255,43430],{},[966,43432,43433],{},[969,43434,43435],{"scriptlevel":740,"displaystyle":269},[255,43436,43437,43439,43441,43447,43449,43455,43461],{},[255,43438],{},[332,43440,992],{},[337,43442,43443,43445],{},[258,43444,4429],{},[332,43446,22083],{},[332,43448,992],{},[337,43450,43451,43453],{},[258,43452,11971],{"mathvariant":816},[332,43454,22083],{},[337,43456,43457,43459],{},[258,43458,4429],{},[332,43460,22083],{},[258,43462,3102],{"mathvariant":344},[262,43464,43465],{"encoding":264},"\\begin{aligned}\n\\mathcal{T}^\\pi V^*\n&=\\mathcal{T}^\\pi V^\\pi=V^\\pi\\\\\n&=V^*=\\mathcal{T}^*V^*.\n\\end{aligned}",[238,43467,43469],{"className":43468,"ariaHidden":269},[268],[238,43470,43472,43475],{"className":43471},[273],[238,43473],{"className":43474,"style":4573},[277],[238,43476,43478],{"className":43477},[282],[238,43479,43481,43578],{"className":43480},[958],[238,43482,43484],{"className":43483},[1117],[238,43485,43487,43570],{"className":43486},[395,848],[238,43488,43490,43567],{"className":43489},[399],[238,43491,43493,43559],{"className":43492,"style":4592},[403],[238,43494,43495,43498],{"style":4595},[238,43496],{"className":43497,"style":1134},[411],[238,43499,43501,43530],{"className":43500},[282],[238,43502,43504,43507],{"className":43503},[282],[238,43505,11971],{"className":43506,"style":11992},[282,840],[238,43508,43510],{"className":43509},[391],[238,43511,43513],{"className":43512},[395],[238,43514,43516],{"className":43515},[399],[238,43517,43519],{"className":43518,"style":4620},[403],[238,43520,43521,43524],{"style":1226},[238,43522],{"className":43523,"style":412},[411],[238,43525,43527],{"className":43526},[416,417,418,419],[238,43528,2297],{"className":43529,"style":2313},[282,283,419],[238,43531,43533,43536],{"className":43532},[282],[238,43534,4429],{"className":43535,"style":660},[282,283],[238,43537,43539],{"className":43538},[391],[238,43540,43542],{"className":43541},[395],[238,43543,43545],{"className":43544},[399],[238,43546,43548],{"className":43547,"style":22441},[403],[238,43549,43550,43553],{"style":1226},[238,43551],{"className":43552,"style":412},[411],[238,43554,43556],{"className":43555},[416,417,418,419],[238,43557,22083],{"className":43558},[664,419],[238,43560,43561,43564],{"style":4643},[238,43562],{"className":43563,"style":1134},[411],[238,43565],{"className":43566},[282],[238,43568,872],{"className":43569},[871],[238,43571,43573],{"className":43572},[399],[238,43574,43576],{"className":43575,"style":4706},[403],[238,43577],{},[238,43579,43581],{"className":43580},[1422],[238,43582,43584,43828],{"className":43583},[395,848],[238,43585,43587,43825],{"className":43586},[399],[238,43588,43590,43706],{"className":43589,"style":4592},[403],[238,43591,43592,43595],{"style":4595},[238,43593],{"className":43594,"style":1134},[411],[238,43596,43598,43601,43604,43607,43610,43639,43668,43671,43674,43677],{"className":43597},[282],[238,43599],{"className":43600},[282],[238,43602],{"className":43603,"style":430},[429],[238,43605,992],{"className":43606},[434],[238,43608],{"className":43609,"style":430},[429],[238,43611,43613,43616],{"className":43612},[282],[238,43614,11971],{"className":43615,"style":11992},[282,840],[238,43617,43619],{"className":43618},[391],[238,43620,43622],{"className":43621},[395],[238,43623,43625],{"className":43624},[399],[238,43626,43628],{"className":43627,"style":4620},[403],[238,43629,43630,43633],{"style":1226},[238,43631],{"className":43632,"style":412},[411],[238,43634,43636],{"className":43635},[416,417,418,419],[238,43637,2297],{"className":43638,"style":2313},[282,283,419],[238,43640,43642,43645],{"className":43641},[282],[238,43643,4429],{"className":43644,"style":660},[282,283],[238,43646,43648],{"className":43647},[391],[238,43649,43651],{"className":43650},[395],[238,43652,43654],{"className":43653},[399],[238,43655,43657],{"className":43656,"style":4620},[403],[238,43658,43659,43662],{"style":1226},[238,43660],{"className":43661,"style":412},[411],[238,43663,43665],{"className":43664},[416,417,418,419],[238,43666,2297],{"className":43667,"style":2313},[282,283,419],[238,43669],{"className":43670,"style":430},[429],[238,43672,992],{"className":43673},[434],[238,43675],{"className":43676,"style":430},[429],[238,43678,43680,43683],{"className":43679},[282],[238,43681,4429],{"className":43682,"style":660},[282,283],[238,43684,43686],{"className":43685},[391],[238,43687,43689],{"className":43688},[395],[238,43690,43692],{"className":43691},[399],[238,43693,43695],{"className":43694,"style":4620},[403],[238,43696,43697,43700],{"style":1226},[238,43698],{"className":43699,"style":412},[411],[238,43701,43703],{"className":43702},[416,417,418,419],[238,43704,2297],{"className":43705,"style":2313},[282,283,419],[238,43707,43708,43711],{"style":4643},[238,43709],{"className":43710,"style":1134},[411],[238,43712,43714,43717,43720,43723,43726,43755,43758,43761,43764,43793,43822],{"className":43713},[282],[238,43715],{"className":43716},[282],[238,43718],{"className":43719,"style":430},[429],[238,43721,992],{"className":43722},[434],[238,43724],{"className":43725,"style":430},[429],[238,43727,43729,43732],{"className":43728},[282],[238,43730,4429],{"className":43731,"style":660},[282,283],[238,43733,43735],{"className":43734},[391],[238,43736,43738],{"className":43737},[395],[238,43739,43741],{"className":43740},[399],[238,43742,43744],{"className":43743,"style":22441},[403],[238,43745,43746,43749],{"style":1226},[238,43747],{"className":43748,"style":412},[411],[238,43750,43752],{"className":43751},[416,417,418,419],[238,43753,22083],{"className":43754},[664,419],[238,43756],{"className":43757,"style":430},[429],[238,43759,992],{"className":43760},[434],[238,43762],{"className":43763,"style":430},[429],[238,43765,43767,43770],{"className":43766},[282],[238,43768,11971],{"className":43769,"style":11992},[282,840],[238,43771,43773],{"className":43772},[391],[238,43774,43776],{"className":43775},[395],[238,43777,43779],{"className":43778},[399],[238,43780,43782],{"className":43781,"style":22441},[403],[238,43783,43784,43787],{"style":1226},[238,43785],{"className":43786,"style":412},[411],[238,43788,43790],{"className":43789},[416,417,418,419],[238,43791,22083],{"className":43792},[664,419],[238,43794,43796,43799],{"className":43795},[282],[238,43797,4429],{"className":43798,"style":660},[282,283],[238,43800,43802],{"className":43801},[391],[238,43803,43805],{"className":43804},[395],[238,43806,43808],{"className":43807},[399],[238,43809,43811],{"className":43810,"style":22441},[403],[238,43812,43813,43816],{"style":1226},[238,43814],{"className":43815,"style":412},[411],[238,43817,43819],{"className":43818},[416,417,418,419],[238,43820,22083],{"className":43821},[664,419],[238,43823,3102],{"className":43824},[282],[238,43826,872],{"className":43827},[871],[238,43829,43831],{"className":43830},[399],[238,43832,43834],{"className":43833,"style":4706},[403],[238,43835],{},[11,43837,20770,43838,40261,43947,44056,44057,44085,44086,44114,44115,21427],{},[238,43839,43841,43871],{"className":43840,"translate":242},[241],[238,43842,43844],{"className":43843},[246],[248,43845,43846],{"xmlns":250},[252,43847,43848,43868],{},[255,43849,43850,43856,43862,43864,43866],{},[337,43851,43852,43854],{},[258,43853,11971],{"mathvariant":816},[258,43855,2297],{},[337,43857,43858,43860],{},[258,43859,4429],{},[332,43861,22083],{},[332,43863,335],{"stretchy":334},[258,43865,341],{},[332,43867,360],{"stretchy":334},[262,43869,43870],{"encoding":264},"\\mathcal{T}^\\pi V^*(s)",[238,43872,43874],{"className":43873,"ariaHidden":269},[268],[238,43875,43877,43880,43909,43938,43941,43944],{"className":43876},[273],[238,43878],{"className":43879,"style":444},[277],[238,43881,43883,43886],{"className":43882},[282],[238,43884,11971],{"className":43885,"style":11992},[282,840],[238,43887,43889],{"className":43888},[391],[238,43890,43892],{"className":43891},[395],[238,43893,43895],{"className":43894},[399],[238,43896,43898],{"className":43897,"style":6609},[403],[238,43899,43900,43903],{"style":407},[238,43901],{"className":43902,"style":412},[411],[238,43904,43906],{"className":43905},[416,417,418,419],[238,43907,2297],{"className":43908,"style":2313},[282,283,419],[238,43910,43912,43915],{"className":43911},[282],[238,43913,4429],{"className":43914,"style":660},[282,283],[238,43916,43918],{"className":43917},[391],[238,43919,43921],{"className":43920},[395],[238,43922,43924],{"className":43923},[399],[238,43925,43927],{"className":43926,"style":22096},[403],[238,43928,43929,43932],{"style":407},[238,43930],{"className":43931,"style":412},[411],[238,43933,43935],{"className":43934},[416,417,418,419],[238,43936,22083],{"className":43937},[664,419],[238,43939,335],{"className":43940},[381],[238,43942,341],{"className":43943},[282,283],[238,43945,360],{"className":43946},[462],[238,43948,43950,43980],{"className":43949,"translate":242},[241],[238,43951,43953],{"className":43952},[246],[248,43954,43955],{"xmlns":250},[252,43956,43957,43977],{},[255,43958,43959,43965,43971,43973,43975],{},[337,43960,43961,43963],{},[258,43962,11971],{"mathvariant":816},[332,43964,22083],{},[337,43966,43967,43969],{},[258,43968,4429],{},[332,43970,22083],{},[332,43972,335],{"stretchy":334},[258,43974,341],{},[332,43976,360],{"stretchy":334},[262,43978,43979],{"encoding":264},"\\mathcal{T}^*V^*(s)",[238,43981,43983],{"className":43982,"ariaHidden":269},[268],[238,43984,43986,43989,44018,44047,44050,44053],{"className":43985},[273],[238,43987],{"className":43988,"style":444},[277],[238,43990,43992,43995],{"className":43991},[282],[238,43993,11971],{"className":43994,"style":11992},[282,840],[238,43996,43998],{"className":43997},[391],[238,43999,44001],{"className":44000},[395],[238,44002,44004],{"className":44003},[399],[238,44005,44007],{"className":44006,"style":22096},[403],[238,44008,44009,44012],{"style":407},[238,44010],{"className":44011,"style":412},[411],[238,44013,44015],{"className":44014},[416,417,418,419],[238,44016,22083],{"className":44017},[664,419],[238,44019,44021,44024],{"className":44020},[282],[238,44022,4429],{"className":44023,"style":660},[282,283],[238,44025,44027],{"className":44026},[391],[238,44028,44030],{"className":44029},[395],[238,44031,44033],{"className":44032},[399],[238,44034,44036],{"className":44035,"style":22096},[403],[238,44037,44038,44041],{"style":407},[238,44039],{"className":44040,"style":412},[411],[238,44042,44044],{"className":44043},[416,417,418,419],[238,44045,22083],{"className":44046},[664,419],[238,44048,335],{"className":44049},[381],[238,44051,341],{"className":44052},[282,283],[238,44054,360],{"className":44055},[462]," 是这些动作值的最大值，等号成立当且仅当 ",[238,44058,44060,44073],{"className":44059,"translate":242},[241],[238,44061,44063],{"className":44062},[246],[248,44064,44065],{"xmlns":250},[252,44066,44067,44071],{},[255,44068,44069],{},[258,44070,2297],{},[262,44072,2300],{"encoding":264},[238,44074,44076],{"className":44075,"ariaHidden":269},[268],[238,44077,44079,44082],{"className":44078},[273],[238,44080],{"className":44081,"style":489},[277],[238,44083,2297],{"className":44084,"style":2313},[282,283]," 在该状态以概率 ",[238,44087,44089,44102],{"className":44088,"translate":242},[241],[238,44090,44092],{"className":44091},[246],[248,44093,44094],{"xmlns":250},[252,44095,44096,44100],{},[255,44097,44098],{},[738,44099,745],{},[262,44101,745],{"encoding":264},[238,44103,44105],{"className":44104,"ariaHidden":269},[268],[238,44106,44108,44111],{"className":44107},[273],[238,44109],{"className":44110,"style":2672},[277],[238,44112,745],{"className":44113},[282]," 选择取得最大值的动作。对确定性策略，这正是 ",[238,44116,44118,44156],{"className":44117,"translate":242},[241],[238,44119,44121],{"className":44120},[246],[248,44122,44123],{"xmlns":250},[252,44124,44125,44153],{},[255,44126,44127,44129,44131,44133,44135,44137,44139,44141,44151],{},[258,44128,2297],{},[332,44130,335],{"stretchy":334},[258,44132,341],{},[332,44134,360],{"stretchy":334},[332,44136,733],{},[258,44138,39246],{},[332,44140,1889],{},[812,44142,44143,44149],{},[255,44144,44145,44147],{},[258,44146,1886],{},[332,44148,1889],{},[258,44150,357],{},[332,44152,36038],{},[262,44154,44155],{"encoding":264},"\\pi(s)\\in\\arg\\max_a\\cdots",[238,44157,44159,44186],{"className":44158,"ariaHidden":269},[268],[238,44160,44162,44165,44168,44171,44174,44177,44180,44183],{"className":44161},[273],[238,44163],{"className":44164,"style":444},[277],[238,44166,2297],{"className":44167,"style":2313},[282,283],[238,44169,335],{"className":44170},[381],[238,44172,341],{"className":44173},[282,283],[238,44175,360],{"className":44176},[462],[238,44178],{"className":44179,"style":430},[429],[238,44181,733],{"className":44182},[434],[238,44184],{"className":44185,"style":430},[429],[238,44187,44189,44192,44197,44200,44241,44244],{"className":44188},[273],[238,44190],{"className":44191,"style":2853},[277],[238,44193,39405,44195],{"className":44194},[1943],[238,44196,39408],{"style":35277},[238,44198],{"className":44199,"style":455},[429],[238,44201,44203,44206],{"className":44202},[1943],[238,44204,1886],{"className":44205},[1943],[238,44207,44209],{"className":44208},[391],[238,44210,44212,44233],{"className":44211},[395,848],[238,44213,44215,44230],{"className":44214},[399],[238,44216,44218],{"className":44217,"style":1927},[403],[238,44219,44221,44224],{"style":44220},"top:-2.55em;margin-right:0.05em;",[238,44222],{"className":44223,"style":412},[411],[238,44225,44227],{"className":44226},[416,417,418,419],[238,44228,357],{"className":44229},[282,283,419],[238,44231,872],{"className":44232},[871],[238,44234,44236],{"className":44235},[399],[238,44237,44239],{"className":44238,"style":879},[403],[238,44240],{},[238,44242],{"className":44243,"style":455},[429],[238,44245,36038],{"className":44246},[3041],[11,44248,44249,44250,44308],{},"这个证明的结构值得注意。它没有直接对无穷期问题做「先选动作再假设后续最优」的循环论证，而是把最优性分解为两个方向。一方面，任何策略都不如 ",[238,44251,44253,44270],{"className":44252,"translate":242},[241],[238,44254,44256],{"className":44255},[246],[248,44257,44258],{"xmlns":250},[252,44259,44260,44268],{},[255,44261,44262],{},[337,44263,44264,44266],{},[258,44265,11971],{"mathvariant":816},[332,44267,22083],{},[262,44269,22086],{"encoding":264},[238,44271,44273],{"className":44272,"ariaHidden":269},[268],[238,44274,44276,44279],{"className":44275},[273],[238,44277],{"className":44278,"style":22096},[277],[238,44280,44282,44285],{"className":44281},[282],[238,44283,11971],{"className":44284,"style":11992},[282,840],[238,44286,44288],{"className":44287},[391],[238,44289,44291],{"className":44290},[395],[238,44292,44294],{"className":44293},[399],[238,44295,44297],{"className":44296,"style":22096},[403],[238,44298,44299,44302],{"style":407},[238,44300],{"className":44301,"style":412},[411],[238,44303,44305],{"className":44304},[416,417,418,419],[238,44306,22083],{"className":44307},[664,419]," 的不动点。另一方面，关于该不动点的贪婪策略恰好达到它。两个方向合起来，既证明了最优方程，也证明了最优策略的存在性。",[11,44310,16622,44311,1838,44313,44372,44373,44401,44402,44430],{},[357,44312,37451],{"href":37450},[238,44314,44316,44334],{"className":44315,"translate":242},[241],[238,44317,44319],{"className":44318},[246],[248,44320,44321],{"xmlns":250},[252,44322,44323,44331],{},[255,44324,44325],{},[337,44326,44327,44329],{},[258,44328,4494],{},[332,44330,22083],{},[262,44332,44333],{"encoding":264},"Q^*",[238,44335,44337],{"className":44336,"ariaHidden":269},[268],[238,44338,44340,44343],{"className":44339},[273],[238,44341],{"className":44342,"style":30755},[277],[238,44344,44346,44349],{"className":44345},[282],[238,44347,4494],{"className":44348},[282,283],[238,44350,44352],{"className":44351},[391],[238,44353,44355],{"className":44354},[395],[238,44356,44358],{"className":44357},[399],[238,44359,44361],{"className":44360,"style":22096},[403],[238,44362,44363,44366],{"style":407},[238,44364],{"className":44365,"style":412},[411],[238,44367,44369],{"className":44368},[416,417,418,419],[238,44370,22083],{"className":44371},[664,419]," 的定义，还可以得到最优动作值函数的表达式。若在状态 ",[238,44374,44376,44389],{"className":44375,"translate":242},[241],[238,44377,44379],{"className":44378},[246],[248,44380,44381],{"xmlns":250},[252,44382,44383,44387],{},[255,44384,44385],{},[258,44386,341],{},[262,44388,341],{"encoding":264},[238,44390,44392],{"className":44391,"ariaHidden":269},[268],[238,44393,44395,44398],{"className":44394},[273],[238,44396],{"className":44397,"style":489},[277],[238,44399,341],{"className":44400},[282,283]," 先固定动作 ",[238,44403,44405,44418],{"className":44404,"translate":242},[241],[238,44406,44408],{"className":44407},[246],[248,44409,44410],{"xmlns":250},[252,44411,44412,44416],{},[255,44413,44414],{},[258,44415,357],{},[262,44417,357],{"encoding":264},[238,44419,44421],{"className":44420,"ariaHidden":269},[268],[238,44422,44424,44427],{"className":44423},[273],[238,44425],{"className":44426,"style":489},[277],[238,44428,357],{"className":44429},[282,283],"，此后采用最优策略，则",[238,44432,44434],{"className":44433,"translate":242},[945],[238,44435,44437,44619],{"className":44436,"translate":242},[241],[238,44438,44440],{"className":44439},[246],[248,44441,44442],{"xmlns":250,"display":954},[252,44443,44444,44616],{},[958,44445,44446,44558],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,44447,44448,44470],{},[966,44449,44450],{},[969,44451,44452],{"scriptlevel":740,"displaystyle":269},[255,44453,44454,44460,44462,44464,44466,44468],{},[337,44455,44456,44458],{},[258,44457,4494],{},[332,44459,22083],{},[332,44461,335],{"stretchy":334},[258,44463,341],{},[332,44465,354],{"separator":269},[258,44467,357],{},[332,44469,360],{"stretchy":334},[966,44471,44472],{},[969,44473,44474],{"scriptlevel":740,"displaystyle":269},[255,44475,44476,44478,44480,44490,44492,44494,44500,44502,44504,44506,44508,44510,44556],{},[255,44477],{},[332,44479,992],{},[6755,44481,44482,44484],{},[332,44483,3390],{},[337,44485,44486,44488],{},[258,44487,341],{},[332,44489,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,44491,330],{},[332,44493,335],{"stretchy":334},[337,44495,44496,44498],{},[258,44497,341],{},[332,44499,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,44501,349],{},[258,44503,341],{},[332,44505,354],{"separator":269},[258,44507,357],{},[332,44509,360],{"stretchy":334},[255,44511,44512,44514,44516,44518,44520,44522,44524,44526,44532,44534,44536,44538,44544,44546,44552,44554],{},[332,44513,736],{"fence":269},[258,44515,600],{},[332,44517,335],{"stretchy":334},[258,44519,341],{},[332,44521,354],{"separator":269},[258,44523,357],{},[332,44525,354],{"separator":269},[337,44527,44528,44530],{},[258,44529,341],{},[332,44531,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,44533,360],{"stretchy":334},[332,44535,987],{},[258,44537,730],{},[337,44539,44540,44542],{},[258,44541,4429],{},[332,44543,22083],{},[332,44545,335],{"stretchy":334},[337,44547,44548,44550],{},[258,44549,341],{},[332,44551,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,44553,360],{"stretchy":334},[332,44555,4479],{"fence":269},[332,44557,354],{"separator":269},[963,44559,44560,44578],{},[966,44561,44562],{},[969,44563,44564],{"scriptlevel":740,"displaystyle":269},[255,44565,44566,44572,44574,44576],{},[337,44567,44568,44570],{},[258,44569,4429],{},[332,44571,22083],{},[332,44573,335],{"stretchy":334},[258,44575,341],{},[332,44577,360],{"stretchy":334},[966,44579,44580],{},[969,44581,44582],{"scriptlevel":740,"displaystyle":269},[255,44583,44584,44586,44588,44598,44604,44606,44608,44610,44612,44614],{},[255,44585],{},[332,44587,992],{},[6755,44589,44590,44596],{},[255,44591,44592,44594],{},[258,44593,1886],{},[332,44595,1889],{},[258,44597,357],{},[337,44599,44600,44602],{},[258,44601,4494],{},[332,44603,22083],{},[332,44605,335],{"stretchy":334},[258,44607,341],{},[332,44609,354],{"separator":269},[258,44611,357],{},[332,44613,360],{"stretchy":334},[258,44615,3102],{"mathvariant":344},[262,44617,44618],{"encoding":264},"\\begin{aligned}\nQ^*(s,a)&=\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V^*(s')\\right],\\\\\nV^*(s)&=\\max_{a}Q^*(s,a).\n\\end{aligned}",[238,44620,44622],{"className":44621,"ariaHidden":269},[268],[238,44623,44625,44629],{"className":44624},[273],[238,44626],{"className":44627,"style":44628},[277],"height:4.184em;vertical-align:-1.842em;",[238,44630,44632],{"className":44631},[282],[238,44633,44635,44763],{"className":44634},[958],[238,44636,44638],{"className":44637},[1117],[238,44639,44641,44754],{"className":44640},[395,848],[238,44642,44644,44751],{"className":44643},[399],[238,44645,44648,44704],{"className":44646,"style":44647},[403],"height:2.342em;",[238,44649,44651,44654],{"style":44650},"top:-4.342em;",[238,44652],{"className":44653,"style":3510},[411],[238,44655,44657,44686,44689,44692,44695,44698,44701],{"className":44656},[282],[238,44658,44660,44663],{"className":44659},[282],[238,44661,4494],{"className":44662},[282,283],[238,44664,44666],{"className":44665},[391],[238,44667,44669],{"className":44668},[395],[238,44670,44672],{"className":44671},[399],[238,44673,44675],{"className":44674,"style":22441},[403],[238,44676,44677,44680],{"style":1226},[238,44678],{"className":44679,"style":412},[411],[238,44681,44683],{"className":44682},[416,417,418,419],[238,44684,22083],{"className":44685},[664,419],[238,44687,335],{"className":44688},[381],[238,44690,341],{"className":44691},[282,283],[238,44693,354],{"className":44694},[451],[238,44696],{"className":44697,"style":455},[429],[238,44699,357],{"className":44700},[282,283],[238,44702,360],{"className":44703},[462],[238,44705,44707,44710],{"style":44706},"top:-1.908em;",[238,44708],{"className":44709,"style":3510},[411],[238,44711,44713,44742,44745,44748],{"className":44712},[282],[238,44714,44716,44719],{"className":44715},[282],[238,44717,4429],{"className":44718,"style":660},[282,283],[238,44720,44722],{"className":44721},[391],[238,44723,44725],{"className":44724},[395],[238,44726,44728],{"className":44727},[399],[238,44729,44731],{"className":44730,"style":22441},[403],[238,44732,44733,44736],{"style":1226},[238,44734],{"className":44735,"style":412},[411],[238,44737,44739],{"className":44738},[416,417,418,419],[238,44740,22083],{"className":44741},[664,419],[238,44743,335],{"className":44744},[381],[238,44746,341],{"className":44747},[282,283],[238,44749,360],{"className":44750},[462],[238,44752,872],{"className":44753},[871],[238,44755,44757],{"className":44756},[399],[238,44758,44761],{"className":44759,"style":44760},[403],"height:1.842em;",[238,44762],{},[238,44764,44766],{"className":44765},[1422],[238,44767,44769,45215],{"className":44768},[395,848],[238,44770,44772,45212],{"className":44771},[399],[238,44773,44775,45092],{"className":44774,"style":44647},[403],[238,44776,44777,44780],{"style":44650},[238,44778],{"className":44779,"style":3510},[411],[238,44781,44783,44786,44789,44792,44795,44871,44874,44877,44880,44912,44915,44918,44921,44924,44927,44930,44933,44936,44939,45086,45089],{"className":44782},[282],[238,44784],{"className":44785},[282],[238,44787],{"className":44788,"style":430},[429],[238,44790,992],{"className":44791},[434],[238,44793],{"className":44794,"style":430},[429],[238,44796,44798],{"className":44797},[1943,3493],[238,44799,44801,44863],{"className":44800},[395,848],[238,44802,44804,44860],{"className":44803},[399],[238,44805,44807,44850],{"className":44806,"style":6869},[403],[238,44808,44809,44812],{"style":7296},[238,44810],{"className":44811,"style":3510},[411],[238,44813,44815],{"className":44814},[416,417,418,419],[238,44816,44818],{"className":44817},[282,419],[238,44819,44821,44824],{"className":44820},[282,419],[238,44822,341],{"className":44823},[282,283,419],[238,44825,44827],{"className":44826},[391],[238,44828,44830],{"className":44829},[395],[238,44831,44833],{"className":44832},[399],[238,44834,44836],{"className":44835,"style":4196},[403],[238,44837,44838,44841],{"style":4199},[238,44839],{"className":44840,"style":4203},[411],[238,44842,44844],{"className":44843},[416,4207,4208,419],[238,44845,44847],{"className":44846},[282,419],[238,44848,346],{"className":44849},[282,419],[238,44851,44852,44855],{"style":3529},[238,44853],{"className":44854,"style":3510},[411],[238,44856,44857],{},[238,44858,3390],{"className":44859},[1943,3538,3539],[238,44861,872],{"className":44862},[871],[238,44864,44866],{"className":44865},[399],[238,44867,44869],{"className":44868,"style":7357},[403],[238,44870],{},[238,44872],{"className":44873,"style":455},[429],[238,44875,330],{"className":44876,"style":377},[282,283],[238,44878,335],{"className":44879},[381],[238,44881,44883,44886],{"className":44882},[282],[238,44884,341],{"className":44885},[282,283],[238,44887,44889],{"className":44888},[391],[238,44890,44892],{"className":44891},[395],[238,44893,44895],{"className":44894},[399],[238,44896,44898],{"className":44897,"style":1223},[403],[238,44899,44900,44903],{"style":1226},[238,44901],{"className":44902,"style":412},[411],[238,44904,44906],{"className":44905},[416,417,418,419],[238,44907,44909],{"className":44908},[282,419],[238,44910,346],{"className":44911},[282,419],[238,44913],{"className":44914,"style":430},[429],[238,44916,349],{"className":44917},[434],[238,44919],{"className":44920,"style":430},[429],[238,44922,341],{"className":44923},[282,283],[238,44925,354],{"className":44926},[451],[238,44928],{"className":44929,"style":455},[429],[238,44931,357],{"className":44932},[282,283],[238,44934,360],{"className":44935},[462],[238,44937],{"className":44938,"style":455},[429],[238,44940,44942,44945,44948,44951,44954,44957,44960,44963,44966,44969,45001,45004,45007,45010,45013,45016,45045,45048,45080,45083],{"className":44941},[3041],[238,44943,736],{"className":44944,"style":4791},[381,4790],[238,44946,600],{"className":44947,"style":637},[282,283],[238,44949,335],{"className":44950},[381],[238,44952,341],{"className":44953},[282,283],[238,44955,354],{"className":44956},[451],[238,44958],{"className":44959,"style":455},[429],[238,44961,357],{"className":44962},[282,283],[238,44964,354],{"className":44965},[451],[238,44967],{"className":44968,"style":455},[429],[238,44970,44972,44975],{"className":44971},[282],[238,44973,341],{"className":44974},[282,283],[238,44976,44978],{"className":44977},[391],[238,44979,44981],{"className":44980},[395],[238,44982,44984],{"className":44983},[399],[238,44985,44987],{"className":44986,"style":1223},[403],[238,44988,44989,44992],{"style":1226},[238,44990],{"className":44991,"style":412},[411],[238,44993,44995],{"className":44994},[416,417,418,419],[238,44996,44998],{"className":44997},[282,419],[238,44999,346],{"className":45000},[282,419],[238,45002,360],{"className":45003},[462],[238,45005],{"className":45006,"style":660},[429],[238,45008,987],{"className":45009},[664],[238,45011],{"className":45012,"style":660},[429],[238,45014,730],{"className":45015,"style":764},[282,283],[238,45017,45019,45022],{"className":45018},[282],[238,45020,4429],{"className":45021,"style":660},[282,283],[238,45023,45025],{"className":45024},[391],[238,45026,45028],{"className":45027},[395],[238,45029,45031],{"className":45030},[399],[238,45032,45034],{"className":45033,"style":22441},[403],[238,45035,45036,45039],{"style":1226},[238,45037],{"className":45038,"style":412},[411],[238,45040,45042],{"className":45041},[416,417,418,419],[238,45043,22083],{"className":45044},[664,419],[238,45046,335],{"className":45047},[381],[238,45049,45051,45054],{"className":45050},[282],[238,45052,341],{"className":45053},[282,283],[238,45055,45057],{"className":45056},[391],[238,45058,45060],{"className":45059},[395],[238,45061,45063],{"className":45062},[399],[238,45064,45066],{"className":45065,"style":1223},[403],[238,45067,45068,45071],{"style":1226},[238,45069],{"className":45070,"style":412},[411],[238,45072,45074],{"className":45073},[416,417,418,419],[238,45075,45077],{"className":45076},[282,419],[238,45078,346],{"className":45079},[282,419],[238,45081,360],{"className":45082},[462],[238,45084,4479],{"className":45085,"style":4791},[462,4790],[238,45087],{"className":45088,"style":455},[429],[238,45090,354],{"className":45091},[451],[238,45093,45094,45097],{"style":44706},[238,45095],{"className":45096,"style":3510},[411],[238,45098,45100,45103,45106,45109,45112,45159,45162,45191,45194,45197,45200,45203,45206,45209],{"className":45099},[282],[238,45101],{"className":45102},[282],[238,45104],{"className":45105,"style":430},[429],[238,45107,992],{"className":45108},[434],[238,45110],{"className":45111,"style":430},[429],[238,45113,45115],{"className":45114},[1943,3493],[238,45116,45118,45151],{"className":45117},[395,848],[238,45119,45121,45148],{"className":45120},[399],[238,45122,45124,45138],{"className":45123,"style":489},[403],[238,45125,45126,45129],{"style":22498},[238,45127],{"className":45128,"style":1134},[411],[238,45130,45132],{"className":45131},[416,417,418,419],[238,45133,45135],{"className":45134},[282,419],[238,45136,357],{"className":45137},[282,283,419],[238,45139,45140,45143],{"style":21798},[238,45141],{"className":45142,"style":1134},[411],[238,45144,45145],{},[238,45146,1886],{"className":45147},[1943],[238,45149,872],{"className":45150},[871],[238,45152,45154],{"className":45153},[399],[238,45155,45157],{"className":45156,"style":22530},[403],[238,45158],{},[238,45160],{"className":45161,"style":455},[429],[238,45163,45165,45168],{"className":45164},[282],[238,45166,4494],{"className":45167},[282,283],[238,45169,45171],{"className":45170},[391],[238,45172,45174],{"className":45173},[395],[238,45175,45177],{"className":45176},[399],[238,45178,45180],{"className":45179,"style":22441},[403],[238,45181,45182,45185],{"style":1226},[238,45183],{"className":45184,"style":412},[411],[238,45186,45188],{"className":45187},[416,417,418,419],[238,45189,22083],{"className":45190},[664,419],[238,45192,335],{"className":45193},[381],[238,45195,341],{"className":45196},[282,283],[238,45198,354],{"className":45199},[451],[238,45201],{"className":45202,"style":455},[429],[238,45204,357],{"className":45205},[282,283],[238,45207,360],{"className":45208},[462],[238,45210,3102],{"className":45211},[282],[238,45213,872],{"className":45214},[871],[238,45216,45218],{"className":45217},[399],[238,45219,45221],{"className":45220,"style":44760},[403],[238,45222],{},[11,45224,45225,45226,45337,45338,45425,45426,1838,45484,45542],{},"第一式的上界由 ",[238,45227,45229,45255],{"className":45228,"translate":242},[241],[238,45230,45232],{"className":45231},[246],[248,45233,45234],{"xmlns":250},[252,45235,45236,45252],{},[255,45237,45238,45244,45246],{},[337,45239,45240,45242],{},[258,45241,4429],{},[258,45243,2297],{},[332,45245,2023],{},[337,45247,45248,45250],{},[258,45249,4429],{},[332,45251,22083],{},[262,45253,45254],{"encoding":264},"V^\\pi\\le V^*",[238,45256,45258,45302],{"className":45257,"ariaHidden":269},[268],[238,45259,45261,45264,45293,45296,45299],{"className":45260},[273],[238,45262],{"className":45263,"style":23995},[277],[238,45265,45267,45270],{"className":45266},[282],[238,45268,4429],{"className":45269,"style":660},[282,283],[238,45271,45273],{"className":45272},[391],[238,45274,45276],{"className":45275},[395],[238,45277,45279],{"className":45278},[399],[238,45280,45282],{"className":45281,"style":6609},[403],[238,45283,45284,45287],{"style":407},[238,45285],{"className":45286,"style":412},[411],[238,45288,45290],{"className":45289},[416,417,418,419],[238,45291,2297],{"className":45292,"style":2313},[282,283,419],[238,45294],{"className":45295,"style":430},[429],[238,45297,2023],{"className":45298},[434],[238,45300],{"className":45301,"style":430},[429],[238,45303,45305,45308],{"className":45304},[273],[238,45306],{"className":45307,"style":22096},[277],[238,45309,45311,45314],{"className":45310},[282],[238,45312,4429],{"className":45313,"style":660},[282,283],[238,45315,45317],{"className":45316},[391],[238,45318,45320],{"className":45319},[395],[238,45321,45323],{"className":45322},[399],[238,45324,45326],{"className":45325,"style":22096},[403],[238,45327,45328,45331],{"style":407},[238,45329],{"className":45330,"style":412},[411],[238,45332,45334],{"className":45333},[416,417,418,419],[238,45335,22083],{"className":45336},[664,419]," 得到，下界由在后续状态选取最优续策略达到。第二式说明，最优动作正是使 ",[238,45339,45341,45369],{"className":45340,"translate":242},[241],[238,45342,45344],{"className":45343},[246],[248,45345,45346],{"xmlns":250},[252,45347,45348,45366],{},[255,45349,45350,45356,45358,45360,45362,45364],{},[337,45351,45352,45354],{},[258,45353,4494],{},[332,45355,22083],{},[332,45357,335],{"stretchy":334},[258,45359,341],{},[332,45361,354],{"separator":269},[258,45363,357],{},[332,45365,360],{"stretchy":334},[262,45367,45368],{"encoding":264},"Q^*(s,a)",[238,45370,45372],{"className":45371,"ariaHidden":269},[268],[238,45373,45375,45378,45407,45410,45413,45416,45419,45422],{"className":45374},[273],[238,45376],{"className":45377,"style":444},[277],[238,45379,45381,45384],{"className":45380},[282],[238,45382,4494],{"className":45383},[282,283],[238,45385,45387],{"className":45386},[391],[238,45388,45390],{"className":45389},[395],[238,45391,45393],{"className":45392},[399],[238,45394,45396],{"className":45395,"style":22096},[403],[238,45397,45398,45401],{"style":407},[238,45399],{"className":45400,"style":412},[411],[238,45402,45404],{"className":45403},[416,417,418,419],[238,45405,22083],{"className":45406},[664,419],[238,45408,335],{"className":45409},[381],[238,45411,341],{"className":45412},[282,283],[238,45414,354],{"className":45415},[451],[238,45417],{"className":45418,"style":455},[429],[238,45420,357],{"className":45421},[282,283],[238,45423,360],{"className":45424},[462]," 最大的动作。因此，",[238,45427,45429,45446],{"className":45428,"translate":242},[241],[238,45430,45432],{"className":45431},[246],[248,45433,45434],{"xmlns":250},[252,45435,45436,45444],{},[255,45437,45438],{},[337,45439,45440,45442],{},[258,45441,4494],{},[332,45443,22083],{},[262,45445,44333],{"encoding":264},[238,45447,45449],{"className":45448,"ariaHidden":269},[268],[238,45450,45452,45455],{"className":45451},[273],[238,45453],{"className":45454,"style":30755},[277],[238,45456,45458,45461],{"className":45457},[282],[238,45459,4494],{"className":45460},[282,283],[238,45462,45464],{"className":45463},[391],[238,45465,45467],{"className":45466},[395],[238,45468,45470],{"className":45469},[399],[238,45471,45473],{"className":45472,"style":22096},[403],[238,45474,45475,45478],{"style":407},[238,45476],{"className":45477,"style":412},[411],[238,45479,45481],{"className":45480},[416,417,418,419],[238,45482,22083],{"className":45483},[664,419],[238,45485,45487,45504],{"className":45486,"translate":242},[241],[238,45488,45490],{"className":45489},[246],[248,45491,45492],{"xmlns":250},[252,45493,45494,45502],{},[255,45495,45496],{},[337,45497,45498,45500],{},[258,45499,4429],{},[332,45501,22083],{},[262,45503,34919],{"encoding":264},[238,45505,45507],{"className":45506,"ariaHidden":269},[268],[238,45508,45510,45513],{"className":45509},[273],[238,45511],{"className":45512,"style":22096},[277],[238,45514,45516,45519],{"className":45515},[282],[238,45517,4429],{"className":45518,"style":660},[282,283],[238,45520,45522],{"className":45521},[391],[238,45523,45525],{"className":45524},[395],[238,45526,45528],{"className":45527},[399],[238,45529,45531],{"className":45530,"style":22096},[403],[238,45532,45533,45536],{"style":407},[238,45534],{"className":45535,"style":412},[411],[238,45537,45539],{"className":45538},[416,417,418,419],[238,45540,22083],{"className":45541},[664,419]," 的差别只在于第一步是否已经固定动作。",[225,45544,45546],{"id":45545},"_43-max-与期望不可交换","4.3 max 与期望不可交换",[11,45548,45549,45550,45633,45634,45716],{},"最优贝尔曼方程中是 ",[238,45551,45553,45577],{"className":45552,"translate":242},[241],[238,45554,45556],{"className":45555},[246],[248,45557,45558],{"xmlns":250},[252,45559,45560,45574],{},[255,45561,45562,45572],{},[812,45563,45564,45570],{},[255,45565,45566,45568],{},[258,45567,1886],{},[332,45569,1889],{},[258,45571,357],{},[258,45573,4452],{"mathvariant":620},[262,45575,45576],{"encoding":264},"\\max_a\\mathbb{E}",[238,45578,45580],{"className":45579,"ariaHidden":269},[268],[238,45581,45583,45587,45627,45630],{"className":45582},[273],[238,45584],{"className":45585,"style":45586},[277],"height:0.8389em;vertical-align:-0.15em;",[238,45588,45590,45593],{"className":45589},[1943],[238,45591,1886],{"className":45592},[1943],[238,45594,45596],{"className":45595},[391],[238,45597,45599,45619],{"className":45598},[395,848],[238,45600,45602,45616],{"className":45601},[399],[238,45603,45605],{"className":45604,"style":1927},[403],[238,45606,45607,45610],{"style":44220},[238,45608],{"className":45609,"style":412},[411],[238,45611,45613],{"className":45612},[416,417,418,419],[238,45614,357],{"className":45615},[282,283,419],[238,45617,872],{"className":45618},[871],[238,45620,45622],{"className":45621},[399],[238,45623,45625],{"className":45624,"style":879},[403],[238,45626],{},[238,45628],{"className":45629,"style":455},[429],[238,45631,4452],{"className":45632},[282,714],"，而不是 ",[238,45635,45637,45661],{"className":45636,"translate":242},[241],[238,45638,45640],{"className":45639},[246],[248,45641,45642],{"xmlns":250},[252,45643,45644,45658],{},[255,45645,45646,45648],{},[258,45647,4452],{"mathvariant":620},[812,45649,45650,45656],{},[255,45651,45652,45654],{},[258,45653,1886],{},[332,45655,1889],{},[258,45657,357],{},[262,45659,45660],{"encoding":264},"\\mathbb{E}\\max_a",[238,45662,45664],{"className":45663,"ariaHidden":269},[268],[238,45665,45667,45670,45673,45676],{"className":45666},[273],[238,45668],{"className":45669,"style":45586},[277],[238,45671,4452],{"className":45672},[282,714],[238,45674],{"className":45675,"style":455},[429],[238,45677,45679,45682],{"className":45678},[1943],[238,45680,1886],{"className":45681},[1943],[238,45683,45685],{"className":45684},[391],[238,45686,45688,45708],{"className":45687},[395,848],[238,45689,45691,45705],{"className":45690},[399],[238,45692,45694],{"className":45693,"style":1927},[403],[238,45695,45696,45699],{"style":44220},[238,45697],{"className":45698,"style":412},[411],[238,45700,45702],{"className":45701},[416,417,418,419],[238,45703,357],{"className":45704},[282,283,419],[238,45706,872],{"className":45707},[871],[238,45709,45711],{"className":45710},[399],[238,45712,45714],{"className":45713,"style":879},[403],[238,45715],{},"。这两个量一般不相等，原因在于动作必须在观察到下一状态之前选定。",[11,45718,45719,45720,45748,45749,45873,45874,45908,45909,45979,45980,45908,46012,46082],{},"考虑一个状态 ",[238,45721,45723,45736],{"className":45722,"translate":242},[241],[238,45724,45726],{"className":45725},[246],[248,45727,45728],{"xmlns":250},[252,45729,45730,45734],{},[255,45731,45732],{},[258,45733,341],{},[262,45735,341],{"encoding":264},[238,45737,45739],{"className":45738,"ariaHidden":269},[268],[238,45740,45742,45745],{"className":45741},[273],[238,45743],{"className":45744,"style":489},[277],[238,45746,341],{"className":45747},[282,283],"，两个动作 ",[238,45750,45752,45778],{"className":45751,"translate":242},[241],[238,45753,45755],{"className":45754},[246],[248,45756,45757],{"xmlns":250},[252,45758,45759,45775],{},[255,45760,45761,45767,45769],{},[812,45762,45763,45765],{},[258,45764,357],{},[738,45766,745],{},[332,45768,354],{"separator":269},[812,45770,45771,45773],{},[258,45772,357],{},[738,45774,17558],{},[262,45776,45777],{"encoding":264},"a_1,a_2",[238,45779,45781],{"className":45780,"ariaHidden":269},[268],[238,45782,45784,45787,45827,45830,45833],{"className":45783},[273],[238,45785],{"className":45786,"style":2853},[277],[238,45788,45790,45793],{"className":45789},[282],[238,45791,357],{"className":45792},[282,283],[238,45794,45796],{"className":45795},[391],[238,45797,45799,45819],{"className":45798},[395,848],[238,45800,45802,45816],{"className":45801},[399],[238,45803,45805],{"className":45804,"style":1162},[403],[238,45806,45807,45810],{"style":1165},[238,45808],{"className":45809,"style":412},[411],[238,45811,45813],{"className":45812},[416,417,418,419],[238,45814,745],{"className":45815},[282,419],[238,45817,872],{"className":45818},[871],[238,45820,45822],{"className":45821},[399],[238,45823,45825],{"className":45824,"style":879},[403],[238,45826],{},[238,45828,354],{"className":45829},[451],[238,45831],{"className":45832,"style":455},[429],[238,45834,45836,45839],{"className":45835},[282],[238,45837,357],{"className":45838},[282,283],[238,45840,45842],{"className":45841},[391],[238,45843,45845,45865],{"className":45844},[395,848],[238,45846,45848,45862],{"className":45847},[399],[238,45849,45851],{"className":45850,"style":1162},[403],[238,45852,45853,45856],{"style":1165},[238,45854],{"className":45855,"style":412},[411],[238,45857,45859],{"className":45858},[416,417,418,419],[238,45860,17558],{"className":45861},[282,419],[238,45863,872],{"className":45864},[871],[238,45866,45868],{"className":45867},[399],[238,45869,45871],{"className":45870,"style":879},[403],[238,45872],{},"。下一状态以 ",[238,45875,45877,45896],{"className":45876,"translate":242},[241],[238,45878,45880],{"className":45879},[246],[248,45881,45882],{"xmlns":250},[252,45883,45884,45893],{},[255,45885,45886,45888,45891],{},[738,45887,745],{},[258,45889,45890],{"mathvariant":344},"\u002F",[738,45892,17558],{},[262,45894,45895],{"encoding":264},"1\u002F2",[238,45897,45899],{"className":45898,"ariaHidden":269},[268],[238,45900,45902,45905],{"className":45901},[273],[238,45903],{"className":45904,"style":444},[277],[238,45906,45895],{"className":45907},[282]," 的概率为 ",[238,45910,45912,45930],{"className":45911,"translate":242},[241],[238,45913,45915],{"className":45914},[246],[248,45916,45917],{"xmlns":250},[252,45918,45919,45927],{},[255,45920,45921],{},[812,45922,45923,45925],{},[258,45924,341],{},[738,45926,745],{},[262,45928,45929],{"encoding":264},"s_1",[238,45931,45933],{"className":45932,"ariaHidden":269},[268],[238,45934,45936,45939],{"className":45935},[273],[238,45937],{"className":45938,"style":3115},[277],[238,45940,45942,45945],{"className":45941},[282],[238,45943,341],{"className":45944},[282,283],[238,45946,45948],{"className":45947},[391],[238,45949,45951,45971],{"className":45950},[395,848],[238,45952,45954,45968],{"className":45953},[399],[238,45955,45957],{"className":45956,"style":1162},[403],[238,45958,45959,45962],{"style":1165},[238,45960],{"className":45961,"style":412},[411],[238,45963,45965],{"className":45964},[416,417,418,419],[238,45966,745],{"className":45967},[282,419],[238,45969,872],{"className":45970},[871],[238,45972,45974],{"className":45973},[399],[238,45975,45977],{"className":45976,"style":879},[403],[238,45978],{},"，以 ",[238,45981,45983,46000],{"className":45982,"translate":242},[241],[238,45984,45986],{"className":45985},[246],[248,45987,45988],{"xmlns":250},[252,45989,45990,45998],{},[255,45991,45992,45994,45996],{},[738,45993,745],{},[258,45995,45890],{"mathvariant":344},[738,45997,17558],{},[262,45999,45895],{"encoding":264},[238,46001,46003],{"className":46002,"ariaHidden":269},[268],[238,46004,46006,46009],{"className":46005},[273],[238,46007],{"className":46008,"style":444},[277],[238,46010,45895],{"className":46011},[282],[238,46013,46015,46033],{"className":46014,"translate":242},[241],[238,46016,46018],{"className":46017},[246],[248,46019,46020],{"xmlns":250},[252,46021,46022,46030],{},[255,46023,46024],{},[812,46025,46026,46028],{},[258,46027,341],{},[738,46029,17558],{},[262,46031,46032],{"encoding":264},"s_2",[238,46034,46036],{"className":46035,"ariaHidden":269},[268],[238,46037,46039,46042],{"className":46038},[273],[238,46040],{"className":46041,"style":3115},[277],[238,46043,46045,46048],{"className":46044},[282],[238,46046,341],{"className":46047},[282,283],[238,46049,46051],{"className":46050},[391],[238,46052,46054,46074],{"className":46053},[395,848],[238,46055,46057,46071],{"className":46056},[399],[238,46058,46060],{"className":46059,"style":1162},[403],[238,46061,46062,46065],{"style":1165},[238,46063],{"className":46064,"style":412},[411],[238,46066,46068],{"className":46067},[416,417,418,419],[238,46069,17558],{"className":46070},[282,419],[238,46072,872],{"className":46073},[871],[238,46075,46077],{"className":46076},[399],[238,46078,46080],{"className":46079,"style":879},[403],[238,46081],{},"，并设未来价值为零。奖励为",[238,46084,46086],{"className":46085,"translate":242},[945],[238,46087,46089,46225],{"className":46088,"translate":242},[241],[238,46090,46092],{"className":46091},[246],[248,46093,46094],{"xmlns":250,"display":954},[252,46095,46096,46222],{},[255,46097,46098,46100,46102,46104,46106,46112,46114,46120,46122,46124,46126,46128,46130,46132,46134,46136,46138,46144,46146,46152,46154,46156,46158,46160,46162,46164,46166,46168,46170,46176,46178,46184,46186,46188,46190,46192,46194,46196,46198,46200,46202,46208,46210,46216,46218,46220],{},[258,46099,600],{},[332,46101,335],{"stretchy":334},[258,46103,341],{},[332,46105,354],{"separator":269},[812,46107,46108,46110],{},[258,46109,357],{},[738,46111,745],{},[332,46113,354],{"separator":269},[812,46115,46116,46118],{},[258,46117,341],{},[738,46119,745],{},[332,46121,360],{"stretchy":334},[332,46123,992],{},[738,46125,745],{},[332,46127,354],{"separator":269},[429,46129],{"width":7090},[258,46131,600],{},[332,46133,335],{"stretchy":334},[258,46135,341],{},[332,46137,354],{"separator":269},[812,46139,46140,46142],{},[258,46141,357],{},[738,46143,745],{},[332,46145,354],{"separator":269},[812,46147,46148,46150],{},[258,46149,341],{},[738,46151,17558],{},[332,46153,360],{"stretchy":334},[332,46155,992],{},[738,46157,740],{},[332,46159,354],{"separator":269},[429,46161],{"width":10450},[258,46163,600],{},[332,46165,335],{"stretchy":334},[258,46167,341],{},[332,46169,354],{"separator":269},[812,46171,46172,46174],{},[258,46173,357],{},[738,46175,17558],{},[332,46177,354],{"separator":269},[812,46179,46180,46182],{},[258,46181,341],{},[738,46183,745],{},[332,46185,360],{"stretchy":334},[332,46187,992],{},[738,46189,740],{},[332,46191,354],{"separator":269},[429,46193],{"width":7090},[258,46195,600],{},[332,46197,335],{"stretchy":334},[258,46199,341],{},[332,46201,354],{"separator":269},[812,46203,46204,46206],{},[258,46205,357],{},[738,46207,17558],{},[332,46209,354],{"separator":269},[812,46211,46212,46214],{},[258,46213,341],{},[738,46215,17558],{},[332,46217,360],{"stretchy":334},[332,46219,992],{},[738,46221,26056],{},[262,46223,46224],{"encoding":264},"r(s,a_1,s_1)=1,\\quad r(s,a_1,s_2)=0,\n\\qquad\nr(s,a_2,s_1)=0,\\quad r(s,a_2,s_2)=1.",[238,46226,46228,46347,46478,46609,46740],{"className":46227,"ariaHidden":269},[268],[238,46229,46231,46234,46237,46240,46243,46246,46249,46289,46292,46295,46335,46338,46341,46344],{"className":46230},[273],[238,46232],{"className":46233,"style":444},[277],[238,46235,600],{"className":46236,"style":637},[282,283],[238,46238,335],{"className":46239},[381],[238,46241,341],{"className":46242},[282,283],[238,46244,354],{"className":46245},[451],[238,46247],{"className":46248,"style":455},[429],[238,46250,46252,46255],{"className":46251},[282],[238,46253,357],{"className":46254},[282,283],[238,46256,46258],{"className":46257},[391],[238,46259,46261,46281],{"className":46260},[395,848],[238,46262,46264,46278],{"className":46263},[399],[238,46265,46267],{"className":46266,"style":1162},[403],[238,46268,46269,46272],{"style":1165},[238,46270],{"className":46271,"style":412},[411],[238,46273,46275],{"className":46274},[416,417,418,419],[238,46276,745],{"className":46277},[282,419],[238,46279,872],{"className":46280},[871],[238,46282,46284],{"className":46283},[399],[238,46285,46287],{"className":46286,"style":879},[403],[238,46288],{},[238,46290,354],{"className":46291},[451],[238,46293],{"className":46294,"style":455},[429],[238,46296,46298,46301],{"className":46297},[282],[238,46299,341],{"className":46300},[282,283],[238,46302,46304],{"className":46303},[391],[238,46305,46307,46327],{"className":46306},[395,848],[238,46308,46310,46324],{"className":46309},[399],[238,46311,46313],{"className":46312,"style":1162},[403],[238,46314,46315,46318],{"style":1165},[238,46316],{"className":46317,"style":412},[411],[238,46319,46321],{"className":46320},[416,417,418,419],[238,46322,745],{"className":46323},[282,419],[238,46325,872],{"className":46326},[871],[238,46328,46330],{"className":46329},[399],[238,46331,46333],{"className":46332,"style":879},[403],[238,46334],{},[238,46336,360],{"className":46337},[462],[238,46339],{"className":46340,"style":430},[429],[238,46342,992],{"className":46343},[434],[238,46345],{"className":46346,"style":430},[429],[238,46348,46350,46353,46356,46359,46362,46365,46368,46371,46374,46377,46380,46420,46423,46426,46466,46469,46472,46475],{"className":46349},[273],[238,46351],{"className":46352,"style":444},[277],[238,46354,745],{"className":46355},[282],[238,46357,354],{"className":46358},[451],[238,46360],{"className":46361,"style":7439},[429],[238,46363],{"className":46364,"style":455},[429],[238,46366,600],{"className":46367,"style":637},[282,283],[238,46369,335],{"className":46370},[381],[238,46372,341],{"className":46373},[282,283],[238,46375,354],{"className":46376},[451],[238,46378],{"className":46379,"style":455},[429],[238,46381,46383,46386],{"className":46382},[282],[238,46384,357],{"className":46385},[282,283],[238,46387,46389],{"className":46388},[391],[238,46390,46392,46412],{"className":46391},[395,848],[238,46393,46395,46409],{"className":46394},[399],[238,46396,46398],{"className":46397,"style":1162},[403],[238,46399,46400,46403],{"style":1165},[238,46401],{"className":46402,"style":412},[411],[238,46404,46406],{"className":46405},[416,417,418,419],[238,46407,745],{"className":46408},[282,419],[238,46410,872],{"className":46411},[871],[238,46413,46415],{"className":46414},[399],[238,46416,46418],{"className":46417,"style":879},[403],[238,46419],{},[238,46421,354],{"className":46422},[451],[238,46424],{"className":46425,"style":455},[429],[238,46427,46429,46432],{"className":46428},[282],[238,46430,341],{"className":46431},[282,283],[238,46433,46435],{"className":46434},[391],[238,46436,46438,46458],{"className":46437},[395,848],[238,46439,46441,46455],{"className":46440},[399],[238,46442,46444],{"className":46443,"style":1162},[403],[238,46445,46446,46449],{"style":1165},[238,46447],{"className":46448,"style":412},[411],[238,46450,46452],{"className":46451},[416,417,418,419],[238,46453,17558],{"className":46454},[282,419],[238,46456,872],{"className":46457},[871],[238,46459,46461],{"className":46460},[399],[238,46462,46464],{"className":46463,"style":879},[403],[238,46465],{},[238,46467,360],{"className":46468},[462],[238,46470],{"className":46471,"style":430},[429],[238,46473,992],{"className":46474},[434],[238,46476],{"className":46477,"style":430},[429],[238,46479,46481,46484,46487,46490,46493,46496,46499,46502,46505,46508,46511,46551,46554,46557,46597,46600,46603,46606],{"className":46480},[273],[238,46482],{"className":46483,"style":444},[277],[238,46485,740],{"className":46486},[282],[238,46488,354],{"className":46489},[451],[238,46491],{"className":46492,"style":10986},[429],[238,46494],{"className":46495,"style":455},[429],[238,46497,600],{"className":46498,"style":637},[282,283],[238,46500,335],{"className":46501},[381],[238,46503,341],{"className":46504},[282,283],[238,46506,354],{"className":46507},[451],[238,46509],{"className":46510,"style":455},[429],[238,46512,46514,46517],{"className":46513},[282],[238,46515,357],{"className":46516},[282,283],[23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l":740,"displaystyle":269},[255,46826,46827,46829],{},[258,46828,4452],{"mathvariant":620},[255,46830,46831,46833,46843,46845,46847,46849,46851,46853,46855,46861,46863,46865,46867],{},[332,46832,736],{"fence":269},[6755,46834,46835,46841],{},[255,46836,46837,46839],{},[258,46838,1886],{},[332,46840,1889],{},[258,46842,357],{},[258,46844,600],{},[332,46846,335],{"stretchy":334},[258,46848,341],{},[332,46850,354],{"separator":269},[258,46852,357],{},[332,46854,354],{"separator":269},[337,46856,46857,46859],{},[258,46858,341],{},[332,46860,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,46862,360],{"stretchy":334},[332,46864,349],{},[258,46866,341],{},[332,46868,4479],{"fence":269},[966,46870,46871],{},[969,46872,46873],{"scriptlevel":740,"displaystyle":269},[255,46874,46875,46877,46879],{},[255,46876],{},[332,46878,992],{},[738,46880,26056],{},[262,46882,46883],{"encoding":264},"\\begin{aligned}\n\\max_{a}\\mathbb{E}[r\\mid s,a]&=\\frac12,\\\\\n\\mathbb{E}\\left[\\max_{a}r(s,a,s')\\mid s\\right]&=1.\n\\end{aligned}",[238,46885,46887],{"className":46886,"ariaHidden":269},[268],[238,46888,46890,46894],{"className":46889},[273],[238,46891],{"className":46892,"style":46893},[277],"height:4.1714em;vertical-align:-1.8357em;",[238,46895,46897],{"className":46896},[282],[238,46898,46900,47169],{"className":46899},[958],[238,46901,46903],{"className":46902},[1117],[238,46904,46906,47160],{"className":46905},[395,848],[238,46907,46909,47157],{"className":46908},[399],[238,46910,46913,47006],{"className":46911,"style":46912},[403],"height:2.3357em;",[238,46914,46916,46920],{"style":46915},"top:-4.3357em;",[238,46917],{"className":46918,"style":46919},[411],"height:3.3214em;",[238,46921,46923,46970,46973,46976,46979,46982,46985,46988,46991,46994,46997,47000,47003],{"className":46922},[282],[238,46924,46926],{"className":46925},[1943,3493],[238,46927,46929,46962],{"className":46928},[395,848],[238,46930,46932,46959],{"className":46931},[399],[238,46933,46935,46949],{"className":46934,"style":489},[403],[238,46936,46937,46940],{"style":22498},[238,46938],{"className":46939,"style":1134},[411],[238,46941,46943],{"className":46942},[416,417,418,419],[238,46944,46946],{"className":46945},[282,419],[238,46947,357],{"className":46948},[282,283,419],[238,46950,46951,46954],{"style":21798},[238,46952],{"className":46953,"style":1134},[411],[238,46955,46956],{},[238,46957,1886],{"className":46958},[1943],[238,46960,872],{"className":46961},[871],[238,46963,46965],{"className":46964},[399],[238,46966,46968],{"className":46967,"style":22530},[403],[238,46969],{},[238,46971],{"className":46972,"style":455},[429],[238,46974,4452],{"className":46975},[282,714],[238,46977,736],{"className":46978},[381],[238,46980,600],{"className":46981,"style":637},[282,283],[238,46983],{"className":46984,"style":430},[429],[238,46986,349],{"className":46987},[434],[238,46989],{"className":46990,"style":430},[429],[238,46992,341],{"className":46993},[282,283],[238,46995,354],{"className":46996},[451],[238,46998],{"className":46999,"style":455},[429],[238,47001,357],{"className":47002},[282,283],[238,47004,4479],{"className":47005},[462],[238,47007,47009,47012],{"style":47008},"top:-2.1857em;",[238,47010],{"className":47011,"style":46919},[411],[238,47013,47015,47018,47021],{"className":47014},[282],[238,47016,4452],{"className":47017},[282,714],[238,47019],{"className":47020,"style":455},[429],[238,47022,47024,47030,47077,47080,47083,47086,47089,47092,47095,47098,47101,47104,47136,47139,47142,47145,47148,47151],{"className":47023},[3041],[238,47025,47027],{"className":47026,"style":4791},[381,4790],[238,47028,736],{"className":47029},[18719,33403],[238,47031,47033],{"className":47032},[1943,3493],[238,47034,47036,47069],{"className":47035},[395,848],[238,47037,47039,47066],{"className":47038},[399],[238,47040,47042,47056],{"className":47041,"style":489},[403],[238,47043,47044,47047],{"style":22498},[238,47045],{"className":47046,"style":1134},[411],[238,47048,47050],{"className":47049},[416,417,418,419],[238,47051,47053],{"className":47052},[282,419],[238,47054,357],{"className":47055},[282,283,419],[238,47057,47058,47061],{"style":21798},[238,47059],{"className":47060,"style":1134},[411],[238,47062,47063],{},[238,47064,1886],{"className":47065},[1943],[238,47067,872],{"className":47068},[871],[238,47070,47072],{"className":47071},[399],[238,47073,47075],{"className":47074,"style":22530},[403],[238,47076],{},[238,47078],{"className":47079,"style":455},[429],[238,47081,600],{"className":47082,"style":637},[282,283],[238,47084,335],{"className":47085},[381],[238,47087,341],{"className":47088},[282,283],[238,47090,354],{"className":47091},[451],[238,47093],{"className":47094,"style":455},[429],[238,47096,357],{"className":47097},[282,283],[238,47099,354],{"className":47100},[451],[238,47102],{"className":47103,"style":455},[429],[238,47105,47107,47110],{"className":47106},[282],[238,47108,341],{"className":47109},[282,283],[238,47111,47113],{"className":47112},[391],[238,47114,47116],{"className":47115},[395],[238,47117,47119],{"className":47118},[399],[238,47120,47122],{"className":47121,"style":1223},[403],[238,47123,47124,47127],{"style":1226},[238,47125],{"className":47126,"style":412},[411],[238,47128,47130],{"className":47129},[416,417,418,419],[238,47131,47133],{"className":47132},[282,419],[238,47134,346],{"className":47135},[282,419],[238,47137,360],{"className":47138},[462],[238,47140],{"className":47141,"style":430},[429],[238,47143,349],{"className":47144},[434],[238,47146],{"className":47147,"style":430},[429],[238,47149,341],{"className":47150},[282,283],[238,47152,47154],{"className":47153,"style":4791},[462,4790],[238,47155,4479],{"className":47156},[18719,33403],[238,47158,872],{"className":47159},[871],[238,47161,47163],{"className":47162},[399],[238,47164,47167],{"className":47165,"style":47166},[403],"height:1.8357em;",[238,47168],{},[238,47170,47172],{"className":47171},[1422],[238,47173,47175,47292],{"className":47174},[395,848],[238,47176,47178,47289],{"className":47177},[399],[238,47179,47181,47266],{"className":47180,"style":46912},[403],[238,47182,47183,47186],{"style":46915},[238,47184],{"className":47185,"style":46919},[411],[238,47187,47189,47192,47195,47198,47201,47263],{"className":47188},[282],[238,47190],{"className":47191},[282],[238,47193],{"className":47194,"style":430},[429],[238,47196,992],{"className":47197},[434],[238,47199],{"className":47200,"style":430},[429],[238,47202,47204,47207,47260],{"className":47203},[282],[238,47205],{"className":47206},[381,5662],[238,47208,47210],{"className":47209},[5561],[238,47211,47213,47252],{"className":47212},[395,848],[238,47214,47216,47249],{"className":47215},[399],[238,47217,47219,47230,47238],{"className":47218,"style":18439},[403],[238,47220,47221,47224],{"style":5678},[238,47222],{"className":47223,"style":1134},[411],[238,47225,47227],{"className":47226},[282],[238,47228,17558],{"className":47229},[282],[238,47231,47232,47235],{"style":5702},[238,47233],{"className":47234,"style":1134},[411],[238,47236],{"className":47237,"style":5710},[5709],[238,47239,47240,47243],{"style":5713},[238,47241],{"className":47242,"style":1134},[411],[238,47244,47246],{"className":47245},[282],[238,47247,745],{"className":47248},[282],[238,47250,872],{"className":47251},[871],[238,47253,47255],{"className":47254},[399],[238,47256,47258],{"className":47257,"style":19686},[403],[238,47259],{},[238,47261],{"className":47262},[462,5662],[238,47264,354],{"className":47265},[451],[238,47267,47268,47271],{"style":47008},[238,47269],{"className":47270,"style":46919},[411],[238,47272,47274,47277,47280,47283,47286],{"className":47273},[282],[238,47275],{"className":47276},[282],[238,47278],{"className":47279,"style":430},[429],[238,47281,992],{"className":47282},[434],[238,47284],{"className":47285,"style":430},[429],[238,47287,26056],{"className":47288},[282],[238,47290,872],{"className":47291},[871],[238,47293,47295],{"className":47294},[399],[238,47296,47298],{"className":47297,"style":47166},[403],[238,47299],{},[11,47301,47302,47303,47364],{},"后者相当于在观察到 ",[238,47304,47306,47323],{"className":47305,"translate":242},[241],[238,47307,47309],{"className":47308},[246],[248,47310,47311],{"xmlns":250},[252,47312,47313,47321],{},[255,47314,47315],{},[337,47316,47317,47319],{},[258,47318,341],{},[332,47320,346],{"mathvariant":344,"lspace":345,"rspace":345},[262,47322,543],{"encoding":264},[238,47324,47326],{"className":47325,"ariaHidden":269},[268],[238,47327,47329,47332],{"className":47328},[273],[238,47330],{"className":47331,"style":404},[277],[238,47333,47335,47338],{"className":47334},[282],[238,47336,341],{"className":47337},[282,283],[238,47339,47341],{"className":47340},[391],[238,47342,47344],{"className":47343},[395],[238,47345,47347],{"className":47346},[399],[238,47348,47350],{"className":47349,"style":404},[403],[238,47351,47352,47355],{"style":407},[238,47353],{"className":47354,"style":412},[411],[238,47356,47358],{"className":47357},[416,417,418,419],[238,47359,47361],{"className":47360},[282,419],[238,47362,346],{"className":47363},[282,419]," 之后才选动作，因此获得了更大的灵活性，但它不是原问题的可行决策顺序。贝尔曼最优方程采用前者，是对决策时序的正确编码。",[225,47366,47368],{"id":47367},"_44-验证定理与最优策略","4.4 验证定理与最优策略",[230,47370,47373,47666,48153],{"id":47371,"title":47372},"corollary-optimal-verification","推论 4.3（最优性的验证定理）",[11,47374,236,47375,16685,47440,14826,47526,47607,47608,47636,47637,47665],{},[238,47376,47378,47401],{"className":47377,"translate":242},[241],[238,47379,47381],{"className":47380},[246],[248,47382,47383],{"xmlns":250},[252,47384,47385,47399],{},[255,47386,47387,47389,47391,47393,47395,47397],{},[258,47388,4429],{},[332,47390,733],{},[258,47392,19971],{},[332,47394,335],{"stretchy":334},[258,47396,260],{},[332,47398,360],{"stretchy":334},[262,47400,19980],{"encoding":264},[238,47402,47404,47422],{"className":47403,"ariaHidden":269},[268],[238,47405,47407,47410,47413,47416,47419],{"className":47406},[273],[238,47408],{"className":47409,"style":13260},[277],[238,47411,4429],{"className":47412,"style":660},[282,283],[238,47414],{"className":47415,"style":430},[429],[238,47417,733],{"className":47418},[434],[238,47420],{"className":47421,"style":430},[429],[238,47423,47425,47428,47431,47434,47437],{"className":47424},[273],[238,47426],{"className":47427,"style":444},[277],[238,47429,19971],{"className":47430,"style":20011},[282,283],[238,47432,335],{"className":47433},[381],[238,47435,260],{"className":47436,"style":284},[282,283],[238,47438,360],{"className":47439},[462],[238,47441,47443,47467],{"className":47442,"translate":242},[241],[238,47444,47446],{"className":47445},[246],[248,47447,47448],{"xmlns":250},[252,47449,47450,47464],{},[255,47451,47452,47454,47456,47462],{},[258,47453,4429],{},[332,47455,992],{},[337,47457,47458,47460],{},[258,47459,11971],{"mathvariant":816},[332,47461,22083],{},[258,47463,4429],{},[262,47465,47466],{"encoding":264},"V=\\mathcal{T}^*V",[238,47468,47470,47488],{"className":47469,"ariaHidden":269},[268],[238,47471,47473,47476,47479,47482,47485],{"className":47472},[273],[238,47474],{"className":47475,"style":278},[277],[238,47477,4429],{"className":47478,"style":660},[282,283],[238,47480],{"className":47481,"style":430},[429],[238,47483,992],{"className":47484},[434],[238,47486],{"className":47487,"style":430},[429],[238,47489,47491,47494,47523],{"className":47490},[273],[238,47492],{"className":47493,"style":22096},[277],[238,47495,47497,47500],{"className":47496},[282],[238,47498,11971],{"className":47499,"style":11992},[282,840],[238,47501,47503],{"className":47502},[391],[238,47504,47506],{"className":47505},[395],[238,47507,47509],{"className":47508},[399],[238,47510,47512],{"className":47511,"style":22096},[403],[238,47513,47514,47517],{"style":407},[238,47515],{"className":47516,"style":412},[411],[238,47518,47520],{"className":47519},[416,417,418,419],[238,47521,22083],{"className":47522},[664,419],[238,47524,4429],{"className":47525,"style":660},[282,283],[238,47527,47529,47551],{"className":47528,"translate":242},[241],[238,47530,47532],{"className":47531},[246],[248,47533,47534],{"xmlns":250},[252,47535,47536,47548],{},[255,47537,47538,47540,47542],{},[258,47539,4429],{},[332,47541,992],{},[337,47543,47544,47546],{},[258,47545,4429],{},[332,47547,22083],{},[262,47549,47550],{"encoding":264},"V=V^*",[238,47552,47554,47572],{"className":47553,"ariaHidden":269},[268],[238,47555,47557,47560,47563,47566,47569],{"className":47556},[273],[238,47558],{"className":47559,"style":278},[277],[238,47561,4429],{"className":47562,"style":660},[282,283],[238,47564],{"className":47565,"style":430},[429],[238,47567,992],{"className":47568},[434],[238,47570],{"className":47571,"style":430},[429],[238,47573,47575,47578],{"className":47574},[273],[238,47576],{"className":47577,"style":22096},[277],[238,47579,47581,47584],{"className":47580},[282],[238,47582,4429],{"className":47583,"style":660},[282,283],[238,47585,47587],{"className":47586},[391],[238,47588,47590],{"className":47589},[395],[238,47591,47593],{"className":47592},[399],[238,47594,47596],{"className":47595,"style":22096},[403],[238,47597,47598,47601],{"style":407},[238,47599],{"className":47600,"style":412},[411],[238,47602,47604],{"className":47603},[416,417,418,419],[238,47605,22083],{"className":47606},[664,419],"。若策略 ",[238,47609,47611,47624],{"className":47610,"translate":242},[241],[238,47612,47614],{"className":47613},[246],[248,47615,47616],{"xmlns":250},[252,47617,47618,47622],{},[255,47619,47620],{},[258,47621,2297],{},[262,47623,2300],{"encoding":264},[238,47625,47627],{"className":47626,"ariaHidden":269},[268],[238,47628,47630,47633],{"className":47629},[273],[238,47631],{"className":47632,"style":489},[277],[238,47634,2297],{"className":47635,"style":2313},[282,283]," 关于 ",[238,47638,47640,47653],{"className":47639,"translate":242},[241],[238,47641,47643],{"className":47642},[246],[248,47644,47645],{"xmlns":250},[252,47646,47647,47651],{},[255,47648,47649],{},[258,47650,4429],{},[262,47652,4429],{"encoding":264},[238,47654,47656],{"className":47655,"ariaHidden":269},[268],[238,47657,47659,47662],{"className":47658},[273],[238,47660],{"className":47661,"style":278},[277],[238,47663,4429],{"className":47664,"style":660},[282,283]," 贪婪，即",[238,47667,47669],{"className":47668,"translate":242},[945],[238,47670,47672,47782],{"className":47671,"translate":242},[241],[238,47673,47675],{"className":47674},[246],[248,47676,47677],{"xmlns":250,"display":954},[252,47678,47679,47779],{},[255,47680,47681,47683,47685,47687,47689,47691,47693,47695,47705,47715,47717,47719,47725,47727,47729,47731,47733,47735,47777],{},[258,47682,2297],{},[332,47684,335],{"stretchy":334},[258,47686,341],{},[332,47688,360],{"stretchy":334},[332,47690,733],{},[258,47692,39246],{},[332,47694,1889],{},[6755,47696,47697,47703],{},[255,47698,47699,47701],{},[258,47700,1886],{},[332,47702,1889],{},[258,47704,357],{},[6755,47706,47707,47709],{},[332,47708,3390],{},[337,47710,47711,47713],{},[258,47712,341],{},[332,47714,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,47716,330],{},[332,47718,335],{"stretchy":334},[337,47720,47721,47723],{},[258,47722,341],{},[332,47724,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,47726,349],{},[258,47728,341],{},[332,47730,354],{"separator":269},[258,47732,357],{},[332,47734,360],{"stretchy":334},[255,47736,47737,47739,47741,47743,47745,47747,47749,47751,47757,47759,47761,47763,47765,47767,47773,47775],{},[332,47738,736],{"fence":269},[258,47740,600],{},[332,47742,335],{"stretchy":334},[258,47744,341],{},[332,47746,354],{"separator":269},[258,47748,357],{},[332,47750,354],{"separator":269},[337,47752,47753,47755],{},[258,47754,341],{},[332,47756,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,47758,360],{"stretchy":334},[332,47760,987],{},[258,47762,730],{},[258,47764,4429],{},[332,47766,335],{"stretchy":334},[337,47768,47769,47771],{},[258,47770,341],{},[332,47772,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,47774,360],{"stretchy":334},[332,47776,4479],{"fence":269},[332,47778,354],{"separator":269},[262,47780,47781],{"encoding":264},"\\pi(s)\\in\\arg\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')\\right],",[238,47783,47785,47812,48002],{"className":47784,"ariaHidden":269},[268],[238,47786,47788,47791,47794,47797,47800,47803,47806,47809],{"className":47787},[273],[238,47789],{"className":47790,"style":444},[277],[238,47792,2297],{"className":47793,"style":2313},[282,283],[238,47795,335],{"className":47796},[381],[238,47798,341],{"className":47799},[282,283],[238,47801,360],{"className":47802},[462],[238,47804],{"className":47805,"style":430},[429],[238,47807,733],{"className":47808},[434],[238,47810],{"className":47811,"style":430},[429],[238,47813,47815,47818,47823,47826,47873,47876,47952,47955,47958,47961,47993,47996,47999],{"className":47814},[273],[238,47816],{"className":47817,"style":22483},[277],[238,47819,39405,47821],{"className":47820},[1943],[238,47822,39408],{"style":35277},[238,47824],{"className":47825,"style":455},[429],[238,47827,47829],{"className":47828},[1943,3493],[238,47830,47832,47865],{"className":47831},[395,848],[238,47833,47835,47862],{"className":47834},[399],[238,47836,47838,47852],{"className":47837,"style":489},[403],[238,47839,47840,47843],{"style":22498},[238,47841],{"className":47842,"style":1134},[411],[238,47844,47846],{"className":47845},[416,417,418,419],[238,47847,47849],{"className":47848},[282,419],[238,47850,357],{"className":47851},[282,283,419],[238,47853,47854,47857],{"style":21798},[238,47855],{"className":47856,"style":1134},[411],[238,47858,47859],{},[238,47860,1886],{"className":47861},[1943],[238,47863,872],{"className":47864},[871],[238,47866,47868],{"className":47867},[399],[238,47869,47871],{"className":47870,"style":22530},[403],[238,47872],{},[238,47874],{"className":47875,"style":455},[429],[238,47877,47879],{"className":47878},[1943,3493],[238,47880,47882,47944],{"className":47881},[395,848],[238,47883,47885,47941],{"className":47884},[399],[238,47886,47888,47931],{"className":47887,"style":6869},[403],[238,47889,47890,47893],{"style":7296},[238,47891],{"className":47892,"style":3510},[411],[238,47894,47896],{"className":47895},[416,417,418,419],[238,47897,47899],{"className":47898},[282,419],[238,47900,47902,47905],{"className":47901},[282,419],[238,47903,341],{"className":47904},[282,283,419],[238,47906,47908],{"className":47907},[391],[238,47909,47911],{"className":47910},[395],[238,47912,47914],{"className":47913},[399],[238,47915,47917],{"className":47916,"style":4196},[403],[238,47918,47919,47922],{"style":4199},[238,47920],{"className":47921,"style":4203},[411],[238,47923,47925],{"className":47924},[416,4207,4208,419],[238,47926,47928],{"className":47927},[282,419],[238,47929,346],{"className":47930},[282,419],[238,47932,47933,47936],{"style":3529},[238,47934],{"className":47935,"style":3510},[411],[238,47937,47938],{},[238,47939,3390],{"className":47940},[1943,3538,3539],[238,47942,872],{"className":47943},[871],[238,47945,47947],{"className":47946},[399],[238,47948,47950],{"className":47949,"style":7357},[403],[238,47951],{},[238,47953],{"className":47954,"style":455},[429],[238,47956,330],{"className":47957,"style":377},[282,283],[238,47959,335],{"className":47960},[381],[238,47962,47964,47967],{"className":47963},[282],[238,47965,341],{"className":47966},[282,283],[238,47968,47970],{"className":47969},[391],[238,47971,47973],{"className":47972},[395],[238,47974,47976],{"className":47975},[399],[238,47977,47979],{"className":47978,"style":1223},[403],[238,47980,47981,47984],{"style":1226},[238,47982],{"className":47983,"style":412},[411],[238,47985,47987],{"className":47986},[416,417,418,419],[238,47988,47990],{"className":47989},[282,419],[238,47991,346],{"className":47992},[282,419],[238,47994],{"className":47995,"style":430},[429],[238,47997,349],{"className":47998},[434],[238,48000],{"className":48001,"style":430},[429],[238,48003,48005,48008,48011,48014,48017,48020,48023,48026,48147,48150],{"className":48004},[273],[238,48006],{"className":48007,"style":22668},[277],[238,48009,341],{"className":48010},[282,283],[238,48012,354],{"className":48013},[451],[238,48015],{"className":48016,"style":455},[429],[238,48018,357],{"className":48019},[282,283],[238,48021,360],{"className":48022},[462],[238,48024],{"className":48025,"style":455},[429],[238,48027,48029,48032,48035,48038,48041,48044,48047,48050,48053,48056,48088,48091,48094,48097,48100,48103,48106,48109,48141,48144],{"className":48028},[3041],[238,48030,736],{"className":48031,"style":4791},[381,4790],[238,48033,600],{"className":48034,"style":637},[282,283],[238,48036,335],{"className":48037},[381],[238,48039,341],{"className":48040},[282,283],[238,48042,354],{"className":48043},[451],[238,48045],{"className":48046,"style":455},[429],[238,48048,357],{"className":48049},[282,283],[238,48051,354],{"className":48052},[451],[238,48054],{"className":48055,"style":455},[429],[238,48057,48059,48062],{"className":48058},[282],[238,48060,341],{"className":48061},[282,283],[238,48063,48065],{"className":48064},[391],[238,48066,48068],{"className":48067},[395],[238,48069,48071],{"clas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是最优策略。",[11,48266,48267,48269,48270,48328,48329,47636,48357,48385,48386,42481,48529,941,48531,21427],{},[1688,48268,5794],{}," 第一式由 ",[238,48271,48273,48290],{"className":48272,"translate":242},[241],[238,48274,48276],{"className":48275},[246],[248,48277,48278],{"xmlns":250},[252,48279,48280,48288],{},[255,48281,48282],{},[337,48283,48284,48286],{},[258,48285,11971],{"mathvariant":816},[332,48287,22083],{},[262,48289,22086],{"encoding":264},[238,48291,48293],{"className":48292,"ariaHidden":269},[268],[238,48294,48296,48299],{"className":48295},[273],[238,48297],{"className":48298,"style":22096},[277],[238,48300,48302,48305],{"className":48301},[282],[238,48303,11971],{"className":48304,"style":11992},[282,840],[238,48306,48308],{"className":48307},[391],[238,48309,48311],{"className":48310},[395],[238,48312,48314],{"className":48313},[399],[238,48315,48317],{"className":48316,"style":22096},[403],[238,48318,48319,48322],{"style":407},[238,48320],{"className":48321,"style":412},[411],[238,48323,48325],{"className":48324},[416,417,418,419],[238,48326,22083],{"className":48327},[664,419]," 的不动点唯一性立即得到。若 ",[238,48330,48332,48345],{"className":48331,"translate":242},[241],[238,48333,48335],{"className":48334},[246],[248,48336,48337],{"xmlns":250},[252,48338,48339,48343],{},[255,48340,48341],{},[258,48342,2297],{},[262,48344,2300],{"encoding":264},[238,48346,48348],{"className":48347,"ariaHidden":269},[268],[238,48349,48351,48354],{"className":48350},[273],[238,48352],{"className":48353,"style":489},[277],[238,48355,2297],{"className":48356,"style":2313},[282,283],[238,48358,48360,48373],{"className":48359,"translate":242},[241],[238,48361,48363],{"className":48362},[246],[248,48364,48365],{"xmlns":250},[252,48366,48367,48371],{},[255,48368,48369],{},[258,48370,4429],{},[262,48372,4429],{"encoding":264},[238,48374,48376],{"className":48375,"ariaHidden":269},[268],[238,48377,48379,48382],{"className":48378},[273],[238,48380],{"className":48381,"style":278},[277],[238,48383,4429],{"className":48384,"style":660},[282,283]," 贪婪，则 ",[238,48387,48389,48423],{"className":48388,"translate":242},[241],[238,48390,48392],{"className":48391},[246],[248,48393,48394],{"xmlns":250},[252,48395,48396,48420],{},[255,48397,48398,48404,48406,48408,48414,48416,48418],{},[337,48399,48400,48402],{},[258,48401,11971],{"mathvariant":816},[258,48403,2297],{},[258,48405,4429],{},[332,48407,992],{},[337,48409,48410,48412],{},[258,48411,11971],{"mathvariant":816},[332,48413,22083],{},[258,48415,4429],{},[332,48417,992],{},[258,48419,4429],{},[262,48421,48422],{"encoding":264},"\\mathcal{T}^\\pi V=\\mathcal{T}^*V=V",[238,48424,48426,48473,48520],{"className":48425,"ariaHidden":269},[268],[238,48427,48429,48432,48461,48464,48467,48470],{"className":48428},[273],[238,48430],{"className":48431,"style":278},[277],[238,48433,48435,48438],{"className":48434},[282],[238,48436,11971],{"className":48437,"style":11992},[282,840],[238,48439,48441],{"className":48440},[391],[238,48442,48444],{"className":48443},[395],[238,48445,48447],{"className":48446},[399],[238,48448,48450],{"className":48449,"style":6609},[403],[238,48451,48452,48455],{"style":407},[238,48453],{"className":48454,"style":412},[411],[238,48456,48458],{"className":48457},[416,417,418,419],[238,48459,2297],{"className":48460,"style":2313},[282,283,419],[238,48462,4429],{"className":48463,"style":660},[282,283],[238,48465],{"className":48466,"style":430},[429],[238,48468,992],{"className":48469},[434],[238,48471],{"className":48472,"style":430},[429],[238,48474,48476,48479,48508,48511,48514,48517],{"className":48475},[273],[238,48477],{"className":48478,"style":22096},[277],[238,48480,48482,48485],{"className":48481},[282],[238,48483,11971],{"className":48484,"style":11992},[282,840],[238,48486,48488],{"className":48487},[391],[238,48489,48491],{"className":48490},[395],[238,48492,48494],{"className":48493},[399],[238,48495,48497],{"className":48496,"style":22096},[403],[238,48498,48499,48502],{"style":407},[238,48500],{"className":48501,"style":412},[411],[238,48503,48505],{"className":48504},[416,417,418,419],[238,48506,22083],{"className":48507},[664,419],[238,48509,4429],{"className":48510,"style":660},[282,283],[238,48512],{"className":48513,"style":430},[429],[238,48515,992],{"className":48516},[434],[238,48518],{"className":48519,"style":430},[429],[238,48521,48523,48526],{"className":48522},[273],[238,48524],{"className":48525,"style":278},[277],[238,48527,4429],{"className":48528,"style":660},[282,283],[357,48530,21493],{"href":21492},[238,48532,48534,48555],{"className":48533,"translate":242},[241],[238,48535,48537],{"className":48536},[246],[248,48538,48539],{"xmlns":250},[252,48540,48541,48553],{},[255,48542,48543,48549,48551],{},[337,48544,48545,48547],{},[258,48546,4429],{},[258,48548,2297],{},[332,48550,992],{},[258,48552,4429],{},[262,48554,48179],{"encoding":264},[238,48556,48558,48602],{"className":48557,"ariaHidden":269},[268],[238,48559,48561,48564,48593,48596,48599],{"className":48560},[273],[238,48562],{"className":48563,"style":278},[277],[238,48565,48567,48570],{"className":48566},[282],[238,48568,4429],{"className":48569,"style":660},[282,283],[238,48571,48573],{"className":48572},[391],[238,48574,48576],{"className":48575},[395],[238,48577,48579],{"className":48578},[399],[238,48580,48582],{"className":48581,"style":6609},[403],[238,48583,48584,48587],{"style":407},[238,48585],{"className":48586,"style":412},[411],[238,48588,48590],{"className":48589},[416,417,418,419],[238,48591,2297],{"className":48592,"style":2313},[282,283,419],[238,48594],{"className":48595,"style":430},[429],[238,48597,992],{"className":48598},[434],[238,48600],{"className":48601,"style":430},[429],[238,48603,48605,48608],{"className":48604},[273],[238,48606],{"className":48607,"style":278},[277],[238,48609,4429],{"className":48610,"style":660},[282,283],[11,48612,48613],{},"这个推论是贝尔曼方程「充分性」的精确表述。它说明，只要能够找到一个有界函数满足最优方程，就不需要再与其他策略比较，因为它必然等于最优值函数。反过来，最优值函数也必然满足该方程。因此，在折扣有界 MDP 中，贝尔曼最优方程既必要又充分。",[225,48615,48617],{"id":48616},"_45-例带决策的路径","4.5 例：带决策的路径",[11,48619,48620,48621,2382,48705,48733],{},"考虑如下路径选择问题。状态为 ",[238,48622,48624,48656],{"className":48623,"translate":242},[241],[238,48625,48627],{"className":48626},[246],[248,48628,48629],{"xmlns":250},[252,48630,48631,48653],{},[255,48632,48633,48635,48637,48639,48641,48644,48646,48649,48651],{},[258,48634,300],{},[332,48636,354],{"separator":269},[258,48638,19971],{},[332,48640,354],{"separator":269},[258,48642,48643],{},"C",[332,48645,354],{"separator":269},[258,48647,48648],{},"D",[332,48650,354],{"separator":269},[258,48652,4452],{},[262,48654,48655],{"encoding":264},"A,B,C,D,E",[238,48657,48659],{"className":48658,"ariaHidden":269},[268],[238,48660,48662,48665,48668,48671,48674,48677,48680,48683,48687,48690,48693,48696,48699,48702],{"className":48661},[273],[238,48663],{"className":48664,"style":6651},[277],[238,48666,300],{"className":48667},[282,283],[238,48669,354],{"className":48670},[451],[238,48672],{"className":48673,"style":455},[429],[238,48675,19971],{"className":48676,"style":20011},[282,283],[238,48678,354],{"className":48679},[451],[238,48681],{"className":48682,"style":455},[429],[238,48684,48643],{"className":48685,"style":48686},[282,283],"margin-right:0.0715em;",[238,48688,354],{"className":48689},[451],[238,48691],{"className":48692,"style":455},[429],[238,48694,48648],{"className":48695,"style":637},[282,283],[238,48697,354],{"className":48698},[451],[238,48700],{"className":48701,"style":455},[429],[238,48703,4452],{"className":48704,"style":284},[282,283],[238,48706,48708,48721],{"className":48707,"translate":242},[241],[238,48709,48711],{"className":48710},[246],[248,48712,48713],{"xmlns":250},[252,48714,48715,48719],{},[255,48716,48717],{},[258,48718,48643],{},[262,48720,48643],{"encoding":264},[238,48722,48724],{"className":48723,"ariaHidden":269},[268],[238,48725,48727,48730],{"className":48726},[273],[238,48728],{"className":48729,"style":278},[277],[238,48731,48643],{"className":48732,"style":48686},[282,283]," 为终止状态。转移关系为",[238,48735,48737],{"className":48736,"translate":242},[945],[238,48738,48740,48830],{"className":48739,"translate":242},[241],[238,48741,48743],{"className":48742},[246],[248,48744,48745],{"xmlns":250,"display":954},[252,48746,48747,48827],{},[958,48748,48749,48793],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,48750,48751,48757],{},[966,48752,48753],{},[969,48754,48755],{"scriptlevel":740,"displaystyle":269},[255,48756],{},[966,48758,48759],{},[969,48760,48761],{"scriptlevel":740,"displaystyle":269},[255,48762,48763,48765,48767,48769,48771,48773,48775,48777,48779,48781,48783,48785,48787,48789,48791],{},[255,48764],{},[258,48766,300],{},[332,48768,617],{},[258,48770,19971],{},[332,48772,354],{"separator":269},[429,48774],{"width":7090},[258,48776,19971],{},[332,48778,617],{},[258,48780,48643],{},[332,48782,354],{"separator":269},[429,48784],{"width":7090},[258,48786,300],{},[332,48788,617],{},[258,48790,48648],{},[332,48792,354],{"separator":269},[963,48794,48795,48801],{},[966,48796,48797],{},[969,48798,48799],{"scriptlevel":740,"displaystyle":269},[255,48800],{},[966,48802,48803],{},[969,48804,48805],{"scriptlevel":740,"displaystyle":269},[255,48806,48807,48809,48811,48813,48815,48817,48819,48821,48823,48825],{},[255,48808],{},[258,48810,48648],{},[332,48812,617],{},[258,48814,4452],{},[332,48816,354],{"separator":269},[429,48818],{"width":7090},[258,48820,4452],{},[332,48822,617],{},[258,48824,48643],{},[258,48826,3102],{"mathvariant":344},[262,48828,48829],{"encoding":264},"\\begin{aligned}\n&A\\to B,\\quad B\\to C,\\quad A\\to D,\\\\\n&D\\to E,\\quad E\\to C.\n\\end{aligned}",[238,48831,48833],{"className":48832,"ariaHidden":269},[268],[238,48834,48836,48839],{"className":48835},[273],[238,48837],{"className":48838,"style":4573},[277],[238,48840,48842],{"className":48841},[282],[238,48843,48845,48884],{"className":48844},[958],[238,48846,48848],{"className":48847},[1117],[238,48849,48851,48876],{"className":48850},[395,848],[238,48852,48854,48873],{"className":48853},[399],[238,48855,48857,48865],{"className":48856,"style":4592},[403],[238,48858,48859,48862],{"style":28565},[238,48860],{"className":48861,"style":28569},[411],[238,48863],{"className":48864},[282],[238,48866,48867,48870],{"style":6033},[238,48868],{"className":48869,"style":28569},[411],[238,48871],{"className":48872},[282],[238,48874,872],{"className":48875},[871],[238,48877,48879],{"className":48878},[399],[238,48880,48882],{"className":48881,"style":4706},[403],[238,48883],{},[238,48885,48887],{"className":48886},[1422],[238,48888,48890,49029],{"className":48889},[395,848],[238,48891,48893,49026],{"className":48892},[399],[238,48894,48896,48973],{"className":48895,"style":4592},[403],[238,48897,48898,48901],{"style":4595},[238,48899],{"className":48900,"style":1134},[411],[238,48902,48904,48907,48910,48913,48916,48919,48922,48925,48928,48931,48934,48937,48940,48943,48946,48949,48952,48955,48958,48961,48964,48967,48970],{"className":48903},[282],[238,48905],{"className":48906},[282],[238,48908,300],{"className":48909},[282,283],[238,48911],{"className":48912,"style":430},[429],[238,48914,617],{"className":48915},[434],[238,48917],{"className":48918,"style":430},[429],[238,48920,19971],{"className":48921,"style":20011},[282,283],[238,48923,354],{"className":48924},[451],[238,48926],{"className":48927,"style":7439},[429],[238,48929],{"className":48930,"style":455},[429],[238,48932,19971],{"className":48933,"style":20011},[282,283],[238,48935],{"className":48936,"style":430},[429],[238,48938,617],{"className":48939},[434],[238,48941],{"className":48942,"style":430},[429],[238,48944,48643],{"className":48945,"style":48686},[282,283],[238,48947,354],{"className":48948},[451],[238,48950],{"className":48951,"style":7439},[429],[238,48953],{"className":48954,"style":455},[429],[238,48956,300],{"className":48957},[282,283],[238,48959],{"className":48960,"style":430},[429],[238,48962,617],{"className":48963},[434],[238,48965],{"className":48966,"style":430},[429],[238,48968,48648],{"className":48969,"style":637},[282,283],[238,48971,354],{"className":48972},[451],[238,48974,48975,48978],{"style":4643},[238,48976],{"className":48977,"style":1134},[411],[238,48979,48981,48984,48987,48990,48993,48996,48999,49002,49005,49008,49011,49014,49017,49020,49023],{"className":48980},[282],[238,48982],{"className":48983},[282],[238,48985,48648],{"className":48986,"style":637},[282,283],[238,48988],{"className":48989,"style":430},[429],[238,48991,617],{"className":48992},[434],[238,48994],{"className":48995,"style":430},[429],[238,48997,4452],{"className":48998,"style":284},[282,283],[238,49000,354],{"className":49001},[451],[238,49003],{"className":49004,"style":7439},[429],[238,49006],{"className":49007,"style":455},[429],[238,49009,4452],{"className":49010,"style":284},[282,283],[238,49012],{"className":49013,"style":430},[429],[238,49015,617],{"className":49016},[434],[238,49018],{"className":49019,"style":430},[429],[238,49021,48643],{"className":49022,"style":48686},[282,283],[238,49024,3102],{"className":49025},[282],[238,49027,872],{"className":49028},[871],[238,49030,49032],{"className":49031},[399],[238,49033,49035],{"className":49034,"style":4706},[403],[238,49036],{},[11,49038,49039,49040,49074,49075,49103,49104,49133,49134,49184,49185,1806],{},"每次转移获得奖励 ",[238,49041,49043,49059],{"className":49042,"translate":242},[241],[238,49044,49046],{"className":49045},[246],[248,49047,49048],{"xmlns":250},[252,49049,49050,49056],{},[255,49051,49052,49054],{},[332,49053,5577],{},[738,49055,745],{},[262,49057,49058],{"encoding":264},"-1",[238,49060,49062],{"className":49061,"ariaHidden":269},[268],[238,49063,49065,49068,49071],{"className":49064},[273],[238,49066],{"className":49067,"style":18081},[277],[238,49069,5577],{"className":49070},[282],[238,49072,745],{"className":49073},[282],"，进入终止状态 ",[238,49076,49078,49091],{"className":49077,"translate":242},[241],[238,49079,49081],{"className":49080},[246],[248,49082,49083],{"xmlns":250},[252,49084,49085,49089],{},[255,49086,49087],{},[258,49088,48643],{},[262,49090,48643],{"encoding":264},[238,49092,49094],{"className":49093,"ariaHidden":269},[268],[238,49095,49097,49100],{"className":49096},[273],[238,49098],{"className":49099,"style":278},[277],[238,49101,48643],{"className":49102,"style":48686},[282,283]," 时额外获得奖励 ",[238,49105,49107,49121],{"className":49106,"translate":242},[241],[238,49108,49110],{"className":49109},[246],[248,49111,49112],{"xmlns":250},[252,49113,49114,49119],{},[255,49115,49116],{},[738,49117,49118],{},"10",[262,49120,49118],{"encoding":264},[238,49122,49124],{"className":49123,"ariaHidden":269},[268],[238,49125,49127,49130],{"className":49126},[273],[238,49128],{"className":49129,"style":2672},[277],[238,49131,49118],{"className":49132},[282],"。设 ",[238,49135,49137,49154],{"className":49136,"translate":242},[241],[238,49138,49140],{"className":49139},[246],[248,49141,49142],{"xmlns":250},[252,49143,49144,49152],{},[255,49145,49146,49148,49150],{},[258,49147,730],{},[332,49149,992],{},[738,49151,17938],{},[262,49153,17941],{"encoding":264},[238,49155,49157,49175],{"className":49156,"ariaHidden":269},[268],[238,49158,49160,49163,49166,49169,49172],{"className":49159},[273],[238,49161],{"className":49162,"style":2853},[277],[238,49164,730],{"className":49165,"style":764},[282,283],[238,49167],{"className":49168,"style":430},[429],[238,49170,992],{"className":49171},[434],[238,49173],{"className":49174,"style":430},[429],[238,49176,49178,49181],{"className":49177},[273],[238,49179],{"className":49180,"style":2672},[277],[238,49182,17938],{"className":49183},[282],"，并令 ",[238,49186,49188,49212],{"className":49187,"translate":242},[241],[238,49189,49191],{"className":49190},[246],[248,49192,49193],{"xmlns":250},[252,49194,49195,49209],{},[255,49196,49197,49199,49201,49203,49205,49207],{},[258,49198,4429],{},[332,49200,335],{"stretchy":334},[258,49202,48643],{},[332,49204,360],{"stretchy":334},[332,49206,992],{},[738,49208,740],{},[262,49210,49211],{"encoding":264},"V(C)=0",[238,49213,49215,49242],{"className":49214,"ariaHidden":269},[268],[238,49216,49218,49221,49224,49227,49230,49233,49236,49239],{"className":49217},[273],[238,49219],{"className":49220,"style":444},[277],[238,49222,4429],{"className":49223,"style":660},[282,283],[238,49225,335],{"className":49226},[381],[238,49228,48643],{"className":49229,"style":48686},[282,283],[238,49231,360],{"className":49232},[462],[238,49234],{"className":49235,"style":430},[429],[238,49237,992],{"className":49238},[434],[238,49240],{"className":49241,"style":430},[429],[238,49243,49245,49248],{"className":49244},[273],[238,49246],{"className":49247,"style":2672},[277],[238,49249,740],{"className":49250},[282],[11,49252,49253,49254,1838,49282,49310,49311,49339],{},"在 ",[238,49255,49257,49270],{"className":49256,"translate":242},[241],[238,49258,49260],{"className":49259},[246],[248,49261,49262],{"xmlns":250},[252,49263,49264,49268],{},[255,49265,49266],{},[258,49267,19971],{},[262,49269,19971],{"encoding":264},[238,49271,49273],{"className":49272,"ariaHidden":269},[268],[238,49274,49276,49279],{"className":49275},[273],[238,49277],{"className":49278,"style":278},[277],[238,49280,19971],{"className":49281,"style":20011},[282,283],[238,49283,49285,49298],{"className":49284,"translate":242},[241],[238,49286,49288],{"className":49287},[246],[248,49289,49290],{"xmlns":250},[252,49291,49292,49296],{},[255,49293,49294],{},[258,49295,4452],{},[262,49297,4452],{"encoding":264},[238,49299,49301],{"className":49300,"ariaHidden":269},[268],[238,49302,49304,49307],{"className":49303},[273],[238,49305],{"className":49306,"style":278},[277],[238,49308,4452],{"className":49309,"style":284},[282,283],"，只有一条通往 ",[238,49312,49314,49327],{"className":49313,"translate":242},[241],[238,49315,49317],{"className":49316},[246],[248,49318,49319],{"xmlns":250},[252,49320,49321,49325],{},[255,49322,49323],{},[258,49324,48643],{},[262,49326,48643],{"encoding":264},[238,49328,49330],{"className":49329,"ariaHidden":269},[268],[238,49331,49333,49336],{"className":49332},[273],[238,49334],{"className":49335,"style":278},[277],[238,49337,48643],{"className":49338,"style":48686},[282,283]," 的动作，因此",[238,49341,49343],{"className":49342,"translate":242},[945],[238,49344,49346,49409],{"className":49345,"translate":242},[241],[238,49347,49349],{"className":49348},[246],[248,49350,49351],{"xmlns":250,"display":954},[252,49352,49353,49406],{},[255,49354,49355,49357,49359,49361,49363,49365,49367,49369,49371,49373,49375,49377,49379,49381,49383,49385,49387,49389,49391,49393,49395,49397,49399,49401,49403],{},[258,49356,4429],{},[332,49358,335],{"stretchy":334},[258,49360,19971],{},[332,49362,360],{"stretchy":334},[332,49364,992],{},[332,49366,5577],{},[738,49368,745],{},[332,49370,987],{},[738,49372,49118],{},[332,49374,987],{},[738,49376,17938],{},[258,49378,4429],{},[332,49380,335],{"stretchy":334},[258,49382,48643],{},[332,49384,360],{"stretchy":334},[332,49386,992],{},[738,49388,29072],{},[332,49390,354],{"separator":269},[429,49392],{"width":10450},[258,49394,4429],{},[332,49396,335],{"stretchy":334},[258,49398,4452],{},[332,49400,360],{"stretchy":334},[332,49402,992],{},[738,49404,49405],{},"9.",[262,49407,49408],{"encoding":264},"V(B)=-1+10+0.9V(C)=9,\n\\qquad\nV(E)=9.",[238,49410,49412,49439,49460,49478,49508,49547],{"className":49411,"ariaHidden":269},[268],[238,49413,49415,49418,49421,49424,49427,49430,49433,49436],{"className":49414},[273],[238,49416],{"className":49417,"style":444},[277],[238,49419,4429],{"className":49420,"style":660},[282,283],[238,49422,335],{"className":49423},[381],[238,49425,19971],{"className":49426,"style":20011},[282,283],[238,49428,360],{"className":49429},[462],[238,49431],{"className":49432,"style":430},[429],[238,49434,992],{"className":49435},[434],[238,49437],{"className":49438,"style":430},[429],[238,49440,49442,49445,49448,49451,49454,49457],{"className":49441},[273],[238,49443],{"className":49444,"style":18081},[277],[238,49446,5577],{"className":49447},[282],[238,49449,745],{"className":49450},[282],[238,49452],{"className":49453,"style":660},[429],[238,49455,987],{"className":49456},[664],[238,49458],{"className":49459,"style":660},[429],[238,49461,49463,49466,49469,49472,49475],{"className":49462},[273],[238,49464],{"className":49465,"style":18081},[277],[238,49467,49118],{"className":49468},[282],[238,49470],{"className":49471,"style":660},[429],[238,49473,987],{"className":49474},[664],[238,49476],{"className":49477,"style":660},[429],[238,49479,49481,49484,49487,49490,49493,49496,49499,49502,49505],{"className":49480},[273],[238,49482],{"className":49483,"style":444},[277],[238,49485,17938],{"className":49486},[282],[238,49488,4429],{"className":49489,"style":660},[282,283],[238,49491,335],{"className":49492},[381],[238,49494,48643],{"className":49495,"style":48686},[282,283],[238,49497,360],{"className":49498},[462],[238,49500],{"className":49501,"style":430},[429],[238,49503,992],{"className":49504},[434],[238,49506],{"className":49507,"style":430},[429],[238,49509,49511,49514,49517,49520,49523,49526,49529,49532,49535,49538,49541,49544],{"className":49510},[273],[238,49512],{"className":49513,"style":444},[277],[238,49515,29072],{"className":49516},[282],[238,49518,354],{"className":49519},[451],[238,49521],{"className":49522,"style":10986},[429],[238,49524],{"className":49525,"style":455},[429],[238,49527,4429],{"className":49528,"style":660},[282,283],[238,49530,335],{"className":49531},[381],[238,49533,4452],{"className":49534,"style":284},[282,283],[238,49536,360],{"className":49537},[462],[238,49539],{"className":49540,"style":430},[429],[238,49542,992],{"className":49543},[434],[238,49545],{"className":49546,"style":430},[429],[238,49548,49550,49553],{"className":49549},[273],[238,49551],{"className":49552,"style":2672},[277],[238,49554,49405],{"className":49555},[282],[11,49557,49253,49558,49586,49587,49615],{},[238,49559,49561,49574],{"className":49560,"translate":242},[241],[238,49562,49564],{"className":49563},[246],[248,49565,49566],{"xmlns":250},[252,49567,49568,49572],{},[255,49569,49570],{},[258,49571,48648],{},[262,49573,48648],{"encoding":264},[238,49575,49577],{"className":49576,"ariaHidden":269},[268],[238,49578,49580,49583],{"className":49579},[273],[238,49581],{"className":49582,"style":278},[277],[238,49584,48648],{"className":49585,"style":637},[282,283],"，唯一动作通向 ",[238,49588,49590,49603],{"className":49589,"translate":242},[241],[238,49591,49593],{"className":49592},[246],[248,49594,49595],{"xmlns":250},[252,49596,49597,49601],{},[255,49598,49599],{},[258,49600,4452],{},[262,49602,4452],{"encoding":264},[238,49604,49606],{"className":49605,"ariaHidden":269},[268],[238,49607,49609,49612],{"className":49608},[273],[238,49610],{"className":49611,"style":278},[277],[238,49613,4452],{"className":49614,"style":284},[282,283],"，故",[238,49617,49619],{"className":49618,"translate":242},[945],[238,49620,49622,49679],{"className":49621,"translate":242},[241],[238,49623,49625],{"className":49624},[246],[248,49626,49627],{"xmlns":250,"display":954},[252,49628,49629,49676],{},[255,49630,49631,49633,49635,49637,49639,49641,49643,49645,49647,49649,49651,49653,49655,49657,49659,49661,49663,49665,49667,49669,49671,49673],{},[258,49632,4429],{},[332,49634,335],{"stretchy":334},[258,49636,48648],{},[332,49638,360],{"stretchy":334},[332,49640,992],{},[332,49642,5577],{},[738,49644,745],{},[332,49646,987],{},[738,49648,17938],{},[258,49650,4429],{},[332,49652,335],{"stretchy":334},[258,49654,4452],{},[332,49656,360],{"stretchy":334},[332,49658,992],{},[332,49660,5577],{},[738,49662,745],{},[332,49664,987],{},[738,49666,17938],{},[332,49668,608],{},[738,49670,29072],{},[332,49672,992],{},[738,49674,49675],{},"7.1.",[262,49677,49678],{"encoding":264},"V(D)=-1+0.9V(E)=-1+0.9\\times9=7.1.",[238,49680,49682,49709,49730,49760,49781,49799,49817],{"className":49681,"ariaHidden":269},[268],[238,49683,49685,49688,49691,49694,49697,49700,49703,49706],{"className":49684},[273],[238,49686],{"className":49687,"style":444},[277],[238,49689,4429],{"className":49690,"style":660},[282,283],[238,49692,335],{"className":49693},[381],[238,49695,48648],{"className":49696,"style":637},[282,283],[238,49698,360],{"className":49699},[462],[238,49701],{"className":49702,"style":430},[429],[238,49704,992],{"className":49705},[434],[238,49707],{"className":49708,"style":430},[429],[238,49710,49712,49715,49718,49721,49724,49727],{"className":49711},[273],[238,49713],{"className":49714,"style":18081},[277],[238,49716,5577],{"className":49717},[282],[238,49719,745],{"className":49720},[282],[238,49722],{"className":49723,"style":660},[429],[238,49725,987],{"className":49726},[664],[238,49728],{"className":49729,"style":660},[429],[238,49731,49733,49736,49739,49742,49745,49748,49751,49754,49757],{"className":49732},[273],[238,49734],{"className":49735,"style":444},[277],[238,49737,17938],{"className":49738},[282],[238,49740,4429],{"className":49741,"style":660},[282,283],[238,49743,335],{"className":49744},[381],[238,49746,4452],{"className":49747,"style":284},[282,283],[238,49749,360],{"className":49750},[462],[238,49752],{"className":49753,"style":430},[429],[238,49755,992],{"className":49756},[434],[238,49758],{"className":49759,"style":430},[429],[238,49761,49763,49766,49769,49772,49775,49778],{"className":49762},[273],[238,49764],{"className":49765,"style":18081},[277],[238,49767,5577],{"className":49768},[282],[238,49770,745],{"className":49771},[282],[238,49773],{"className":49774,"style":660},[429],[238,49776,987],{"className":49777},[664],[238,49779],{"className":49780,"style":660},[429],[238,49782,49784,49787,49790,49793,49796],{"className":49783},[273],[238,49785],{"className":49786,"style":18081},[277],[238,49788,17938],{"className":49789},[282],[238,49791],{"className":49792,"style":660},[429],[238,49794,608],{"className":49795},[664],[238,49797],{"className":49798,"style":660},[429],[238,49800,49802,49805,49808,49811,49814],{"className":49801},[273],[238,49803],{"className":49804,"style":2672},[277],[238,49806,29072],{"className":49807},[282],[238,49809],{"className":49810,"style":430},[429],[238,49812,992],{"className":49813},[434],[238,49815],{"className":49816,"style":430},[429],[238,49818,49820,49823],{"className":49819},[273],[238,49821],{"className":49822,"style":2672},[277],[238,49824,49675],{"className":49825},[282],[11,49827,49253,49828,49856,49857,1838,49923,49989,49990,50018],{},[238,49829,49831,49844],{"className":49830,"translate":242},[241],[238,49832,49834],{"className":49833},[246],[248,49835,49836],{"xmlns":250},[252,49837,49838,49842],{},[255,49839,49840],{},[258,49841,300],{},[262,49843,300],{"encoding":264},[238,49845,49847],{"className":49846,"ariaHidden":269},[268],[238,49848,49850,49853],{"className":49849},[273],[238,49851],{"className":49852,"style":278},[277],[238,49854,300],{"className":49855},[282,283],"，有两条动作。记 ",[238,49858,49860,49884],{"className":49859,"translate":242},[241],[238,49861,49863],{"className":49862},[246],[248,49864,49865],{"xmlns":250},[252,49866,49867,49881],{},[255,49868,49869,49871,49873,49875,49877,49879],{},[258,49870,4494],{},[332,49872,335],{"stretchy":334},[258,49874,300],{},[332,49876,617],{},[258,49878,19971],{},[332,49880,360],{"stretchy":334},[262,49882,49883],{"encoding":264},"Q(A\\to B)",[238,49885,49887,49911],{"className":49886,"ariaHidden":269},[268],[238,49888,49890,49893,49896,49899,49902,49905,49908],{"className":49889},[273],[238,49891],{"className":49892,"style":444},[277],[238,49894,4494],{"className":49895},[282,283],[238,49897,335],{"className":49898},[381],[238,49900,300],{"className":49901},[282,283],[238,49903],{"className":49904,"style":430},[429],[238,49906,617],{"className":49907},[434],[238,49909],{"className":49910,"style":430},[429],[238,49912,49914,49917,49920],{"className":49913},[273],[238,49915],{"className":49916,"style":444},[277],[238,49918,19971],{"className":49919,"style":20011},[282,283],[238,49921,360],{"className":49922},[462],[238,49924,49926,49950],{"className":49925,"translate":242},[241],[238,49927,49929],{"className":49928},[246],[248,49930,49931],{"xmlns":250},[252,49932,49933,49947],{},[255,49934,49935,49937,49939,49941,49943,49945],{},[258,49936,4494],{},[332,49938,335],{"stretchy":334},[258,49940,300],{},[332,49942,617],{},[258,49944,48648],{},[332,49946,360],{"stretchy":334},[262,49948,49949],{"encoding":264},"Q(A\\to D)",[238,49951,49953,49977],{"className":49952,"ariaHidden":269},[268],[238,49954,49956,49959,49962,49965,49968,49971,49974],{"className":49955},[273],[238,49957],{"className":49958,"style":444},[277],[238,49960,4494],{"className":49961},[282,283],[238,49963,335],{"className":49964},[381],[238,49966,300],{"className":49967},[282,283],[238,49969],{"className":49970,"style":430},[429],[238,49972,617],{"className":49973},[434],[238,49975],{"className":49976,"style":430},[429],[238,49978,49980,49983,49986],{"className":49979},[273],[238,49981],{"className":49982,"style":444},[277],[238,49984,48648],{"className":49985,"style":637},[282,283],[238,49987,360],{"className":49988},[462]," 分别为在 ",[238,49991,49993,50006],{"className":49992,"translate":242},[241],[238,49994,49996],{"className":49995},[246],[248,49997,49998],{"xmlns":250},[252,49999,50000,50004],{},[255,50001,50002],{},[258,50003,300],{},[262,50005,300],{"encoding":264},[238,50007,50009],{"className":50008,"ariaHidden":269},[268],[238,50010,50012,50015],{"className":50011},[273],[238,50013],{"className":50014,"style":278},[277],[238,50016,300],{"className":50017},[282,283]," 选择这两条动作的动作值，则",[238,50020,50022],{"className":50021,"translate":242},[945],[238,50023,50025,50088],{"className":50024,"translate":242},[241],[238,50026,50028],{"className":50027},[246],[248,50029,50030],{"xmlns":250,"display":954},[252,50031,50032,50085],{},[255,50033,50034,50036,50038,50040,50042,50044,50046,50048,50050,50052,50054,50056,50058,50060,50062,50064,50066,50068,50070,50072,50074,50076,50078,50080,50083],{},[258,50035,4494],{},[332,50037,335],{"stretchy":334},[258,50039,300],{},[332,50041,617],{},[258,50043,19971],{},[332,50045,360],{"stretchy":334},[332,50047,992],{},[332,50049,5577],{},[738,50051,745],{},[332,50053,987],{},[738,50055,17938],{},[258,50057,4429],{},[332,50059,335],{"stretchy":334},[258,50061,19971],{},[332,50063,360],{"stretchy":334},[332,50065,992],{},[332,50067,5577],{},[738,50069,745],{},[332,50071,987],{},[738,50073,17938],{},[332,50075,608],{},[738,50077,29072],{},[332,50079,992],{},[738,50081,50082],{},"7.1",[332,50084,354],{"separator":269},[262,50086,50087],{"encoding":264},"Q(A\\to B)=-1+0.9V(B)=-1+0.9\\times9=7.1,",[238,50089,50091,50115,50136,50157,50187,50208,50226,50244],{"className":50090,"ariaHidden":269},[268],[238,50092,50094,50097,50100,50103,50106,50109,50112],{"className":50093},[273],[238,50095],{"className":50096,"style":444},[277],[238,50098,4494],{"className":50099},[282,283],[238,50101,335],{"className":50102},[381],[238,50104,300],{"className":50105},[282,283],[238,50107],{"className":50108,"style":430},[429],[238,50110,617],{"className":50111},[434],[238,50113],{"className":50114,"style":430},[429],[238,50116,50118,50121,50124,50127,50130,50133],{"className":50117},[273],[238,50119],{"className":50120,"style":444},[277],[238,50122,19971],{"className":50123,"style":20011},[282,283],[238,50125,360],{"className":50126},[462],[238,50128],{"className":50129,"style":430},[429],[238,50131,992],{"className":50132},[434],[238,50134],{"className":50135,"style":430},[429],[238,50137,50139,50142,50145,50148,50151,50154],{"className":50138},[273],[238,50140],{"className":50141,"style":18081},[277],[238,50143,5577],{"className":50144},[282],[238,50146,745],{"className":50147},[282],[238,50149],{"className":50150,"style":660},[429],[238,50152,987],{"className":50153},[664],[238,50155],{"className":50156,"style":660},[429],[238,50158,50160,50163,50166,50169,50172,50175,50178,50181,50184],{"className":50159},[273],[238,50161],{"className":50162,"style":444},[277],[238,50164,17938],{"className":50165},[282],[238,50167,4429],{"className":50168,"style":660},[282,283],[238,50170,335],{"className":50171},[381],[238,50173,19971],{"className":50174,"style":20011},[282,283],[238,50176,360],{"className":50177},[462],[238,50179],{"className":50180,"style":430},[429],[238,50182,992],{"className":50183},[434],[238,50185],{"className":50186,"style":430},[429],[238,50188,50190,50193,50196,50199,50202,50205],{"className":50189},[273],[238,50191],{"className":50192,"style":18081},[277],[238,50194,5577],{"className":50195},[282],[238,50197,745],{"className":50198},[282],[238,50200],{"className":50201,"style":660},[429],[238,50203,987],{"className":50204},[664],[238,50206],{"className":50207,"style":660},[429],[238,50209,50211,50214,50217,50220,50223],{"className":50210},[273],[238,50212],{"className":50213,"style":18081},[277],[238,50215,17938],{"className":50216},[282],[238,50218],{"className":50219,"style":660},[429],[238,50221,608],{"className":50222},[664],[238,50224],{"className":50225,"style":660},[429],[238,50227,50229,50232,50235,50238,50241],{"className":50228},[273],[238,50230],{"className":50231,"style":2672},[277],[238,50233,29072],{"className":50234},[282],[238,50236],{"className":50237,"style":430},[429],[238,50239,992],{"className":50240},[434],[238,50242],{"className":50243,"style":430},[429],[238,50245,50247,50250,50253],{"className":50246},[273],[238,50248],{"className":50249,"style":17777},[277],[238,50251,50082],{"className":50252},[282],[238,50254,354],{"className":50255},[451],[238,50257,50259],{"className":50258,"translate":242},[945],[238,50260,50262,50323],{"className":50261,"translate":242},[241],[238,50263,50265],{"className":50264},[246],[248,50266,50267],{"xmlns":250,"display":954},[252,50268,50269,50320],{},[255,50270,50271,50273,50275,50277,50279,50281,50283,50285,50287,50289,50291,50293,50295,50297,50299,50301,50303,50305,50307,50309,50311,50313,50315,50317],{},[258,50272,4494],{},[332,50274,335],{"stretchy":334},[258,50276,300],{},[332,50278,617],{},[258,50280,48648],{},[332,50282,360],{"stretchy":334},[332,50284,992],{},[332,50286,5577],{},[738,50288,745],{},[332,50290,987],{},[738,50292,17938],{},[258,50294,4429],{},[332,50296,335],{"stretchy":334},[258,50298,48648],{},[332,50300,360],{"stretchy":334},[332,50302,992],{},[332,50304,5577],{},[738,50306,745],{},[332,50308,987],{},[738,50310,17938],{},[332,50312,608],{},[738,50314,50082],{},[332,50316,992],{},[738,50318,50319],{},"5.39.",[262,50321,50322],{"encoding":264},"Q(A\\to D)=-1+0.9V(D)=-1+0.9\\times7.1=5.39.",[238,50324,50326,50350,50371,50392,50422,50443,50461,50479],{"className":50325,"ariaHidden":269},[268],[238,50327,50329,50332,50335,50338,50341,50344,50347],{"className":50328},[273],[238,50330],{"className":50331,"style":444},[277],[238,50333,4494],{"className":50334},[282,283],[238,50336,335],{"className":50337},[381],[238,50339,300],{"className":50340},[282,283],[238,50342],{"className":50343,"style":430},[429],[238,50345,617],{"className":50346},[434],[238,50348],{"className":50349,"style":430},[429],[238,50351,50353,50356,50359,50362,50365,50368],{"className":50352},[273],[238,50354],{"className":50355,"style":444},[277],[238,50357,48648],{"className":50358,"style":637},[282,283],[238,50360,360],{"className":50361},[462],[238,50363],{"className":50364,"style":430},[429],[238,50366,992],{"className":50367},[434],[238,50369],{"className":50370,"style":430},[429],[238,50372,50374,50377,50380,50383,50386,50389],{"className":50373},[273],[238,50375],{"className":50376,"style":18081},[277],[238,50378,5577],{"className":50379},[282],[238,50381,745],{"className":50382},[282],[238,50384],{"className":50385,"style":660},[429],[238,50387,987],{"className":50388},[664],[238,50390],{"className":50391,"style":660},[429],[238,50393,50395,50398,50401,50404,50407,50410,50413,50416,50419],{"className":50394},[273],[238,50396],{"className":50397,"style":444},[277],[238,50399,17938],{"className":50400},[282],[238,50402,4429],{"className":50403,"style":660},[282,283],[238,50405,335],{"className":50406},[381],[238,50408,48648],{"className":50409,"style":637},[282,283],[238,50411,360],{"className":50412},[462],[238,50414],{"className":50415,"style":430},[429],[238,50417,992],{"className":50418},[434],[238,50420],{"className":50421,"style":430},[429],[238,50423,50425,50428,50431,50434,50437,50440],{"className":50424},[273],[238,50426],{"className":50427,"style":18081},[277],[238,50429,5577],{"className":50430},[282],[238,50432,745],{"className":50433},[282],[238,50435],{"className":50436,"style":660},[429],[238,50438,987],{"className":50439},[664],[238,50441],{"className":50442,"style":660},[429],[238,50444,50446,50449,50452,50455,50458],{"className":50445},[273],[238,50447],{"className":50448,"style":18081},[277],[238,50450,17938],{"className":50451},[282],[238,50453],{"className":50454,"style":660},[429],[238,50456,608],{"className":50457},[664],[238,50459],{"className":50460,"style":660},[429],[238,50462,50464,50467,50470,50473,50476],{"className":50463},[273],[238,50465],{"className":50466,"style":2672},[277],[238,50468,50082],{"className":50469},[282],[238,50471],{"className":50472,"style":430},[429],[238,50474,992],{"className":50475},[434],[238,50477],{"className":50478,"style":430},[429],[238,50480,50482,50485],{"className":50481},[273],[238,50483],{"className":50484,"style":2672},[277],[238,50486,50319],{"className":50487},[282],[11,50489,18309],{},[238,50491,50493],{"className":50492,"translate":242},[945],[238,50494,50496,50539],{"className":50495,"translate":242},[241],[238,50497,50499],{"className":50498},[246],[248,50500,50501],{"xmlns":250,"display":954},[252,50502,50503,50536],{},[255,50504,50505,50507,50509,50511,50513,50515,50517,50519,50521,50523,50525,50528,50530,50532,50534],{},[258,50506,4429],{},[332,50508,335],{"stretchy":334},[258,50510,300],{},[332,50512,360],{"stretchy":334},[332,50514,992],{},[258,50516,1886],{},[332,50518,1889],{},[332,50520,13150],{"stretchy":334},[738,50522,50082],{},[332,50524,354],{"separator":269},[738,50526,50527],{},"5.39",[332,50529,13164],{"stretchy":334},[332,50531,992],{},[738,50533,50082],{},[332,50535,354],{"separator":269},[262,50537,50538],{"encoding":264},"V(A)=\\max\\{7.1,5.39\\}=7.1,",[238,50540,50542,50569,50605],{"className":50541,"ariaHidden":269},[268],[238,50543,50545,50548,50551,50554,50557,50560,50563,50566],{"className":50544},[273],[238,50546],{"className":50547,"style":444},[277],[238,50549,4429],{"className":50550,"style":660},[282,283],[238,50552,335],{"className":50553},[381],[238,50555,300],{"className":50556},[282,283],[238,50558,360],{"className":50559},[462],[238,50561],{"className":50562,"style":430},[429],[238,50564,992],{"className":50565},[434],[238,50567],{"className":50568,"style":430},[429],[238,50570,50572,50575,50578,50581,50584,50587,50590,50593,50596,50599,50602],{"className":50571},[273],[238,50573],{"className":50574,"style":444},[277],[238,50576,1886],{"className":50577},[1943],[238,50579,13150],{"className":50580},[381],[238,50582,50082],{"className":50583},[282],[238,50585,354],{"className":50586},[451],[238,50588],{"className":50589,"style":455},[429],[238,50591,50527],{"className":50592},[282],[238,50594,13164],{"className":50595},[462],[238,50597],{"className":50598,"style":430},[429],[238,50600,992],{"className":50601},[434],[238,50603],{"className":50604,"style":430},[429],[238,50606,50608,50611,50614],{"className":50607},[273],[238,50609],{"className":50610,"style":17777},[277],[238,50612,50082],{"className":50613},[282],[238,50615,354],{"className":50616},[451],[11,50618,50619,50620,50648,50649,50722,50723,50818,50819,50852],{},"最优动作是走向 ",[238,50621,50623,50636],{"className":50622,"translate":242},[241],[238,50624,50626],{"className":50625},[246],[248,50627,50628],{"xmlns":250},[252,50629,50630,50634],{},[255,50631,50632],{},[258,50633,19971],{},[262,50635,19971],{"encoding":264},[238,50637,50639],{"className":50638,"ariaHidden":269},[268],[238,50640,50642,50645],{"className":50641},[273],[238,50643],{"className":50644,"style":278},[277],[238,50646,19971],{"className":50647,"style":20011},[282,283],"。路线 ",[238,50650,50652,50674],{"className":50651,"translate":242},[241],[238,50653,50655],{"className":50654},[246],[248,50656,50657],{"xmlns":250},[252,50658,50659,50671],{},[255,50660,50661,50663,50665,50667,50669],{},[258,50662,300],{},[332,50664,617],{},[258,50666,19971],{},[332,50668,617],{},[258,50670,48643],{},[262,50672,50673],{"encoding":264},"A\\to B\\to C",[238,50675,50677,50695,50713],{"className":50676,"ariaHidden":269},[268],[238,50678,50680,50683,50686,50689,50692],{"className":50679},[273],[238,50681],{"className":50682,"style":278},[277],[238,50684,300],{"className":50685},[282,283],[238,50687],{"className":50688,"style":430},[429],[238,50690,617],{"className":50691},[434],[238,50693],{"className":50694,"style":430},[429],[238,50696,50698,50701,50704,50707,50710],{"className":50697},[273],[238,50699],{"className":50700,"style":278},[277],[238,50702,19971],{"className":50703,"style":20011},[282,283],[238,50705],{"className":50706,"style":430},[429],[238,50708,617],{"className":50709},[434],[238,50711],{"className":50712,"style":430},[429],[238,50714,50716,50719],{"className":50715},[273],[238,50717],{"className":50718,"style":278},[277],[238,50720,48643],{"className":50721,"style":48686},[282,283]," 虽然与 ",[238,50724,50726,50752],{"className":50725,"translate":242},[241],[238,50727,50729],{"className":50728},[246],[248,50730,50731],{"xmlns":250},[252,50732,50733,50749],{},[255,50734,50735,50737,50739,50741,50743,50745,50747],{},[258,50736,300],{},[332,50738,617],{},[258,50740,48648],{},[332,50742,617],{},[258,50744,4452],{},[332,50746,617],{},[258,50748,48643],{},[262,50750,50751],{"encoding":264},"A\\to D\\to E\\to C",[238,50753,50755,50773,50791,50809],{"className":50754,"ariaHidden":269},[268],[238,50756,50758,50761,50764,50767,50770],{"className":50757},[273],[238,50759],{"className":50760,"style":278},[277],[238,50762,300],{"className":50763},[282,283],[238,50765],{"className":50766,"style":430},[429],[238,50768,617],{"className":50769},[434],[238,50771],{"className":50772,"style":430},[429],[238,50774,50776,50779,50782,50785,50788],{"className":50775},[273],[238,50777],{"className":50778,"style":278},[277],[238,50780,48648],{"className":50781,"style":637},[282,283],[238,50783],{"className":50784,"style":430},[429],[238,50786,617],{"className":50787},[434],[238,50789],{"className":50790,"style":430},[429],[238,50792,50794,50797,50800,50803,50806],{"className":50793},[273],[238,50795],{"className":50796,"style":278},[277],[238,50798,4452],{"className":50799,"style":284},[282,283],[238,50801],{"className":50802,"style":430},[429],[238,50804,617],{"className":50805},[434],[238,50807],{"className":50808,"style":430},[429],[238,50810,50812,50815],{"className":50811},[273],[238,50813],{"className":50814,"style":278},[277],[238,50816,48643],{"className":50817,"style":48686},[282,283]," 具有相同的终止奖励，但后者多走一步，多承担一次 ",[238,50820,50822,50837],{"className":50821,"translate":242},[241],[238,50823,50825],{"className":50824},[246],[248,50826,50827],{"xmlns":250},[252,50828,50829,50835],{},[255,50830,50831,50833],{},[332,50832,5577],{},[738,50834,745],{},[262,50836,49058],{"encoding":264},[238,50838,50840],{"className":50839,"ariaHidden":269},[268],[238,50841,50843,50846,50849],{"className":50842},[273],[238,50844],{"className":50845,"style":18081},[277],[238,50847,5577],{"className":50848},[282],[238,50850,745],{"className":50851},[282]," 的即时奖励，并通过折扣因子进一步缩小了终止奖励的现值，因此其价值更低。",[11,50854,50855,50856,50884,50885,50913,50914,50942,50943,50994,50995,51023,51024,50942,51052,51125,51126,51154],{},"这个例子同时展示了最优子结构的两层含义。第一，从 ",[238,50857,50859,50872],{"className":50858,"translate":242},[241],[238,50860,50862],{"className":50861},[246],[248,50863,50864],{"xmlns":250},[252,50865,50866,50870],{},[255,50867,50868],{},[258,50869,300],{},[262,50871,300],{"encoding":264},[238,50873,50875],{"className":50874,"ariaHidden":269},[268],[238,50876,50878,50881],{"className":50877},[273],[238,50879],{"className":50880,"style":278},[277],[238,50882,300],{"className":50883},[282,283]," 出发的最优路线选择了 ",[238,50886,50888,50901],{"className":50887,"translate":242},[241],[238,50889,50891],{"className":50890},[246],[248,50892,50893],{"xmlns":250},[252,50894,50895,50899],{},[255,50896,50897],{},[258,50898,19971],{},[262,50900,19971],{"encoding":264},[238,50902,50904],{"className":50903,"ariaHidden":269},[268],[238,50905,50907,50910],{"className":50906},[273],[238,50908],{"className":50909,"style":278},[277],[238,50911,19971],{"className":50912,"style":20011},[282,283],"，而从 ",[238,50915,50917,50930],{"className":50916,"translate":242},[241],[238,50918,50920],{"className":50919},[246],[248,50921,50922],{"xmlns":250},[252,50923,50924,50928],{},[255,50925,50926],{},[258,50927,19971],{},[262,50929,19971],{"encoding":264},[238,50931,50933],{"className":50932,"ariaHidden":269},[268],[238,50934,50936,50939],{"className":50935},[273],[238,50937],{"className":50938,"style":278},[277],[238,50940,19971],{"className":50941,"style":20011},[282,283]," 出发的剩余路线 ",[238,50944,50946,50964],{"className":50945,"translate":242},[241],[238,50947,50949],{"className":50948},[246],[248,50950,50951],{"xmlns":250},[252,50952,50953,50961],{},[255,50954,50955,50957,50959],{},[258,50956,19971],{},[332,50958,617],{},[258,50960,48643],{},[262,50962,50963],{"encoding":264},"B\\to C",[238,50965,50967,50985],{"className":50966,"ariaHidden":269},[268],[238,50968,50970,50973,50976,50979,50982],{"className":50969},[273],[238,50971],{"className":50972,"style":278},[277],[238,50974,19971],{"className":50975,"style":20011},[282,283],[238,50977],{"className":50978,"style":430},[429],[238,50980,617],{"className":50981},[434],[238,50983],{"className":50984,"style":430},[429],[238,50986,50988,50991],{"className":50987},[273],[238,50989],{"className":50990,"style":278},[277],[238,50992,48643],{"className":50993,"style":48686},[282,283]," 本身是 ",[238,50996,50998,51011],{"className":50997,"translate":242},[241],[238,50999,51001],{"className":51000},[246],[248,51002,51003],{"xmlns":250},[252,51004,51005,51009],{},[255,51006,51007],{},[258,51008,19971],{},[262,51010,19971],{"encoding":264},[238,51012,51014],{"className":51013,"ariaHidden":269},[268],[238,51015,51017,51020],{"className":51016},[273],[238,51018],{"className":51019,"style":278},[277],[238,51021,19971],{"className":51022,"style":20011},[282,283]," 处子问题的最优解。第二，从 ",[238,51025,51027,51040],{"className":51026,"translate":242},[241],[238,51028,51030],{"className":51029},[246],[248,51031,51032],{"xmlns":250},[252,51033,51034,51038],{},[255,51035,51036],{},[258,51037,48648],{},[262,51039,48648],{"encoding":264},[238,51041,51043],{"className":51042,"ariaHidden":269},[268],[238,51044,51046,51049],{"className":51045},[273],[238,51047],{"className":51048,"style":278},[277],[238,51050,48648],{"className":51051,"style":637},[282,283],[238,51053,51055,51077],{"className":51054,"translate":242},[241],[238,51056,51058],{"className":51057},[246],[248,51059,51060],{"xmlns":250},[252,51061,51062,51074],{},[255,51063,51064,51066,51068,51070,51072],{},[258,51065,48648],{},[332,51067,617],{},[258,51069,4452],{},[332,51071,617],{},[258,51073,48643],{},[262,51075,51076],{"encoding":264},"D\\to E\\to C",[238,51078,51080,51098,51116],{"className":51079,"ariaHidden":269},[268],[238,51081,51083,51086,51089,51092,51095],{"className":51082},[273],[238,51084],{"className":51085,"style":278},[277],[238,51087,48648],{"className":51088,"style":637},[282,283],[238,51090],{"className":51091,"style":430},[429],[238,51093,617],{"className":51094},[434],[238,51096],{"className":51097,"style":430},[429],[238,51099,51101,51104,51107,51110,51113],{"className":51100},[273],[238,51102],{"className":51103,"style":278},[277],[238,51105,4452],{"className":51106,"style":284},[282,283],[238,51108],{"className":51109,"style":430},[429],[238,51111,617],{"className":51112},[434],[238,51114],{"className":51115,"style":430},[429],[238,51117,51119,51122],{"className":51118},[273],[238,51120],{"className":51121,"style":278},[277],[238,51123,48643],{"className":51124,"style":48686},[282,283]," 也是 ",[238,51127,51129,51142],{"className":51128,"translate":242},[241],[238,51130,51132],{"className":51131},[246],[248,51133,51134],{"xmlns":250},[252,51135,51136,51140],{},[255,51137,51138],{},[258,51139,48648],{},[262,51141,48648],{"encoding":264},[238,51143,51145],{"className":51144,"ariaHidden":269},[268],[238,51146,51148,51151],{"className":51147},[273],[238,51149],{"className":51150,"style":278},[277],[238,51152,48648],{"className":51153,"style":637},[282,283]," 处子问题的最优解。整体最优并不是把各段的局部最优任意拼接，而是沿着实际到达的后继状态，取该后继状态的最优续策略。",[11,51156,51157,51158,51224],{},"若改用 ",[238,51159,51161,51185],{"className":51160,"translate":242},[241],[238,51162,51164],{"className":51163},[246],[248,51165,51166],{"xmlns":250},[252,51167,51168,51182],{},[255,51169,51170,51172,51174,51176,51178,51180],{},[258,51171,4429],{},[332,51173,335],{"stretchy":334},[258,51175,48643],{},[332,51177,360],{"stretchy":334},[332,51179,992],{},[738,51181,49118],{},[262,51183,51184],{"encoding":264},"V(C)=10",[238,51186,51188,51215],{"className":51187,"ariaHidden":269},[268],[238,51189,51191,51194,51197,51200,51203,51206,51209,51212],{"className":51190},[273],[238,51192],{"className":51193,"style":444},[277],[238,51195,4429],{"className":51196,"style":660},[282,283],[238,51198,335],{"className":51199},[381],[238,51201,48643],{"className":51202,"style":48686},[282,283],[238,51204,360],{"className":51205},[462],[238,51207],{"className":51208,"style":430},[429],[238,51210,992],{"className":51211},[434],[238,51213],{"className":51214,"style":430},[429],[238,51216,51218,51221],{"className":51217},[273],[238,51219],{"className":51220,"style":2672},[277],[238,51222,49118],{"className":51223},[282]," 的约定，即把终止奖励直接记在终止状态上，而不是记在进入终止状态的转移上，则上述数值会改变，但最优动作不变。两种约定的差别只在于奖励被记在转移上还是被记在状态上，贝尔曼方程的形式必须与所采用的约定保持一致。",[220,51226,51228],{"id":51227},"五收敛性与误差界","五、收敛性与误差界",[225,51230,51232],{"id":51231},"_51-不动点迭代","5.1 不动点迭代",[11,51234,51235,51236,51343],{},"压缩映射不仅证明了解的存在唯一性，还给出了构造解的方法。任取初始函数 ",[238,51237,51239,51267],{"className":51238,"translate":242},[241],[238,51240,51242],{"className":51241},[246],[248,51243,51244],{"xmlns":250},[252,51245,51246,51264],{},[255,51247,51248,51254,51256,51258,51260,51262],{},[812,51249,51250,51252],{},[258,51251,4429],{},[738,51253,740],{},[332,51255,733],{},[258,51257,19971],{},[332,51259,335],{"stretchy":334},[258,51261,260],{},[332,51263,360],{"stretchy":334},[262,51265,51266],{"encoding":264},"V_0\\in B(S)",[238,51268,51270,51325],{"className":51269,"ariaHidden":269},[268],[238,51271,51273,51276,51316,51319,51322],{"className":51272},[273],[238,51274],{"className":51275,"style":833},[277],[238,51277,51279,51282],{"className":51278},[282],[238,51280,4429],{"className":51281,"style":660},[282,283],[238,51283,51285],{"className":51284},[391],[238,51286,51288,51308],{"className":51287},[395,848],[238,51289,51291,51305],{"className":51290},[399],[238,51292,51294],{"className":51293,"style":1162},[403],[238,51295,51296,51299],{"style":13806},[238,51297],{"className":51298,"style":412},[411],[238,51300,51302],{"className":51301},[416,417,418,419],[238,51303,740],{"className":51304},[282,419],[238,51306,872],{"className":51307},[871],[238,51309,51311],{"className":51310},[399],[238,51312,51314],{"className":51313,"style":879},[403],[238,51315],{},[238,51317],{"className":51318,"style":430},[429],[238,51320,733],{"className":51321},[434],[238,51323],{"className":51324,"style":430},[429],[238,51326,51328,51331,51334,51337,51340],{"className":51327},[273],[238,51329],{"className":51330,"style":444},[277],[238,51332,19971],{"className":51333,"style":20011},[282,283],[238,51335,335],{"className":51336},[381],[238,51338,260],{"className":51339,"style":284},[282,283],[238,51341,360],{"className":51342},[462],"，定义迭代",[238,51345,51347],{"className":51346,"translate":242},[945],[238,51348,51350,51406],{"className":51349,"translate":242},[241],[238,51351,51353],{"className":51352},[246],[248,51354,51355],{"xmlns":250,"display":954},[252,51356,51357,51403],{},[255,51358,51359,51371,51373,51375,51381,51383,51385,51387,51389,51391,51393,51395,51397,51399,51401],{},[812,51360,51361,51363],{},[258,51362,4429],{},[255,51364,51365,51367,51369],{},[258,51366,3395],{},[332,51368,987],{},[738,51370,745],{},[332,51372,992],{},[258,51374,11971],{"mathvariant":816},[812,51376,51377,51379],{},[258,51378,4429],{},[258,51380,3395],{},[332,51382,354],{"separator":269},[429,51384],{"width":10450},[258,51386,3395],{},[332,51388,992],{},[738,51390,740],{},[332,51392,354],{"separator":269},[738,51394,745],{},[332,51396,354],{"separator":269},[738,51398,17558],{},[332,51400,354],{"separator":269},[332,51402,2840],{},[262,51404,51405],{"encoding":264},"V_{k+1}=\\mathcal{T}V_k,\n\\qquad k=0,1,2,\\dots",[238,51407,51409,51473,51544],{"className":51408,"ariaHidden":269},[268],[238,51410,51412,51415,51464,51467,51470],{"className":51411},[273],[238,51413],{"className":51414,"style":7783},[277],[238,51416,51418,51421],{"className":51417},[282],[238,51419,4429],{"className":51420,"style":660},[282,283],[238,51422,51424],{"className":51423},[391],[238,51425,51427,51456],{"className":51426},[395,848],[238,51428,51430,51453],{"className":51429},[399],[238,51431,51433],{"className":51432,"style":3618},[403],[238,51434,51435,51438],{"style":13806},[238,51436],{"className":51437,"style":412},[411],[238,51439,51441],{"className":51440},[416,417,418,419],[238,51442,51444,51447,51450],{"className":51443},[282,419],[238,51445,3395],{"className":51446,"style":3520},[282,283,419],[238,51448,987],{"className":51449},[664,419],[238,51451,745],{"className":51452},[282,419],[238,51454,872],{"className":51455},[871],[238,51457,51459],{"className":51458},[399],[238,51460,51462],{"className":51461,"style":1193},[403],[238,51463],{},[238,51465],{"className":51466,"style":430},[429],[238,51468,992],{"className":51469},[434],[238,51471],{"className":51472,"style":430},[429],[238,51474,51476,51480,51483,51523,51526,51529,51532,51535,51538,51541],{"className":51475},[273],[238,51477],{"className":51478,"style":51479},[277],"height:0.8889em;vertical-align:-0.1944em;",[238,51481,11971],{"className":51482,"style":11992},[282,840],[238,51484,51486,51489],{"className":51485},[282],[238,51487,4429],{"className":51488,"style":660},[282,283],[238,51490,51492],{"className":51491},[391],[238,51493,51495,51515],{"className":51494},[395,848],[238,51496,51498,51512],{"className":51497},[399],[238,51499,51501],{"className":51500,"style":3618},[403],[238,51502,51503,51506],{"style":13806},[238,51504],{"className":51505,"style":412},[411],[238,51507,51509],{"className":51508},[416,417,418,419],[238,51510,3395],{"className":51511,"style":3520},[282,283,419],[238,51513,872],{"className":51514},[871],[238,51516,51518],{"className":51517},[399],[238,51519,51521],{"className":51520,"style":879},[403],[238,51522],{},[238,51524,354],{"className":51525},[451],[238,51527],{"className":51528,"style":10986},[429],[238,51530],{"className":51531,"style":455},[429],[238,51533,3395],{"className":51534,"style":3520},[282,283],[238,51536],{"className":51537,"style":430},[429],[238,51539,992],{"className":51540},[434],[238,51542],{"className":51543,"style":430},[429],[238,51545,51547,51550,51553,51556,51559,51562,51565,51568,51571,51574,51577],{"className":51546},[273],[238,51548],{"className":51549,"style":17777},[277],[238,51551,740],{"className":51552},[282],[238,51554,354],{"className":51555},[451],[238,51557],{"className":51558,"style":455},[429],[238,51560,745],{"className":51561},[282],[238,51563,354],{"className":51564},[451],[238,51566],{"className":51567,"style":455},[429],[238,51569,17558],{"className":51570},[282],[238,51572,354],{"className":51573},[451],[238,51575],{"className":51576,"style":455},[429],[238,51578,2840],{"className":51579},[3041],[11,51581,24774,51582,51610,51611,35055,51669,51727],{},[238,51583,51585,51598],{"className":51584,"translate":242},[241],[238,51586,51588],{"className":51587},[246],[248,51589,51590],{"xmlns":250},[252,51591,51592,51596],{},[255,51593,51594],{},[258,51595,11971],{"mathvariant":816},[262,51597,34984],{"encoding":264},[238,51599,51601],{"className":51600,"ariaHidden":269},[268],[238,51602,51604,51607],{"className":51603},[273],[238,51605],{"className":51606,"style":278},[277],[238,51608,11971],{"className":51609,"style":11992},[282,840]," 可以是 ",[238,51612,51614,51631],{"className":51613,"translate":242},[241],[238,51615,51617],{"className":51616},[246],[248,51618,51619],{"xmlns":250},[252,51620,51621,51629],{},[255,51622,51623],{},[337,51624,51625,51627],{},[258,51626,11971],{"mathvariant":816},[258,51628,2297],{},[262,51630,11976],{"encoding":264},[238,51632,51634],{"className":51633,"ariaHidden":269},[268],[238,51635,51637,51640],{"className":51636},[273],[238,51638],{"className":51639,"style":278},[277],[238,51641,51643,51646],{"className":51642},[282],[238,51644,11971],{"className":51645,"style":11992},[282,840],[238,51647,51649],{"className":51648},[391],[238,51650,51652],{"className":51651},[395],[238,51653,51655],{"className":51654},[399],[238,51656,51658],{"className":51657,"style":6609},[403],[238,51659,51660,51663],{"style":407},[238,51661],{"className":51662,"style":412},[411],[238,51664,51666],{"className":51665},[416,417,418,419],[238,51667,2297],{"className":51668,"style":2313},[282,283,419],[238,51670,51672,51689],{"className":51671,"translate":242},[241],[238,51673,51675],{"className":51674},[246],[248,51676,51677],{"xmlns":250},[252,51678,51679,51687],{},[255,51680,51681],{},[337,51682,51683,51685],{},[258,51684,11971],{"mathvariant":816},[332,51686,22083],{},[262,51688,22086],{"encoding":264},[238,51690,51692],{"className":51691,"ariaHidden":269},[268],[238,51693,51695,51698],{"className":51694},[273],[238,51696],{"className":51697,"style":22096},[277],[238,51699,51701,51704],{"className":51700},[282],[238,51702,11971],{"className":51703,"style":11992},[282,840],[238,51705,51707],{"className":51706},[391],[238,51708,51710],{"className":51709},[395],[238,51711,51713],{"className":51712},[399],[238,51714,51716],{"className":51715,"style":22096},[403],[238,51717,51718,51721],{"style":407},[238,51719],{"className":51720,"style":412},[411],[238,51722,51724],{"className":51723},[416,417,418,419],[238,51725,22083],{"className":51726},[664,419],"。在强化学习中，这一迭代称为值迭代。由 Banach 不动点定理，迭代序列必然收敛到相应的不动点。",[230,51729,51732,52126,52530],{"id":51730,"title":51731},"theorem-convergence","定理 5.1（几何收敛）",[11,51733,236,51734,941,51806,21601,51834,35055,51892,35114,51950,52019,52020,941],{},[238,51735,51737,51758],{"className":51736,"translate":242},[241],[238,51738,51740],{"className":51739},[246],[248,51741,51742],{"xmlns":250},[252,51743,51744,51756],{},[255,51745,51746,51748,51750,51752,51754],{},[738,51747,740],{},[332,51749,2023],{},[258,51751,730],{},[332,51753,1892],{},[738,51755,745],{},[262,51757,14776],{"encoding":264},[238,51759,51761,51779,51797],{"className":51760,"ariaHidden":269},[268],[238,51762,51764,51767,51770,51773,51776],{"className":51763},[273],[238,51765],{"className":51766,"style":14786},[277],[238,51768,740],{"className":51769},[282],[238,51771],{"className":51772,"style":430},[429],[238,51774,2023],{"className":51775},[434],[238,51777],{"className":51778,"style":430},[429],[238,51780,51782,51785,51788,51791,51794],{"className":51781},[273],[238,51783],{"className":51784,"style":760},[277],[238,51786,730],{"className":51787,"style":764},[282,283],[238,51789],{"className":51790,"style":430},[429],[238,51792,1892],{"className":51793},[434],[238,51795],{"className":51796,"style":430},[429],[238,51798,51800,51803],{"className":51799},[273],[238,51801],{"className":51802,"style":2672},[277],[238,51804,745],{"className":51805},[282],[238,51807,51809,51822],{"className":51808,"translate":242},[241],[238,51810,51812],{"className":51811},[246],[248,51813,51814],{"xmlns":250},[252,51815,51816,51820],{},[255,51817,51818],{},[258,51819,11971],{"mathvariant":816},[262,51821,34984],{"encoding":264},[238,51823,51825],{"className":51824,"ariaHidden":269},[268],[238,51826,51828,51831],{"className":51827},[273],[238,51829],{"className":51830,"style":278},[277],[238,51832,11971],{"className":51833,"style":11992},[282,840],[238,51835,51837,51854],{"className":51836,"translate":242},[241],[238,51838,51840],{"className":51839},[246],[248,51841,51842],{"xmlns":250},[252,51843,51844,51852],{},[255,51845,51846],{},[337,51847,51848,51850],{},[258,51849,11971],{"mathvariant":816},[258,51851,2297],{},[262,51853,11976],{"encoding":264},[238,51855,51857],{"className":51856,"ariaHidden":269},[268],[238,51858,51860,51863],{"className":51859},[273],[238,51861],{"className":51862,"style":278},[277],[238,51864,51866,51869],{"className":51865},[282],[238,51867,11971],{"className":51868,"style":11992},[282,840],[238,51870,51872],{"className":51871},[391],[238,51873,51875],{"className":51874},[395],[238,51876,51878],{"className":51877},[399],[238,51879,51881],{"className":51880,"style":6609},[403],[238,51882,51883,51886],{"style":407},[238,51884],{"className":51885,"style":412},[411],[238,51887,51889],{"className":51888},[416,417,418,419],[238,51890,2297],{"className":51891,"style":2313},[282,283,419],[238,51893,51895,51912],{"className":51894,"translate":242},[241],[238,51896,51898],{"className":51897},[246],[248,51899,51900],{"xmlns":250},[252,51901,51902,51910],{},[255,51903,51904],{},[337,51905,51906,51908],{},[258,51907,11971],{"mathvariant":816},[332,51909,22083],{},[262,51911,22086],{"encoding":264},[238,51913,51915],{"className":51914,"ariaHidden":269},[268],[238,51916,51918,51921],{"className":51917},[273],[238,51919],{"className":51920,"style":22096},[277],[238,51922,51924,51927],{"className":51923},[282],[238,51925,11971],{"className":51926,"style":11992},[282,840],[238,51928,51930],{"className":51929},[391],[238,51931,51933],{"className":51932},[395],[238,51934,51936],{"className":51935},[399],[238,51937,51939],{"className":51938,"style":22096},[403],[238,51940,51941,51944],{"style":407},[238,51942],{"className":51943,"style":412},[411],[238,51945,51947],{"className":51946},[416,417,418,419],[238,51948,22083],{"className":51949},[664,419],[238,51951,51953,51970],{"className":51952,"translate":242},[241],[238,51954,51956],{"className":51955},[246],[248,51957,51958],{"xmlns":250},[252,51959,51960,51968],{},[255,51961,51962],{},[812,51963,51964,51966],{},[258,51965,4429],{},[258,51967,1895],{"mathvariant":344},[262,51969,35135],{"encoding":264},[238,51971,51973],{"className":51972,"ariaHidden":269},[268],[238,51974,51976,51979],{"className":51975},[273],[238,51977],{"className":51978,"style":833},[277],[238,51980,51982,51985],{"className":51981},[282],[238,51983,4429],{"className":51984,"style":660},[282,283],[238,51986,51988],{"className":51987},[391],[238,51989,51991,52011],{"className":51990},[395,848],[238,51992,51994,52008],{"className":51993},[399],[238,51995,51997],{"className":51996,"style":1927},[403],[238,51998,51999,52002],{"style":13806},[238,52000],{"className":52001,"style":412},[411],[238,52003,52005],{"className":52004},[416,417,418,419],[238,52006,1895],{"className":52007},[282,419],[238,52009,872],{"className":52010},[871],[238,52012,52014],{"className":52013},[399],[238,52015,52017],{"className":52016,"style":879},[403],[238,52018],{},"。则对任意 ",[238,52021,52023,52050],{"className":52022,"translate":242},[241],[238,52024,52026],{"className":52025},[246],[248,52027,52028],{"xmlns":250},[252,52029,52030,52048],{},[255,52031,52032,52038,52040,52042,52044,52046],{},[812,52033,52034,52036],{},[258,52035,4429],{},[738,52037,740],{},[332,52039,733],{},[258,52041,19971],{},[332,52043,335],{"stretchy":334},[258,52045,260],{},[332,52047,360],{"stretchy":334},[262,52049,51266],{"encoding":264},[238,52051,52053,52108],{"className":52052,"ariaHidden":269},[268],[238,52054,52056,52059,52099,52102,52105],{"className":52055},[273],[238,52057],{"className":52058,"style":833},[277],[238,52060,52062,52065],{"className":52061},[282],[238,52063,4429],{"className":52064,"style":660},[282,283],[238,52066,52068],{"className":52067},[391],[238,52069,52071,52091],{"className":52070},[395,848],[238,52072,52074,52088],{"className":52073},[399],[238,52075,52077],{"className":52076,"style":1162},[403],[238,52078,52079,52082],{"style":13806},[238,52080],{"className":52081,"style":412},[411],[238,52083,52085],{"className":52084},[416,417,418,419],[238,52086,740],{"className":52087},[282,419],[238,52089,872],{"className":52090},[871],[238,52092,52094],{"className":52093},[399],[238,52095,52097],{"className":52096,"style":879},[403],[238,52098],{},[238,52100],{"className":52101,"style":430},[429],[238,52103,733],{"className":52104},[434],[238,52106],{"className":52107,"style":430},[429],[238,52109,52111,52114,52117,52120,52123],{"className":52110},[273],[238,52112],{"className":52113,"style":444},[277],[238,52115,19971],{"className":52116,"style":20011},[282,283],[238,52118,335],{"className":52119},[381],[238,52121,260],{"className":52122,"style":284},[282,283],[238,52124,360],{"className":52125},[462],[238,52127,52129],{"className":52128,"translate":242},[945],[238,52130,52132,52198],{"className":52131,"translate":242},[241],[238,52133,52135],{"className":52134},[246],[248,52136,52137],{"xmlns":250,"display":954},[252,52138,52139,52195],{},[255,52140,52141,52143,52149,52151,52157,52163,52165,52171,52173,52179,52181,52187,52193],{},[258,52142,15588],{"mathvariant":344},[812,52144,52145,52147],{},[258,52146,4429],{},[258,52148,3395],{},[332,52150,5577],{},[812,52152,52153,52155],{},[258,52154,4429],{},[258,52156,1895],{"mathvariant":344},[812,52158,52159,52161],{},[258,52160,15588],{"mathvariant":344},[258,52162,1895],{"mathvariant":344},[332,52164,2023],{},[337,52166,52167,52169],{},[258,52168,730],{},[258,52170,3395],{},[258,52172,15588],{"mathvariant":344},[812,52174,52175,52177],{},[258,52176,4429],{},[738,52178,740],{},[332,52180,5577],{},[812,52182,52183,52185],{},[258,52184,4429],{},[258,52186,1895],{"mathvariant":344},[812,52188,52189,52191],{},[258,52190,15588],{"mathvariant":344},[258,52192,1895],{"mathvariant":344},[258,52194,3102],{"mathvariant":344},[262,52196,52197],{"encoding":264},"\\|V_k-V_\\infty\\|_\\infty\n\\le\\gamma^k\\|V_0-V_\\infty\\|_\\infty.",[238,52199,52201,52259,52354,52441],{"className":52200,"ariaHidden":269},[268],[238,52202,52204,52207,52210,52250,52253,52256],{"className":52203},[273],[238,52205],{"className":52206,"style":444},[277],[238,52208,15588],{"className":52209},[282],[238,52211,52213,52216],{"className":52212},[282],[238,52214,4429],{"className":52215,"style":660},[282,283],[238,52217,52219],{"className":52218},[391],[238,52220,52222,52242],{"className":52221},[395,848],[238,52223,52225,52239],{"className":52224},[399],[238,52226,52228],{"className":52227,"style":3618},[403],[238,52229,52230,52233],{"style":13806},[238,52231],{"className":52232,"style":412},[411],[238,52234,52236],{"className":52235},[416,417,418,419],[238,52237,3395],{"className":52238,"style":3520},[282,283,419],[238,52240,872],{"className":52241},[871],[238,52243,52245],{"className":52244},[399],[238,52246,52248],{"className":52247,"style":879},[403],[238,52249],{},[238,52251],{"className":52252,"style":660},[429],[238,52254,5577],{"className":52255},[664],[238,52257],{"className":52258,"style":660},[429],[238,52260,52262,52265,52305,52345,52348,52351],{"className":52261},[273],[238,52263],{"className":52264,"style":444},[277],[238,52266,52268,52271],{"className":52267},[282],[238,52269,4429],{"className":52270,"style":660},[282,283],[238,52272,52274],{"className":52273},[391],[238,52275,52277,52297],{"className":52276},[395,848],[238,52278,52280,52294],{"className":52279},[399],[238,52281,52283],{"className":52282,"style":1927},[403],[238,52284,52285,52288],{"style":13806},[238,52286],{"className":52287,"style":412},[411],[238,52289,52291],{"className":52290},[416,417,418,419],[238,52292,1895],{"className":52293},[282,419],[238,52295,872],{"className":52296},[871],[238,52298,52300],{"className":52299},[399],[238,52301,52303],{"className":52302,"style":879},[403],[238,52304],{},[238,52306,52308,52311],{"className":52307},[282],[238,52309,15588],{"className":52310},[282],[238,52312,52314],{"className":52313},[391],[238,52315,52317,52337],{"className":52316},[395,848],[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V_\\infty",[238,52563,52565,52620],{"className":52564,"ariaHidden":269},[268],[238,52566,52568,52571,52611,52614,52617],{"className":52567},[273],[238,52569],{"className":52570,"style":833},[277],[238,52572,52574,52577],{"className":52573},[282],[238,52575,4429],{"className":52576,"style":660},[282,283],[238,52578,52580],{"className":52579},[391],[238,52581,52583,52603],{"className":52582},[395,848],[238,52584,52586,52600],{"className":52585},[399],[238,52587,52589],{"className":52588,"style":3618},[403],[238,52590,52591,52594],{"style":13806},[238,52592],{"className":52593,"style":412},[411],[238,52595,52597],{"className":52596},[416,417,418,419],[238,52598,3395],{"className":52599,"style":3520},[282,283,419],[238,52601,872],{"className":52602},[871],[238,52604,52606],{"className":52605},[399],[238,52607,52609],{"className":52608,"style":879},[403],[238,52610],{},[238,52612],{"className":52613,"style":430},[429],[238,52615,617],{"className":52616},[434],[238,52618],{"className":52619,"style":430},[429],[238,52621,52623,52626],{"className":52622},[273],[238,52624],{"className":52625,"style":833},[277],[238,52627,52629,52632],{"className":52628},[282],[238,52630,4429],{"className":52631,"style":660},[282,283],[238,52633,52635],{"className":52634},[391],[238,52636,52638,52658],{"className":52637},[395,848],[238,52639,52641,52655],{"className":52640},[399],[238,52642,52644],{"className":52643,"style":1927},[403],[238,52645,52646,52649],{"style":13806},[238,52647],{"className":52648,"style":412},[411],[238,52650,52652],{"className":52651},[416,417,418,419],[238,52653,1895],{"className":52654},[282,419],[238,52656,872],{"className":52657},[871],[238,52659,52661],{"className":52660},[399],[238,52662,52664],{"className":52663,"style":879},[403],[238,52665],{}," 在 sup 范数下成立。",[11,52668,52669,52671],{},[1688,52670,5794],{}," 由压缩性，",[238,52673,52675],{"className":52674,"translate":242},[945],[238,52676,52678,52806],{"className":52677,"translate":242},[241],[238,52679,52681],{"className":52680},[246],[248,52682,52683],{"xmlns":250,"display":954},[252,52684,52685,52803],{},[958,52686,52687,52759],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,52688,52689,52723],{},[966,52690,52691],{},[969,52692,52693],{"scriptlevel":740,"displaystyle":269},[255,52694,52695,52697,52709,52711,52717],{},[258,52696,15588],{"mathvariant":344},[812,52698,52699,52701],{},[258,52700,4429],{},[255,52702,52703,52705,52707],{},[258,52704,3395],{},[332,52706,987],{},[738,52708,745],{},[332,52710,5577],{},[812,52712,52713,52715],{},[258,52714,4429],{},[258,52716,1895],{"mathvariant":344},[812,52718,52719,52721],{},[258,52720,15588],{"mathvariant":344},[258,52722,1895],{"mathvariant":344},[966,52724,52725],{},[969,52726,52727],{"scriptlevel":740,"displaystyle":269},[255,52728,52729,52731,52733,52735,52737,52743,52745,52747,52753],{},[255,52730],{},[332,52732,992],{},[258,52734,15588],{"mathvariant":344},[258,52736,11971],{"mathvariant":816},[812,52738,52739,52741],{},[258,52740,4429],{},[258,52742,3395],{},[332,52744,5577],{},[258,52746,11971],{"mathvariant":816},[812,52748,52749,52751],{},[258,52750,4429],{},[258,52752,1895],{"mathvariant":344},[812,52754,52755,52757],{},[258,52756,15588],{"mathvariant":344},[258,52758,1895],{"mathvariant":344},[963,52760,52761,52767],{},[966,52762,52763],{},[969,52764,52765],{"scriptlevel":740,"displaystyle":269},[255,52766],{},[966,52768,52769],{},[969,52770,52771],{"scriptlevel":740,"displaystyle":269},[255,52772,52773,52775,52777,52779,52781,52787,52789,52795,52801],{},[255,52774],{},[332,52776,2023],{},[258,52778,730],{},[258,52780,15588],{"mathvariant":344},[812,52782,52783,52785],{},[258,52784,4429],{},[258,52786,3395],{},[332,52788,5577],{},[812,52790,52791,52793],{},[258,52792,4429],{},[258,52794,1895],{"mathvariant":344},[812,52796,52797,52799],{},[258,52798,15588],{"mathvariant":344},[258,52800,1895],{"mathvariant":344},[258,52802,3102],{"mathvariant":344},[262,52804,52805],{"encoding":264},"\\begin{aligned}\n\\|V_{k+1}-V_\\infty\\|_\\infty\n&=\\|\\mathcal{T}V_k-\\mathcal{T}V_\\infty\\|_\\infty\\\\\n&\\le\\gamma\\|V_k-V_\\infty\\|_\\infty.\n\\end{aligned}",[238,52807,52809],{"className":52808,"ariaHidden":269},[268],[238,52810,52812,52815],{"className":52811},[273],[238,52813],{"className":52814,"style":4573},[277],[238,52816,52818],{"className":52817},[282],[238,52819,52821,53001],{"className":52820},[958],[238,52822,52824],{"className":52823},[1117],[238,52825,52827,52993],{"className":52826},[395,848],[238,52828,52830,52990],{"className":52829},[399],[238,52831,52833,52982],{"className":52832,"style":4592},[403],[238,52834,52835,52838],{"style":4595},[238,52836],{"className":52837,"style":1134},[411],[238,52839,52841,52844,52893,52896,52899,52902,52942],{"className":52840},[282],[238,52842,15588],{"className":52843},[282],[238,52845,52847,52850],{"className":52846},[282],[238,52848,4429],{"className":52849,"style":660},[282,283],[238,52851,52853],{"className":52852},[391],[238,52854,52856,52885],{"className":52855},[395,848],[238,52857,52859,52882],{"className":52858},[399],[238,52860,52862],{"className":52861,"style":3618},[403],[238,52863,52864,52867],{"style":13806},[238,52865],{"className":52866,"style":412},[411],[238,52868,52870],{"className":52869},[416,417,418,419],[238,52871,52873,52876,52879],{"className":52872},[282,419],[238,52874,3395],{"className":52875,"style":3520},[282,283,419],[238,52877,987],{"className":52878},[664,419],[238,52880,745],{"className":52881},[282,419],[238,52883,872],{"className":52884},[871],[238,52886,52888],{"className":52887},[399],[238,52889,52891],{"className":52890,"style":1193},[403],[238,52892],{},[238,52894],{"className":52895,"style":660},[429],[238,52897,5577],{"className":52898},[664],[238,52900],{"className":52901,"style":660},[429],[238,52903,52905,52908],{"className":52904},[282],[238,52906,4429],{"className":52907,"style":660},[282,283],[238,52909,52911],{"className":52910},[391],[238,52912,52914,52934],{"className":52913},[395,848],[238,52915,52917,52931],{"className":52916},[399],[238,52918,52920],{"className":52919,"style":1927},[403],[238,52921,52922,52925],{"style":13806},[238,52923],{"className":52924,"style":412},[411],[238,52926,52928],{"className":52927},[416,417,418,419],[238,52929,1895],{"className":52930},[282,419],[238,52932,872],{"className":52933},[871],[238,52935,52937],{"className":52936},[399],[238,52938,52940],{"className":52939,"style":879},[403],[238,52941],{},[238,52943,52945,52948],{"className":52944},[282],[238,52946,15588],{"className":52947},[282],[238,52949,52951],{"className":52950},[391],[238,52952,52954,52974],{"className":52953},[395,848],[238,52955,52957,52971],{"className":52956},[399],[238,52958,52960],{"className":52959,"style":1927},[403],[238,52961,52962,52965],{"style":1165},[238,52963],{"className":52964,"style":412},[411],[238,52966,52968],{"className":52967},[416,417,418,419],[238,52969,1895],{"className":52970},[282,419],[238,52972,872],{"className":52973},[871],[238,52975,52977],{"className":52976},[399],[238,52978,52980],{"className":52979,"style":879},[403],[238,52981],{},[238,52983,52984,52987],{"style":4643},[238,52985],{"className":52986,"style":1134},[411],[238,52988],{"className":52989},[282],[238,52991,872],{"className":52992},[871],[238,52994,52996],{"className":52995},[399],[238,52997,52999],{"className":52998,"style":4706},[403],[238,53000],{},[238,53002,53004],{"className":53003},[1422],[238,53005,53007,53332],{"className":53006},[395,848],[238,53008,53010,53329],{"className":53009},[399],[238,53011,53013,53171],{"className":53012,"style":4592},[403],[238,53014,53015,53018],{"style":4595},[238,53016],{"className":53017,"style":1134},[411],[238,53019,53021,53024,53027,53030,53033,53036,53039,53079,53082,53085,53088,53091,53131],{"className":53020},[282],[238,53022],{"className":53023},[282],[238,53025],{"className":53026,"style":430},[429],[238,53028,992],{"className":53029},[434],[238,53031],{"className":53032,"style":430},[429],[238,53034,15588],{"className":53035},[282],[238,53037,11971],{"className":53038,"style":11992},[282,840],[238,53040,53042,53045],{"className":53041},[282],[238,53043,4429],{"className":53044,"style":660},[282,283],[238,53046,53048],{"className":53047},[391],[238,53049,53051,53071],{"className":53050},[395,848],[238,53052,53054,53068],{"className":53053},[399],[238,53055,53057],{"className":53056,"style":3618},[403],[238,53058,53059,53062],{"style":13806},[238,53060],{"className":53061,"style":412},[411],[238,53063,53065],{"className":53064},[416,417,418,419],[238,53066,3395],{"className":53067,"style":3520},[282,283,419],[238,53069,872],{"className":53070},[871],[238,53072,53074],{"className":53073},[399],[238,53075,53077],{"className":53076,"style":879},[403],[238,53078],{},[238,53080],{"className":53081,"style":660},[429],[238,53083,5577],{"className":53084},[664],[238,53086],{"className":53087,"style":660},[429],[238,53089,11971],{"className":53090,"style":11992},[282,840],[238,53092,53094,53097],{"className":53093},[282],[238,53095,4429],{"className":53096,"style":660},[282,283],[238,53098,53100],{"className":53099},[391],[238,53101,53103,53123],{"className":53102},[395,848],[238,53104,53106,53120],{"className":53105},[399],[238,53107,53109],{"className":53108,"style":1927},[403],[238,53110,53111,53114],{"style":13806},[238,53112],{"className":53113,"style":412},[411],[238,53115,53117],{"className":53116},[416,417,418,419],[238,53118,1895],{"className":53119},[282,419],[238,53121,872],{"className":53122},[871],[238,53124,53126],{"className":53125},[399],[238,53127,53129],{"className":53128,"style":879},[403],[238,53130],{},[238,53132,53134,53137],{"className":53133},[282],[238,53135,15588],{"className":53136},[282],[238,53138,53140],{"className":53139},[391],[238,53141,53143,53163],{"className":53142},[395,848],[238,53144,53146,53160],{"className":53145},[399],[238,53147,53149],{"className":53148,"style":1927},[403],[238,53150,53151,53154],{"style":1165},[238,53152],{"className":53153,"style":412},[411],[238,53155,53157],{"className":53156},[416,417,418,419],[238,53158,1895],{"className":53159},[282,419],[238,53161,872],{"className":53162},[871],[238,53164,53166],{"className":53165},[399],[238,53167,53169],{"className":53168,"style":879},[403],[238,53170],{},[238,53172,53173,53176],{"style":4643},[238,53174],{"className":53175,"style":1134},[411],[238,53177,53179,53182,53185,53188,53191,53194,53197,53237,53240,53243,53246,53286,53326],{"className":53178},[282],[238,53180],{"className":53181},[282],[238,53183],{"className":53184,"style":430},[429],[238,53186,2023],{"className":53187},[434],[238,53189],{"className":53190,"style":430},[429],[238,53192,730],{"className":53193,"style":764},[282,283],[238,53195,15588],{"className":53196},[282],[238,53198,53200,53203],{"className":53199},[282],[238,53201,4429],{"className":53202,"style":660},[282,283],[238,53204,53206],{"className":53205},[391],[238,53207,53209,53229],{"className":53208},[395,848],[238,53210,53212,53226],{"className":53211},[399],[238,53213,53215],{"className":53214,"style":3618},[403],[238,53216,53217,53220],{"style":13806},[238,53218],{"className":53219,"style":412},[411],[238,53221,53223],{"className":53222},[416,417,418,419],[238,53224,3395],{"className":53225,"style":3520},[282,283,419],[238,53227,872],{"className":53228},[871],[238,53230,53232],{"className":53231},[399],[238,53233,53235],{"className":53234,"style":879},[403],[238,53236],{},[238,53238],{"className":53239,"style":660},[429],[238,53241,5577],{"className":53242},[664],[238,53244],{"className":53245,"style":660},[429],[238,53247,53249,53252],{"className":53248},[282],[238,53250,4429],{"className":53251,"style":660},[282,283],[238,53253,53255],{"className":53254},[391],[238,53256,53258,53278],{"className":53257},[395,848],[238,53259,53261,53275],{"className":53260},[399],[238,53262,53264],{"className":53263,"style":1927},[403],[238,53265,53266,53269],{"style":13806},[238,53267],{"className":53268,"style":412},[411],[238,53270,53272],{"className":53271},[416,417,418,419],[238,53273,1895],{"className":53274},[282,419],[238,53276,872],{"className":53277},[871],[238,53279,53281],{"className":53280},[399],[238,53282,53284],{"className":53283,"style":879},[403],[238,53285],{},[238,53287,53289,53292],{"className":53288},[282],[238,53290,15588],{"className":53291},[282],[238,53293,53295],{"className":53294},[391],[238,53296,53298,53318],{"className":53297},[395,848],[238,53299,53301,53315],{"className":53300},[399],[238,53302,53304],{"className":53303,"style":1927},[403],[238,53305,53306,53309],{"style":1165},[238,53307],{"className":53308,"style":412},[411],[238,53310,53312],{"className":53311},[416,417,418,419],[238,53313,1895],{"className":53314},[282,419],[238,53316,872],{"className":53317},[871],[238,53319,53321],{"className":53320},[399],[238,53322,53324],{"className":53323,"style":879},[403],[238,53325],{},[238,53327,3102],{"className":53328},[282],[238,53330,872],{"className":53331},[871],[238,53333,53335],{"className":53334},[399],[238,53336,53338],{"className":53337,"style":4706},[403],[238,53339],{},[11,53341,26642,53342,53370],{},[238,53343,53345,53358],{"className":53344,"translate":242},[241],[238,53346,53348],{"className":53347},[246],[248,53349,53350],{"xmlns":250},[252,53351,53352,53356],{},[255,53353,53354],{},[258,53355,3395],{},[262,53357,3395],{"encoding":264},[238,53359,53361],{"className":53360,"ariaHidden":269},[268],[238,53362,53364,53367],{"className":53363},[273],[238,53365],{"className":53366,"style":29023},[277],[238,53368,3395],{"className":53369,"style":3520},[282,283]," 反复迭代即得。证毕。",[11,53372,53373,53374,53444,53445,53504,53505,53533,53534,53562],{},"这个定理说明，初始估计 ",[238,53375,53377,53395],{"className":53376,"translate":242},[241],[238,53378,53380],{"className":53379},[246],[248,53381,53382],{"xmlns":250},[252,53383,53384,53392],{},[255,53385,53386],{},[812,53387,53388,53390],{},[258,53389,4429],{},[738,53391,740],{},[262,53393,53394],{"encoding":264},"V_0",[238,53396,53398],{"className":53397,"ariaHidden":269},[268],[238,53399,53401,53404],{"className":53400},[273],[238,53402],{"className":53403,"style":833},[277],[238,53405,53407,53410],{"className":53406},[282],[238,53408,4429],{"className":53409,"style":660},[282,283],[238,53411,53413],{"className":53412},[391],[238,53414,53416,53436],{"className":53415},[395,848],[238,53417,53419,53433],{"className":53418},[399],[238,53420,53422],{"className":53421,"style":1162},[403],[238,53423,53424,53427],{"style":13806},[238,53425],{"className":53426,"style":412},[411],[238,53428,53430],{"className":53429},[416,417,418,419],[238,53431,740],{"className":53432},[282,419],[238,53434,872],{"className":53435},[871],[238,53437,53439],{"className":53438},[399],[238,53440,53442],{"className":53441,"style":879},[403],[238,53443],{}," 不需要很准确。无论从哪个有界函数出发，误差都会按 ",[238,53446,53448,53466],{"className":53447,"translate":242},[241],[238,53449,53451],{"className":53450},[246],[248,53452,53453],{"xmlns":250},[252,53454,53455,53463],{},[255,53456,53457],{},[337,53458,53459,53461],{},[258,53460,730],{},[258,53462,3395],{},[262,53464,53465],{"encoding":264},"\\gamma^k",[238,53467,53469],{"className":53468,"ariaHidden":269},[268],[238,53470,53472,53475],{"className":53471},[273],[238,53473],{"className":53474,"style":15843},[277],[238,53476,53478,53481],{"className":53477},[282],[238,53479,730],{"className":53480,"style":764},[282,283],[238,53482,53484],{"className":53483},[391],[238,53485,53487],{"className":53486},[395],[238,53488,53490],{"className":53489},[399],[238,53491,53493],{"className":53492,"style":15776},[403],[238,53494,53495,53498],{"style":407},[238,53496],{"className":53497,"style":412},[411],[238,53499,53501],{"className":53500},[416,417,418,419],[238,53502,3395],{"className":53503,"style":3520},[282,283,419]," 的速度衰减。",[238,53506,53508,53521],{"className":53507,"translate":242},[241],[238,53509,53511],{"className":53510},[246],[248,53512,53513],{"xmlns":250},[252,53514,53515,53519],{},[255,53516,53517],{},[258,53518,730],{},[262,53520,26785],{"encoding":264},[238,53522,53524],{"className":53523,"ariaHidden":269},[268],[238,53525,53527,53530],{"className":53526},[273],[238,53528],{"className":53529,"style":2853},[277],[238,53531,730],{"className":53532,"style":764},[282,283]," 越接近 ",[238,53535,53537,53550],{"className":53536,"translate":242},[241],[238,53538,53540],{"className":53539},[246],[248,53541,53542],{"xmlns":250},[252,53543,53544,53548],{},[255,53545,53546],{},[738,53547,745],{},[262,53549,745],{"encoding":264},[238,53551,53553],{"className":53552,"ariaHidden":269},[268],[238,53554,53556,53559],{"className":53555},[273],[238,53557],{"className":53558,"style":2672},[277],[238,53560,745],{"className":53561},[282],"，衰减越慢，所需的迭代次数越多。这也是折扣因子在数值上既影响问题定义、又影响求解难度的体现。",[225,53564,53566],{"id":53565},"_52-误差上界","5.2 误差上界",[11,53568,53569,53573,53574,53643],{},[357,53570,53572],{"href":53571},"#theorem-convergence","定理 5.1"," 给出的误差界含有未知的 ",[238,53575,53577,53594],{"className":53576,"translate":242},[241],[238,53578,53580],{"className":53579},[246],[248,53581,53582],{"xmlns":250},[252,53583,53584,53592],{},[255,53585,53586],{},[812,53587,53588,53590],{},[258,53589,4429],{},[258,53591,1895],{"mathvariant":344},[262,53593,35135],{"encoding":264},[238,53595,53597],{"className":53596,"ariaHidden":269},[268],[238,53598,53600,53603],{"className":53599},[273],[238,53601],{"className":53602,"style":833},[277],[238,53604,53606,53609],{"className":53605},[282],[238,53607,4429],{"className":53608,"style":660},[282,283],[238,53610,53612],{"className":53611},[391],[238,53613,53615,53635],{"className":53614},[395,848],[238,53616,53618,53632],{"className":53617},[399],[238,53619,53621],{"className":53620,"style":1927},[403],[238,53622,53623,53626],{"style":13806},[238,53624],{"className":53625,"style":412},[411],[238,53627,53629],{"className":53628},[416,417,418,419],[238,53630,1895],{"className":53631},[282,419],[238,53633,872],{"className":53634},[871],[238,53636,53638],{"className":53637},[399],[238,53639,53641],{"className":53640,"style":879},[403],[238,53642],{},"，在实际计算中不能直接使用。下面的命题给出只依赖迭代差分与初始差分的界。",[230,53645,53648,53982,54469,54471],{"id":53646,"title":53647},"proposition-error-bounds","命题 5.2（先验与后验误差界）",[11,53649,236,53650,21601,53678,53706,53707,941,53860,53929,53930,941],{},[238,53651,53653,53666],{"className":53652,"translate":242},[241],[238,53654,53656],{"className":53655},[246],[248,53657,53658],{"xmlns":250},[252,53659,53660,53664],{},[255,53661,53662],{},[258,53663,11971],{"mathvariant":816},[262,53665,34984],{"encoding":264},[238,53667,53669],{"className":53668,"ariaHidden":269},[268],[238,53670,53672,53675],{"className":53671},[273],[238,53673],{"className":53674,"style":278},[277],[238,53676,11971],{"className":53677,"style":11992},[282,840],[238,53679,53681,53694],{"className":53680,"translate":242},[241],[238,53682,53684],{"className":53683},[246],[248,53685,53686],{"xmlns":250},[252,53687,53688,53692],{},[255,53689,53690],{},[258,53691,730],{},[262,53693,26785],{"encoding":264},[238,53695,53697],{"className":53696,"ariaHidden":269},[268],[238,53698,53700,53703],{"className":53699},[273],[238,53701],{"className":53702,"style":2853},[277],[238,53704,730],{"className":53705,"style":764},[282,283],"-压缩映射，",[238,53708,53710,53744],{"className":53709,"translate":242},[241],[238,53711,53713],{"className":53712},[246],[248,53714,53715],{"xmlns":250},[252,53716,53717,53741],{},[255,53718,53719,53731,53733,53735],{},[812,53720,53721,53723],{},[258,53722,4429],{},[255,53724,53725,53727,53729],{},[258,53726,3395],{},[332,53728,987],{},[738,53730,745],{},[332,53732,992],{},[258,53734,11971],{"mathvariant":816},[812,53736,53737,53739],{},[258,53738,4429],{},[258,53740,3395],{},[262,53742,53743],{"encoding":264},"V_{k+1}=\\mathcal{T}V_k",[238,53745,53747,53811],{"className":53746,"ariaHidden":269},[268],[238,53748,53750,53753,53802,53805,53808],{"className":53749},[273],[238,53751],{"className":53752,"style":7783},[277],[238,53754,53756,53759],{"className":53755},[282],[238,53757,4429],{"className":53758,"style":660},[282,283],[238,53760,53762],{"className":53761},[391],[238,53763,53765,53794],{"className":53764},[395,848],[238,53766,53768,53791],{"className":53767},[399],[238,53769,53771],{"className":53770,"style":3618},[403],[238,53772,53773,53776],{"style":13806},[238,53774],{"className":53775,"style":412},[411],[238,53777,53779],{"className":53778},[416,417,418,419],[238,53780,53782,53785,53788],{"className":53781},[282,419],[238,53783,3395],{"className":53784,"style":3520},[282,283,419],[238,53786,987],{"className":53787},[664,419],[238,53789,745],{"className":53790},[282,419],[238,53792,872],{"className":53793},[871],[238,53795,53797],{"className":53796},[399],[238,53798,53800],{"className":53799,"style":1193},[403],[238,53801],{},[238,53803],{"className":53804,"style":430},[429],[238,53806,992],{"className":53807},[434],[238,53809],{"className":53810,"style":430},[429],[238,53812,53814,53817,53820],{"className":53813},[273],[238,53815],{"className":53816,"style":833},[277],[238,53818,11971],{"className":53819,"style":11992},[282,840],[238,53821,53823,53826],{"className":53822},[282],[238,53824,4429],{"className":53825,"style":660},[282,283],[238,53827,53829],{"className":53828},[391],[238,53830,53832,53852],{"className":53831},[395,848],[238,53833,53835,53849],{"className":53834},[399],[238,53836,53838],{"className":53837,"style":3618},[403],[238,53839,53840,53843],{"style":13806},[238,53841],{"className":53842,"style":412},[411],[238,53844,53846],{"className":53845},[416,417,418,419],[238,53847,3395],{"className":53848,"style":3520},[282,283,419],[238,53850,872],{"className":53851},[871],[238,53853,53855],{"className":53854},[399],[238,53856,53858],{"className":53857,"style":879},[403],[238,53859],{},[238,53861,53863,53880],{"className":53862,"translate":242},[241],[238,53864,53866],{"className":53865},[246],[248,53867,53868],{"xmlns":250},[252,53869,53870,53878],{},[255,53871,53872],{},[812,53873,53874,53876],{},[258,53875,4429],{},[258,53877,1895],{"mathvariant":344},[262,53879,35135],{"encoding":264},[238,53881,53883],{"className":53882,"ariaHidden":269},[268],[238,53884,53886,53889],{"className":53885},[273],[238,53887],{"className":53888,"style":833},[277],[238,53890,53892,53895],{"className":53891},[282],[238,53893,4429],{"className":53894,"style":660},[282,283],[238,53896,53898],{"className":53897},[391],[238,53899,53901,53921],{"className":53900},[395,848],[238,53902,53904,53918],{"className":53903},[399],[238,53905,53907],{"className":53906,"style":1927},[403],[238,53908,53909,53912],{"style":13806},[238,53910],{"className":53911,"style":412},[411],[238,53913,53915],{"className":53914},[416,417,418,419],[238,53916,1895],{"className":53917},[282,419],[238,53919,872],{"className":53920},[871],[238,53922,53924],{"className":53923},[399],[238,53925,53927],{"className":53926,"style":879},[403],[238,53928],{}," 为不动点。则对任意 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先证第一式。由三角不等式与压缩性，",[238,54949,54951],{"className":54950,"translate":242},[945],[238,54952,54954,55150],{"className":54953,"translate":242},[241],[238,54955,54957],{"className":54956},[246],[248,54958,54959],{"xmlns":250,"display":954},[252,54960,54961,55147],{},[958,54962,54963,55045],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,54964,54965,54993],{},[966,54966,54967],{},[969,54968,54969],{"scriptlevel":740,"displaystyle":269},[255,54970,54971,54973,54979,54981,54987],{},[258,54972,15588],{"mathvariant":344},[812,54974,54975,54977],{},[258,54976,4429],{},[258,54978,3395],{},[332,54980,5577],{},[812,54982,54983,54985],{},[258,54984,4429],{},[258,54986,1895],{"mathvariant":344},[812,54988,54989,54991],{},[258,54990,15588],{"mathvariant":344},[258,54992,1895],{"mathvariant":344},[966,54994,54995],{},[969,54996,54997],{"scriptlevel":740,"displaystyle":269},[255,54998,54999,55001,55003,55017,55019,55031,55033,55039],{},[255,55000],{},[332,55002,2023],{},[3386,55004,55005,55007,55015],{},[332,55006,3390],{},[255,55008,55009,55011,55013],{},[258,55010,13645],{},[332,55012,992],{},[258,55014,3395],{},[258,55016,1895],{"mathvariant":344},[258,55018,15588],{"mathvariant":344},[812,55020,55021,55023],{},[258,55022,4429],{},[255,55024,55025,55027,55029],{},[258,55026,13645],{},[332,55028,987],{},[738,55030,745],{},[332,55032,5577],{},[812,55034,55035,55037],{},[258,55036,4429],{},[258,55038,13645],{},[812,55040,55041,55043],{},[258,55042,15588],{"mathvariant":344},[258,55044,1895],{"mathvariant":344},[963,55046,55047,55053],{},[966,55048,55049],{},[969,55050,55051],{"scriptlevel":740,"displaystyle":269},[255,55052],{},[966,55054,55055],{},[969,55056,55057],{"scriptlevel":740,"displaystyle":269},[255,55058,55059,55061,55063,55077,55083,55085,55091,55093,55099,55105,55107,55123,55125,55131,55133,55139,55145],{},[255,55060],{},[332,55062,2023],{},[3386,55064,55065,55067,55075],{},[332,55066,3390],{},[255,55068,55069,55071,55073],{},[258,55070,13645],{},[332,55072,992],{},[258,55074,3395],{},[258,55076,1895],{"mathvariant":344},[337,55078,55079,55081],{},[258,55080,730],{},[258,55082,13645],{},[258,55084,15588],{"mathvariant":344},[812,55086,55087,55089],{},[258,55088,4429],{},[738,55090,745],{},[332,55092,5577],{},[812,55094,55095,55097],{},[258,55096,4429],{},[738,55098,740],{},[812,55100,55101,55103],{},[258,55102,15588],{"mathvariant":344},[258,55104,1895],{"mathvariant":344},[332,55106,992],{},[5561,55108,55109,55115],{},[337,55110,55111,55113],{},[258,55112,730],{},[258,55114,3395],{},[255,55116,55117,55119,55121],{},[738,55118,745],{},[332,55120,5577],{},[258,55122,730],{},[258,55124,15588],{"mathvariant":344},[812,55126,55127,55129],{},[258,55128,4429],{},[738,55130,745],{},[332,55132,5577],{},[812,55134,55135,55137],{},[258,55136,4429],{},[738,55138,740],{},[812,55140,55141,55143],{},[258,55142,15588],{"mathvariant":344},[258,55144,1895],{"mathvariant":344},[258,55146,3102],{"mathvariant":344},[262,55148,55149],{"encoding":264},"\\begin{aligned}\n\\|V_k-V_\\infty\\|_\\infty\n&\\le\\sum_{j=k}^{\\infty}\\|V_{j+1}-V_j\\|_\\infty\\\\\n&\\le\\sum_{j=k}^{\\infty}\\gamma^j\\|V_1-V_0\\|_\\infty\n=\\frac{\\gamma^k}{1-\\gamma}\\|V_1-V_0\\|_\\infty.\n\\end{aligned}",[238,55151,55153],{"className":55152,"ariaHidden":269},[268],[238,55154,55156,55160],{"className":55155},[273],[238,55157],{"className":55158,"style":55159},[277],"height:6.4792em;vertical-align:-2.9896em;",[238,55161,55163],{"className":55162},[282],[238,55164,55166,55340],{"className":55165},[958],[238,55167,55169],{"className":55168},[1117],[238,55170,55172,55331],{"className":55171},[395,848],[238,55173,55175,55328],{"className":55174},[399],[238,55176,55179,55320],{"className":55177,"style":55178},[403],"height:3.4896em;",[238,55180,55182,55185],{"style":55181},"top:-5.4896em;",[238,55183],{"className":55184,"style":5981},[411],[238,55186,55188,55191,55231,55234,55237,55240,55280],{"className":55187},[282],[238,55189,15588],{"className":55190},[282],[238,55192,55194,55197],{"className":55193},[282],[238,55195,4429],{"className":55196,"style":660},[282,283],[238,55198,55200],{"className":55199},[391],[238,55201,55203,55223],{"className":55202},[395,848],[238,55204,55206,55220],{"className":55205},[399],[238,55207,55209],{"className":55208,"style":3618},[403],[238,55210,55211,55214],{"style":13806},[238,55212],{"className":55213,"style":412},[411],[238,55215,55217],{"className":55216},[416,417,418,419],[238,55218,3395],{"className":55219,"style":3520},[282,283,419],[238,55221,872],{"className":55222},[871],[238,55224,55226],{"className":55225},[399],[238,55227,55229],{"className":55228,"style":879},[403],[238,55230],{},[238,55232],{"className":55233,"style":660},[429],[238,55235,5577],{"className":55236},[664],[238,55238],{"className":55239,"style":660},[429],[238,55241,55243,55246],{"className":55242},[282],[238,55244,4429],{"className":55245,"style":660},[282,283],[238,55247,55249],{"className":55248},[391],[238,55250,55252,55272],{"classNam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me":57033},[399],[238,57035,57037],{"className":57036,"style":1927},[403],[238,57038,57039,57042],{"style":1165},[238,57040],{"className":57041,"style":412},[411],[238,57043,57045],{"className":57044},[416,417,418,419],[238,57046,1895],{"className":57047},[282,419],[238,57049,872],{"className":57050},[871],[238,57052,57054],{"className":57053},[399],[238,57055,57057],{"className":57056,"style":879},[403],[238,57058],{},[238,57060,3102],{"className":57061},[282],[11,57063,57064],{},"合并两式得",[238,57066,57068],{"className":57067,"translate":242},[945],[238,57069,57071,57165],{"className":57070,"translate":242},[241],[238,57072,57074],{"className":57073},[246],[248,57075,57076],{"xmlns":250,"display":954},[252,57077,57078,57162],{},[255,57079,57080,57082,57088,57090,57096,57102,57104,57106,57108,57120,57122,57128,57134,57136,57138,57140,57146,57148,57154,57160],{},[258,57081,15588],{"mathvariant":344},[812,57083,57084,57086],{},[258,57085,4429],{},[258,57087,3395],{},[332,57089,5577],{},[812,57091,57092,57094],{},[258,57093,4429],{},[258,57095,1895],{"mathvariant":344},[812,57097,57098,57100],{},[258,57099,15588],{"mathvariant":344},[258,57101,1895],{"mathvariant":344},[332,57103,2023],{},[258,57105,730],{},[258,57107,15588],{"mathvariant":344},[812,57109,57110,57112],{},[258,57111,4429],{},[255,57113,57114,57116,57118],{},[258,57115,3395],{},[332,57117,5577],{},[738,57119,745],{},[332,57121,5577],{},[812,57123,57124,57126],{},[258,57125,4429],{},[258,57127,3395],{},[812,57129,57130,57132],{},[258,57131,15588],{"mathvariant":344},[258,57133,1895],{"mathvariant":344},[332,57135,987],{},[258,57137,730],{},[258,57139,15588],{"mathvariant":344},[812,57141,57142,57144],{},[258,57143,4429],{},[258,57145,3395],{},[332,57147,5577],{},[812,57149,57150,57152],{},[258,57151,4429],{},[258,57153,1895],{"mathvariant":344},[812,57155,57156,57158],{},[258,57157,15588],{"mathvariant":344},[258,57159,1895],{"mathvariant":344},[258,57161,3102],{"mathvariant":344},[262,57163,57164],{"encoding":264},"\\|V_k-V_\\infty\\|_\\infty\n\\le\\gamma\\|V_{k-1}-V_k\\|_\\infty\n+\\gamma\\|V_k-V_\\infty\\|_\\infty.",[238,57166,57168,57226,57321,57391,57486,57547],{"className":57167,"ariaHidden":269},[268],[238,57169,57171,57174,57177,57217,57220,57223],{"className":57170},[273],[238,57172],{"className":57173,"style":444},[277],[238,57175,15588],{"className":57176},[282],[238,57178,57180,57183],{"className":57179},[282],[238,57181,4429],{"className":57182,"style":660},[282,283],[238,57184,57186],{"className":57185},[391],[238,57187,57189,57209],{"className":57188},[395,848],[238,57190,57192,57206],{"className":57191},[399],[238,57193,57195],{"className":57194,"style":3618},[403],[238,57196,57197,57200],{"style":13806},[238,57198],{"className":57199,"style":412},[411],[238,57201,57203],{"className":57202},[416,417,418,419],[238,57204,3395],{"className":57205,"style":3520},[282,283,419],[238,57207,872],{"className":57208},[871],[238,57210,57212],{"className":57211},[399],[238,57213,57215],{"className":57214,"style":879},[403],[238,57216],{},[238,57218],{"className":57219,"style":660},[429],[238,57221,5577],{"className":57222},[664],[238,57224],{"className":57225,"style":660},[429],[238,57227,57229,57232,57272,57312,57315,57318],{"className":57228},[273],[238,57230],{"className":57231,"style":444},[277],[238,57233,57235,57238],{"className":57234},[282],[238,57236,4429],{"className":57237,"style":660},[282,283],[238,57239,57241],{"className":57240},[391],[238,57242,57244,57264],{"className":57243},[395,848],[238,57245,57247,57261],{"className":57246},[399],[238,57248,57250],{"className":57249,"style":1927},[403],[238,57251,57252,57255],{"style":13806},[238,57253],{"className":57254,"style":412},[411],[238,57256,57258],{"className":57257},[416,417,418,419],[238,57259,1895],{"className":57260},[282,419],[238,57262,872],{"className":57263},[871],[238,57265,57267],{"className":57266},[399],[238,57268,57270],{"className":57269,"style":879},[403],[238,57271],{},[238,57273,57275,57278],{"className":57274},[282],[238,57276,15588],{"className":57277},[282],[238,57279,57281],{"className":57280},[391],[238,57282,57284,57304],{"className":57283},[395,848],[238,57285,57287,57301],{"className":57286},[399],[238,57288,57290],{"className":57289,"style":1927},[403],[238,57291,57292,57295],{"style":1165},[238,57293],{"className":57294,"style":412},[411],[238,57296,57298],{"className":57297},[416,417,418,419],[238,57299,1895],{"className":57300},[282,419],[238,57302,872],{"className":57303},[871],[238,57305,57307],{"className":57306},[399],[238,57308,57310],{"className":57309,"style":879},[403],[238,57311],{},[238,57313],{"className":57314,"style":430},[429],[238,57316,2023],{"className":57317},[434],[238,57319],{"className":57320,"style":430},[429],[238,57322,57324,57327,57330,57333,57382,57385,57388],{"className":57323},[273],[238,57325],{"className":57326,"style":444},[277],[238,57328,730],{"className":57329,"style":764},[282,283],[238,57331,15588],{"className":5733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时，若",[238,58124,58126],{"className":58125,"translate":242},[945],[238,58127,58129,58226],{"className":58128,"translate":242},[241],[238,58130,58132],{"className":58131},[246],[248,58133,58134],{"xmlns":250,"display":954},[252,58135,58136,58223],{},[958,58137,58138,58182],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,58139,58140,58146],{},[966,58141,58142],{},[969,58143,58144],{"scriptlevel":740,"displaystyle":269},[255,58145],{},[966,58147,58148],{},[969,58149,58150],{"scriptlevel":740,"displaystyle":269},[255,58151,58152,58154,58156,58162,58164,58176],{},[255,58153],{},[258,58155,15588],{"mathvariant":344},[812,58157,58158,58160],{},[258,58159,4429],{},[258,58161,3395],{},[332,58163,5577],{},[812,58165,58166,58168],{},[258,58167,4429],{},[255,58169,58170,58172,58174],{},[258,58171,3395],{},[332,58173,5577],{},[738,58175,745],{},[812,58177,58178,58180],{},[258,58179,15588],{"mathvariant":344},[258,58181,1895],{"mathvariant":344},[963,58183,58184,58190],{},[966,58185,58186],{},[969,58187,58188],{"scriptlevel":740,"displaystyle":269},[255,58189],{},[966,58191,58192],{},[969,58193,58194],{"scriptlevel":740,"displaystyle":269},[255,58195,58196,58198,58200,58202,58221],{},[255,58197],{},[429,58199],{"width":7090},[332,58201,2023],{},[5561,58203,58204,58219],{},[255,58205,58206,58209,58211,58213,58215,58217],{},[258,58207,58208],{},"ε",[332,58210,335],{"stretchy":334},[738,58212,745],{},[332,58214,5577],{},[258,58216,730],{},[332,58218,360],{"stretchy":334},[258,58220,730],{},[332,58222,354],{"separator":269},[262,58224,58225],{"encoding":264},"\\begin{aligned}\n&\\|V_k-V_{k-1}\\|_\\infty\\\\\n&\\quad\\le\\frac{\\varepsilon(1-\\gamma)}{\\gamma},\n\\end{aligned}",[238,58227,58229],{"className":58228,"ariaHidden":269},[268],[238,58230,58232,58236],{"className":58231},[273],[238,58233],{"className":58234,"style":58235},[277],"height:3.8074em;vertical-align:-1.6537em;",[238,58237,58239],{"className":58238},[282],[238,58240,58242,58286],{"className":58241},[958],[238,58243,58245],{"className":58244},[1117],[238,58246,58248,58277],{"className":58247},[395,848],[238,58249,58251,58274],{"className":58250},[399],[238,58252,58255,58265],{"className":58253,"style":58254},[403],"height:2.1537em;",[238,58256,58258,58262],{"style":58257},"top:-4.7407em;",[238,58259],{"className":58260,"style":58261},[411],"height:3.427em;",[238,58263],{"className":58264},[282],[238,58266,58268,58271],{"style":58267},"top:-2.6537em;",[238,58269],{"className":58270,"style":58261},[411],[238,58272],{"className":58273},[282],[238,58275,872],{"className":58276},[871],[238,58278,58280],{"className":58279},[399],[238,58281,58284],{"className":58282,"style":58283},[403],"height:1.6537em;",[238,58285],{},[238,58287,58289],{"className":58288},[1422],[238,58290,58292,58563],{"className":58291},[395,848],[238,58293,58295,58560],{"className":58294},[399],[238,58296,58298,58450],{"className":58297,"style":58254},[403],[238,58299,58300,58303],{"style":58257},[238,58301],{"className":58302,"style":58261},[411],[238,58304,58306,58309,58312,58352,58355,58358,58361,58410],{"className":58305},[282],[238,58307],{"className":58308},[282],[238,58310,15588],{"className":58311},[282],[238,58313,58315,58318],{"className":58314},[282],[238,58316,4429],{"className":58317,"style":660},[282,283],[238,58319,58321],{"className":58320},[391],[238,58322,58324,58344],{"className":58323},[395,848],[238,58325,58327,58341],{"className":58326},[399],[238,58328,58330],{"className":58329,"style":3618},[403],[238,58331,58332,58335],{"style":13806},[238,58333],{"className":58334,"style":412},[411],[238,58336,58338],{"className":58337},[416,417,418,419],[238,58339,3395],{"className":58340,"style":3520},[282,283,419],[238,58342,872],{"className":58343},[871],[238,58345,58347],{"className":58346},[399],[238,58348,58350],{"className":58349,"style":879},[403],[238,58351],{},[238,58353],{"className":58354,"style":660},[429],[238,58356,5577],{"className":58357},[664],[238,58359],{"className":58360,"style":660},[429],[238,58362,58364,58367],{"className":58363},[282],[238,58365,4429],{"className":58366,"style":660},[282,283],[238,58368,58370],{"className":58369},[391],[238,58371,58373,58402],{"className":58372},[395,848],[238,58374,58376,58399],{"className":58375},[399],[238,58377,58379],{"className":58378,"style":3618},[403],[238,58380,58381,58384],{"style":13806},[238,58382],{"className":58383,"style":412},[411],[238,58385,58387],{"className":58386},[416,417,418,419],[238,58388,58390,58393,58396],{"className":58389},[282,419],[238,58391,3395],{"className":58392,"style":3520},[282,283,419],[238,58394,5577],{"className":58395},[664,419],[238,58397,745],{"className":58398},[282,419],[238,58400,872],{"className":58401},[871],[238,58403,58405],{"className":58404},[399],[238,58406,58408],{"className":58407,"style":1193},[403],[238,58409],{},[238,58411,58413,58416],{"className":58412},[282],[238,58414,15588],{"className":58415},[282],[238,58417,58419],{"className":58418},[391],[238,5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",[238,58575,58577,58615],{"className":58576,"translate":242},[241],[238,58578,58580],{"className":58579},[246],[248,58581,58582],{"xmlns":250},[252,58583,58584,58612],{},[255,58585,58586,58588,58594,58596,58602,58608,58610],{},[258,58587,15588],{"mathvariant":344},[812,58589,58590,58592],{},[258,58591,4429],{},[258,58593,3395],{},[332,58595,5577],{},[812,58597,58598,58600],{},[258,58599,4429],{},[258,58601,1895],{"mathvariant":344},[812,58603,58604,58606],{},[258,58605,15588],{"mathvariant":344},[258,58607,1895],{"mathvariant":344},[332,58609,2023],{},[258,58611,58208],{},[262,58613,58614],{"encoding":264},"\\|V_k-V_\\infty\\|_\\infty\\le\\varepsilon",[238,58616,58618,58676,58771],{"className":58617,"ariaHidden":269},[268],[238,58619,58621,58624,58627,58667,58670,58673],{"className":58620},[273],[238,58622],{"className":58623,"style":444},[277],[238,58625,15588],{"className":58626},[282],[238,58628,58630,58633],{"className":58629},[282],[238,58631,4429],{"className":58632,"style":660},[282,283],[238,58634,58636],{"className":58635},[391],[238,58637,58639,58659],{"className":58638},[395,848],[238,58640,58642,58656],{"className":58641},[399],[238,58643,58645],{"className":58644,"style":3618},[403],[238,58646,58647,58650],{"style":13806},[238,58648],{"className":58649,"style":412},[411],[238,58651,58653],{"className":58652},[416,417,418,419],[238,58654,3395],{"className":58655,"style":3520},[282,283,419],[238,58657,872],{"className":58658},[871],[238,58660,58662],{"className":58661},[399],[238,58663,58665],{"className":58664,"style":879},[403],[238,58666],{},[238,58668],{"className":58669,"style":660},[429],[238,58671,5577],{"className":58672},[664],[238,58674],{"className":58675,"style":660},[429],[238,58677,58679,58682,58722,58762,58765,58768],{"className":58678},[273],[238,58680],{"className":58681,"style":444},[277],[238,58683,58685,58688],{"className":58684},[282],[238,58686,4429],{"className":58687,"style":660},[282,283],[238,58689,58691],{"className":58690},[391],[238,58692,58694,58714],{"className":58693},[395,848],[238,58695,58697,58711],{"className":58696},[399],[238,58698,58700],{"className":58699,"style":1927},[403],[238,58701,58702,58705],{"style":13806},[238,58703],{"className":58704,"style":412},[411],[238,58706,58708],{"className":58707},[416,417,418,419],[238,58709,1895],{"className":58710},[282,419],[238,58712,872],{"className":58713},[871],[238,58715,58717],{"className":58716},[399],[238,58718,58720],{"className":58719,"style":879},[403],[238,58721],{},[238,58723,58725,58728],{"className":58724},[282],[238,58726,15588],{"className":58727},[282],[238,58729,58731],{"className":58730},[391],[238,58732,58734,58754],{"className":58733},[395,848],[238,58735,58737,58751],{"className":58736},[399],[238,58738,58740],{"className":58739,"style":1927},[403],[238,58741,58742,58745],{"style":1165},[238,58743],{"className":58744,"style":412},[411],[238,58746,58748],{"className":58747},[416,417,418,419],[238,58749,1895],{"className":58750},[282,419],[238,58752,872],{"className":58753},[871],[238,58755,58757],{"className":58756},[399],[238,58758,58760],{"className":58759,"style":879},[403],[238,58761],{},[238,58763],{"className":58764,"style":430},[429],[238,58766,2023],{"className":58767},[434],[238,58769],{"className":58770,"style":430},[429],[238,58772,58774,58777],{"className":58773},[273],[238,58775],{"className":58776,"style":489},[277],[238,58778,58208],{"className":58779},[282,283],[225,58781,58783],{"id":58782},"_53-例显式迭代","5.3 例：显式迭代",[11,58785,58786,58787,58902],{},"回到 2.4 节的两状态马尔可夫链。令 ",[238,58788,58790,58820],{"className":58789,"translate":242},[241],[238,58791,58793],{"className":58792},[246],[248,58794,58795],{"xmlns":250},[252,58796,58797,58817],{},[255,58798,58799,58805,58807,58809,58811,58813,58815],{},[812,58800,58801,58803],{},[258,58802,4429],{},[738,58804,740],{},[332,58806,992],{},[332,58808,335],{"stretchy":334},[738,58810,740],{},[332,58812,354],{"separator":269},[738,58814,740],{},[332,58816,360],{"stretchy":334},[262,58818,58819],{"encoding":264},"V_0=(0,0)",[238,58821,58823,58878],{"className":58822,"ariaHidden":269},[268],[238,58824,58826,58829,58869,58872,58875],{"className":58825},[273],[238,58827],{"className":58828,"style":833},[277],[238,58830,58832,58835],{"className":58831},[282],[238,58833,4429],{"className":58834,"style":660},[282,283],[238,58836,58838],{"className":58837},[391],[238,58839,58841,58861],{"className":58840},[395,848],[238,58842,58844,58858],{"className":58843},[399],[238,58845,58847],{"className":58846,"style":1162},[403],[238,58848,58849,58852],{"style":13806},[238,58850],{"className":58851,"style":412},[411],[238,58853,58855],{"className":58854},[416,417,418,419],[238,58856,740],{"className":58857},[282,419],[238,58859,872],{"className":58860},[871],[238,58862,58864],{"className":58863},[399],[238,58865,58867],{"className":58866,"style":879},[403],[238,58868],{},[238,58870],{"className":58871,"style":430},[429],[238,58873,992],{"className":58874},[434],[238,58876],{"className":58877,"style":430},[429],[238,58879,58881,58884,58887,58890,58893,58896,58899],{"className":58880},[273],[238,58882],{"className":58883,"style":444},[277],[238,58885,335],{"className":58886},[381],[238,58888,740],{"className":58889},[282],[238,58891,354],{"className":58892},[451],[238,58894],{"className":58895,"style":455},[429],[238,58897,740],{"className":58898},[282],[238,58900,360],{"className":58901},[462],"，迭代为",[238,58904,58906],{"className":58905,"translate":242},[945],[238,58907,58909,58999],{"className":58908,"translate":242},[241],[238,58910,58912],{"className":58911},[246],[248,58913,58914],{"xmlns":250,"display":954},[252,58915,58916,58996],{},[255,58917,58918,58930,58932,58934,58936,58938,58940,58942,58944,58950,58952,58954,58956,58958,58960,58972,58974,58976,58978,58980,58982,58988,58990,58992,58994],{},[812,58919,58920,58922],{},[258,58921,4429],{},[255,58923,58924,58926,58928],{},[258,58925,3395],{},[332,58927,987],{},[738,58929,745],{},[332,58931,335],{"stretchy":334},[738,58933,745],{},[332,58935,360],{"stretchy":334},[332,58937,992],{},[738,58939,745],{},[332,58941,987],{},[738,58943,17938],{},[812,58945,58946,58948],{},[258,58947,4429],{},[258,58949,3395],{},[332,58951,335],{"stretchy":334},[738,58953,17558],{},[332,58955,360],{"stretchy":334},[332,58957,354],{"separator":269},[429,58959],{"width":10450},[812,58961,58962,58964],{},[258,58963,4429],{},[255,58965,58966,58968,58970],{},[258,58967,3395],{},[332,58969,987],{},[738,58971,745],{},[332,58973,335],{"stretchy":334},[738,58975,17558],{},[332,58977,360],{"stretchy":334},[332,58979,992],{},[738,58981,17938],{},[812,58983,58984,58986],{},[258,58985,4429],{},[258,58987,3395],{},[332,58989,335],{"stretchy":334},[738,58991,745],{},[332,58993,360],{"stretchy":334},[258,58995,3102],{"mathvariant":344},[262,58997,58998],{"encoding":264},"V_{k+1}(1)=1+0.9V_k(2),\n\\qquad\nV_{k+1}(2)=0.9V_k(1).",[238,59000,59002,59075,59093,59227],{"className":59001,"ariaHidden":269},[268],[238,59003,59005,59008,59057,59060,59063,59066,59069,59072],{"className":59004},[273],[238,59006],{"className":59007,"style":444},[277],[238,59009,59011,59014],{"className":59010},[282],[238,59012,4429],{"className":59013,"style":660},[282,283],[238,59015,59017],{"className":59016},[391],[238,59018,59020,59049],{"className":59019},[395,848],[238,59021,59023,59046],{"className":59022},[399],[238,59024,59026],{"className":59025,"style":3618},[403],[238,59027,59028,59031],{"style":13806},[238,59029],{"className":59030,"style":412},[411],[238,59032,59034],{"className":59033},[416,417,418,419],[238,59035,59037,59040,59043],{"className":59036},[282,419],[238,59038,3395],{"className":59039,"style":3520},[282,283,419],[238,59041,987],{"className":59042},[664,419],[238,59044,745],{"className":59045},[282,419],[238,59047,872],{"className":59048},[871],[238,59050,59052],{"className":59051},[399],[238,59053,59055],{"className":59054,"style":1193},[403],[238,59056],{},[238,59058,335],{"className":59059},[381],[238,59061,745],{"className":59062},[282],[238,59064,360],{"className":59065},[462],[238,59067],{"className":59068,"style":430},[429],[238,59070,992],{"className":59071},[434],[238,59073],{"className":59074,"style":430},[429],[238,59076,59078,59081,59084,59087,59090],{"className":59077},[273],[238,59079],{"className":59080,"style":18081},[277],[238,59082,745],{"className":59083},[282],[238,59085],{"className":59086,"style":660},[429],[238,59088,987],{"className":59089},[664],[238,59091],{"className":59092,"style":660},[429],[238,59094,59096,59099,59102,59142,59145,59148,59151,59154,59157,59160,59209,59212,59215,59218,59221,59224],{"className":59095},[273],[238,59097],{"className":59098,"style":444},[277],[238,59100,17938],{"className":59101},[282],[238,59103,59105,59108],{"className":59104},[282],[238,59106,4429],{"className":59107,"style":660},[282,283],[238,59109,59111],{"className":59110},[391],[238,59112,59114,59134],{"className":59113},[395,848],[238,59115,59117,59131],{"className":59116},[399],[238,59118,59120],{"className":59119,"style":3618},[403],[238,59121,59122,59125],{"style":13806},[238,59123],{"className":59124,"style":412},[411],[238,59126,59128],{"className":59127},[416,417,418,419],[238,59129,3395],{"className":59130,"style":3520},[282,283,419],[238,59132,872],{"className":59133},[871],[238,59135,59137],{"className":59136},[399],[238,59138,59140],{"className":59139,"style":879},[403],[238,59141],{},[238,59143,335],{"className":59144},[381],[238,59146,17558],{"className":59147},[282],[238,59149,360],{"className":59150},[462],[238,59152,354],{"className":59153},[451],[238,59155],{"className":59156,"style":10986},[429],[238,59158],{"className":59159,"style":455},[429],[238,59161,59163,59166],{"className":59162},[282],[238,59164,4429],{"className":59165,"style":660},[282,283],[238,59167,59169],{"className":59168},[391],[238,59170,59172,59201],{"className":59171},[395,848],[238,59173,59175,59198],{"className":59174},[399],[238,59176,59178],{"className":59177,"style":3618},[403],[238,59179,59180,59183],{"style":13806},[238,59181],{"className":59182,"style":412},[411],[238,59184,59186],{"className":59185},[416,417,418,419],[238,59187,59189,59192,59195],{"className":59188},[282,419],[238,59190,3395],{"className":59191,"style":3520},[282,283,419],[238,59193,987],{"className":59194},[664,419],[238,59196,745],{"className":59197},[282,419],[238,59199,872],{"className":59200},[871],[238,59202,59204],{"className":59203},[399],[238,59205,59207],{"className":59206,"style":1193},[403],[238,59208],{},[238,59210,335],{"className":59211},[381],[238,59213,17558],{"className":59214},[282],[238,59216,360],{"className":59217},[462],[238,59219],{"className":59220,"style":430},[429],[238,59222,992],{"className":59223},[434],[238,59225],{"className":59226,"style":430},[429],[238,59228,59230,59233,59236,59276,59279,59282,59285],{"className":59229},[273],[238,59231],{"className":59232,"style":444},[277],[238,59234,17938],{"className":59235},[282],[238,59237,59239,59242],{"className":59238},[282],[238,59240,4429],{"className":59241,"style":660},[282,283],[238,59243,59245],{"className":59244},[391],[238,59246,59248,59268],{"className":59247},[395,848],[238,59249,59251,59265],{"className":59250},[399],[238,59252,59254],{"className":59253,"style":3618},[403],[238,59255,59256,59259],{"style":13806},[238,59257],{"className":59258,"style":412},[411],[238,59260,59262],{"className":59261},[416,417,418,419],[238,59263,3395],{"className":59264,"style":3520},[282,283,419],[238,59266,872],{"className":59267},[871],[238,59269,59271],{"className":59270},[399],[238,59272,59274],{"className":59273,"style":879},[403],[238,59275],{},[238,59277,335],{"className":59278},[381],[238,59280,745],{"className":59281},[282],[238,59283,360],{"className":59284},[462],[238,59286,3102],{"className":59287},[282],[11,59289,59290],{},"前几步为",[238,59292,59294],{"className":59293,"translate":242},[945],[238,59295,59297,59485],{"className":59296,"translate":242},[241],[238,59298,59300],{"className":59299},[246],[248,59301,59302],{"xmlns":250,"display":954},[252,59303,59304,59482],{},[958,59305,59306,59340,59374,59410,59446],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,59307,59308,59318],{},[966,59309,59310],{},[969,59311,59312],{"scriptlevel":740,"displaystyle":269},[812,59313,59314,59316],{},[258,59315,4429],{},[738,59317,745],{},[966,59319,59320],{},[969,59321,59322],{"scriptlevel":740,"displaystyle":269},[255,59323,59324,59326,59328,59330,59332,59334,59336,59338],{},[255,59325],{},[332,59327,992],{},[332,59329,335],{"stretchy":334},[738,59331,745],{},[332,59333,354],{"separator":269},[738,59335,740],{},[332,59337,360],{"stretchy":334},[332,59339,354],{"separator":269},[963,59341,59342,59352],{},[966,59343,59344],{},[969,59345,59346],{"scriptlevel":740,"displaystyle":269},[812,59347,59348,59350],{},[258,59349,4429],{},[738,59351,17558],{},[966,59353,59354],{},[969,59355,59356],{"scriptlevel":740,"displaystyle":269},[255,59357,59358,59360,59362,59364,59366,59368,59370,59372],{},[255,59359],{},[332,59361,992],{},[332,59363,335],{"stretchy":334},[738,59365,745],{},[332,59367,354],{"separator":269},[738,59369,17938],{},[332,59371,360],{"stretchy":334},[332,59373,354],{"separator":269},[963,59375,59376,59387],{},[966,59377,59378],{},[969,59379,59380],{"scriptlevel":740,"displaystyle":269},[812,59381,59382,59384],{},[258,59383,4429],{},[738,59385,59386],{},"3",[966,59388,59389],{},[969,59390,59391],{"scriptlevel":740,"displaystyle":269},[255,59392,59393,59395,59397,59399,59402,59404,59406,59408],{},[255,59394],{},[332,59396,992],{},[332,59398,335],{"stretchy":334},[738,59400,59401],{},"1.81",[332,59403,354],{"separator":269},[738,59405,17938],{},[332,59407,360],{"stretchy":334},[332,59409,354],{"separator":269},[963,59411,59412,59423],{},[966,59413,59414],{},[969,59415,59416],{"scriptlevel":740,"displaystyle":269},[812,59417,59418,59420],{},[258,59419,4429],{},[738,59421,59422],{},"4",[966,59424,59425],{},[969,59426,59427],{"scriptlevel":740,"displaystyle":269},[255,59428,59429,59431,59433,59435,59437,59439,59442,59444],{},[255,59430],{},[332,59432,992],{},[332,59434,335],{"stretchy":334},[738,59436,59401],{},[332,59438,354],{"separator":269},[738,59440,59441],{},"1.629",[332,59443,360],{"stretchy":334},[332,59445,354],{"separator":269},[963,59447,59448,59459],{},[966,59449,59450],{},[969,59451,59452],{"scriptlevel":740,"displaystyle":269},[812,59453,59454,59456],{},[258,59455,4429],{},[738,59457,59458],{},"5",[966,59460,59461],{},[969,59462,59463],{"scriptlevel":740,"displaystyle":269},[255,59464,59465,59467,59469,59471,59474,59476,59478,59480],{},[255,59466],{},[332,59468,992],{},[332,59470,335],{"stretchy":334},[738,59472,59473],{},"2.4661",[332,59475,354],{"separator":269},[738,59477,59441],{},[332,59479,360],{"stretchy":334},[258,59481,3102],{"mathvariant":344},[262,59483,59484],{"encoding":264},"\\begin{aligned}\nV_1&=(1,0),\\\\\nV_2&=(1,0.9),\\\\\nV_3&=(1.81,0.9),\\\\\nV_4&=(1.81,1.629),\\\\\nV_5&=(2.4661,1.629).\n\\end{aligned}",[238,59486,59488],{"className":59487,"ariaHidden":269},[268],[238,59489,59491,59495],{"className":59490},[273],[238,59492],{"className":59493,"style":59494},[277],"height:7.2em;vertical-align:-3.35em;",[238,59496,59498],{"className":59497},[282],[238,59499,59501,59770],{"className":59500},[958],[238,59502,59504],{"className":59503},[1117],[238,59505,59507,59761],{"className":59506},[395,848],[238,59508,59510,59758],{"className":59509},[399],[238,59511,59514,59563,59612,59660,59709],{"className":59512,"style":59513},[403],"height:3.85em;",[238,59515,59517,59520],{"style":59516},"top:-6.01em;",[238,59518],{"className":59519,"style":1134},[411],[238,59521,59523],{"className":59522},[282],[238,59524,59526,59529],{"className":59525},[282],[238,59527,4429],{"className":59528,"style":660},[282,283],[238,59530,59532],{"className":59531},[391],[238,59533,59535,59555],{"className":59534},[395,848],[238,59536,59538,59552],{"className":59537},[399],[238,59539,59541],{"className":59540,"style":1162},[403],[238,59542,59543,59546],{"style":13806},[238,59544],{"className":59545,"style":412},[411],[238,59547,59549],{"className":59548},[416,417,418,419],[238,59550,745],{"className":59551},[282,419],[238,59553,872],{"className":59554},[871],[238,59556,59558],{"className":59557},[399],[238,59559,59561],{"className":59560,"style":879},[403],[238,59562],{},[238,59564,59566,59569],{"style":59565},"top:-4.51em;",[238,59567],{"className":59568,"style":1134},[411],[238,59570,59572],{"className":59571},[282],[238,59573,59575,59578],{"className":59574},[282],[238,59576,4429],{"className":59577,"style":660},[282,283],[238,59579,59581],{"className":59580},[391],[238,59582,59584,59604],{"className":59583},[395,848],[238,59585,59587,59601],{"className":59586},[399],[238,59588,59590],{"className":59589,"style":1162},[403],[238,59591,59592,59595],{"style":13806},[238,59593],{"className":59594,"style":412},[411],[238,59596,59598],{"className":59597},[416,417,418,419],[238,59599,17558],{"className":59600},[282,419],[238,59602,872],{"className":59603},[871],[238,59605,59607],{"className":59606},[399],[238,59608,59610],{"className":59609,"style":879},[403],[238,59611],{},[238,59613,59614,59617],{"style":1130},[238,59615],{"className":59616,"style":1134},[411],[238,59618,59620],{"className":59619},[282],[238,59621,59623,59626],{"className":59622},[282],[238,59624,4429],{"className":59625,"style":660},[282,283],[238,59627,59629],{"className":59628},[391],[238,59630,59632,59652],{"className":59631},[395,848],[238,59633,59635,59649],{"className":59634},[399],[238,59636,59638],{"className":59637,"style":1162},[403],[238,59639,59640,59643],{"style":13806},[238,59641],{"className":59642,"style":412},[411],[238,59644,59646],{"className":59645},[416,417,418,419],[238,59647,59386],{"className":59648},[282,419],[238,59650,872],{"className":59651},[871],[238,59653,59655],{"className":59654},[399],[238,59656,59658],{"className":59657,"style":879},[403],[238,59659],{},[238,59661,59663,59666],{"style":59662},"top:-1.51em;",[238,59664],{"className":59665,"style":1134},[411],[238,59667,59669],{"className":59668},[282],[238,59670,59672,59675],{"className":59671},[282],[238,59673,4429],{"className":59674,"style":660},[282,283],[238,59676,59678],{"className":59677},[391],[238,59679,59681,59701],{"className":59680},[395,848],[238,59682,59684,59698],{"className":59683},[399],[238,59685,59687],{"className":59686,"style":1162},[403],[238,59688,59689,59692],{"style":13806},[238,59690],{"className":59691,"style":412},[411],[238,59693,59695],{"className":59694},[416,417,418,419],[238,59696,59422],{"className":59697},[282,419],[238,59699,872],{"className":59700},[871],[238,59702,59704],{"className":59703},[399],[238,59705,59707],{"className":59706,"style":879},[403],[238,59708],{},[238,59710,59712,59715],{"style":59711},"top:-0.01em;",[238,59713],{"className":59714,"style":1134},[411],[238,59716,59718],{"className":59717},[282],[238,59719,59721,59724],{"className":59720},[282],[238,59722,4429],{"className":59723,"style":660},[282,283],[238,59725,59727],{"className":59726},[391],[238,59728,59730,59750],{"className":59729},[395,848],[238,59731,59733,59747],{"className":59732},[399],[238,59734,59736],{"className":59735,"style":1162},[403],[238,59737,59738,59741],{"style":13806},[238,59739],{"className":59740,"style":412},[411],[238,59742,59744],{"className":59743},[416,417,418,419],[238,59745,59458],{"className":59746},[282,419],[238,59748,872],{"className":59749},[871],[238,59751,59753],{"className":59752},[399],[238,59754,59756],{"className":59755,"style":879},[403],[238,59757],{},[238,59759,872],{"className":59760},[871],[238,59762,59764],{"className":59763},[399],[238,59765,59768],{"className":59766,"style":59767},[403],"height:3.35em;",[238,59769],{},[238,59771,59773],{"className":59772},[1422],[238,59774,59776,59990],{"className":59775},[395,848],[238,59777,59779,59987],{"className":59778},[399],[238,59780,59782,59823,59864,59905,59946],{"className":59781,"style":59513},[403],[238,59783,59784,59787],{"style":59516},[238,59785],{"className":59786,"style":1134},[411],[238,59788,59790,59793,59796,59799,59802,59805,59808,59811,59814,59817,59820],{"className":59789},[282],[238,59791],{"className":59792},[282],[238,59794],{"className":59795,"style":430},[429],[238,59797,992],{"className":59798},[434],[238,59800],{"className":59801,"style":430},[429],[238,59803,335],{"className":59804},[381],[238,59806,745],{"className":59807},[282],[238,59809,354],{"className":59810},[451],[238,59812],{"className":59813,"style":455},[429],[238,59815,740],{"className":59816},[282],[238,59818,360],{"className":59819},[462],[238,59821,354],{"className":59822},[451],[238,59824,59825,59828],{"style":59565},[238,59826],{"className":59827,"style":1134},[411],[238,59829,59831,59834,59837,59840,59843,59846,59849,59852,59855,59858,59861],{"className":59830},[282],[238,59832],{"className":59833},[282],[238,59835],{"className":59836,"style":430},[429],[238,59838,992],{"className":59839},[434],[238,59841],{"className":59842,"style":430},[429],[238,59844,335],{"className":59845},[381],[238,59847,745],{"className":59848},[282],[238,59850,354],{"className":59851},[451],[238,59853],{"className":59854,"style":455},[429],[238,59856,17938],{"className":59857},[282],[238,59859,360],{"className":59860},[462],[238,59862,354],{"className":59863},[451],[238,59865,59866,59869],{"style":1130},[238,59867],{"className":59868,"style":1134},[411],[238,59870,59872,59875,59878,59881,59884,59887,59890,59893,59896,59899,59902],{"className":59871},[282],[238,59873],{"className":59874},[282],[238,59876],{"className":59877,"style":430},[429],[238,59879,992],{"className":59880},[434],[238,59882],{"className":59883,"style":430},[429],[238,59885,335],{"className":59886},[381],[238,59888,59401],{"className":59889},[282],[238,59891,354],{"className":59892},[451],[238,59894],{"className":59895,"style":455},[429],[238,59897,17938],{"className":59898},[282],[238,59900,360],{"className":59901},[462],[238,59903,354],{"className":59904},[451],[238,59906,59907,59910],{"style":59662},[238,59908],{"className":59909,"style":1134},[411],[238,59911,59913,59916,59919,59922,59925,59928,59931,59934,59937,59940,59943],{"className":59912},[282],[238,59914],{"className":59915},[282],[238,59917],{"className":59918,"style":430},[429],[238,59920,992],{"className":59921},[434],[238,59923],{"className":59924,"style":430},[429],[238,59926,335],{"className":59927},[381],[238,59929,59401],{"className":59930},[282],[238,59932,354],{"className":59933},[451],[238,59935],{"className":59936,"style":455},[429],[238,59938,59441],{"className":59939},[282],[238,59941,360],{"className":59942},[462],[238,59944,354],{"className":59945},[451],[238,59947,59948,59951],{"style":59711},[238,59949],{"className":59950,"style":1134},[411],[238,59952,59954,59957,59960,59963,59966,59969,59972,59975,59978,59981,59984],{"className":59953},[282],[238,59955],{"className":59956},[282],[238,59958],{"className":59959,"style":430},[429],[238,59961,992],{"className":59962},[434],[238,59964],{"className":59965,"style":430},[429],[238,59967,335],{"className":59968},[381],[238,59970,59473],{"className":59971},[282],[238,59973,354],{"className":59974},[451],[238,59976],{"className":59977,"style":455},[429],[238,59979,59441],{"className":59980},[282],[238,59982,360],{"className":59983},[462],[238,59985,3102],{"className":59986},[282],[238,59988,872],{"className":59989},[871],[238,59991,59993],{"className":59992},[399],[238,59994,59996],{"className":59995,"style":59767},[403],[238,59997],{},[11,59999,60000,60001,941],{},"可以看到，奇数步与偶数步分别单调地逼近极限。事实上，对 ",[238,60002,60004,60022],{"className":60003,"translate":242},[241],[238,60005,60007],{"className":60006},[246],[248,60008,60009],{"xmlns":250},[252,60010,60011,60019],{},[255,60012,60013,60015,60017],{},[258,60014,1947],{},[332,60016,25372],{},[738,60018,740],{},[262,60020,60021],{"encoding":264},"m\\ge0",[238,60023,60025,60044],{"className":60024,"ariaHidden":269},[268],[238,60026,60028,60032,60035,60038,60041],{"className":60027},[273],[238,60029],{"className":60030,"style":60031},[277],"height:0.7719em;vertical-align:-0.136em;",[238,60033,1947],{"className":60034},[282,283],[238,60036],{"className":60037,"style":430},[429],[238,60039,25372],{"className":60040},[434],[238,60042],{"className":60043,"style":430},[429],[238,60045,60047,60050],{"className":60046},[273],[238,60048],{"className":60049,"style":2672},[277],[238,60051,740],{"className":60052},[282],[238,60054,60056],{"className":60055,"translate":242},[945],[238,60057,60059,60179],{"className":60058,"translate":242},[241],[238,60060,60062],{"className":60061},[246],[248,60063,60064],{"xmlns":250,"display":954},[252,60065,60066,60176],{},[255,60067,60068,60082,60084,60086,60088,60090,60104,60114,60116,60118,60132,60134,60136,60138,60140,60160,60174],{},[812,60069,60070,60072],{},[258,60071,4429],{},[255,60073,60074,60076,60078,60080],{},[738,60075,17558],{},[258,60077,1947],{},[332,60079,987],{},[738,60081,745],{},[332,60083,335],{"stretchy":334},[738,60085,745],{},[332,60087,360],{"stretchy":334},[332,60089,992],{},[3386,60091,60092,60094,60102],{},[332,60093,3390],{},[255,60095,60096,60098,60100],{},[258,60097,13645],{},[332,60099,992],{},[738,60101,740],{},[258,60103,1947],{},[337,60105,60106,60108],{},[738,60107,17938],{},[255,60109,60110,60112],{},[738,60111,17558],{},[258,60113,13645],{},[332,60115,354],{"separator":269},[429,60117],{"width":10450},[812,60119,60120,60122],{},[258,60121,4429],{},[255,60123,60124,60126,60128,60130],{},[738,60125,17558],{},[258,60127,1947],{},[332,60129,987],{},[738,60131,745],{},[332,60133,335],{"stretchy":334},[738,60135,17558],{},[332,60137,360],{"stretchy":334},[332,60139,992],{},[3386,60141,60142,60144,60152],{},[332,60143,3390],{},[255,60145,60146,60148,60150],{},[258,60147,13645],{},[332,60149,992],{},[738,60151,740],{},[255,60153,60154,60156,60158],{},[258,60155,1947],{},[332,60157,5577],{},[738,60159,745],{},[337,60161,60162,60164],{},[738,60163,17938],{},[255,60165,60166,60168,60170,60172],{},[738,60167,17558],{},[258,60169,13645],{},[332,60171,987],{},[738,60173,745],{},[258,60175,3102],{"mathvariant":344},[262,60177,60178],{"encoding":264},"V_{2m+1}(1)=\\sum_{j=0}^{m}0.9^{2j},\n\\qquad\nV_{2m+1}(2)=\\sum_{j=0}^{m-1}0.9^{2j+1}.",[238,60180,60182,60258,60452],{"className":60181,"ariaHidden":269},[268],[238,60183,60185,60188,60240,60243,60246,60249,60252,60255],{"className":60184},[273],[238,60186],{"className":60187,"style":444},[277],[238,60189,60191,60194],{"className":60190},[282],[238,60192,4429],{"className":60193,"style":660},[282,283],[238,60195,60197],{"className":60196},[391],[238,60198,60200,60232],{"className":60199},[395,848],[238,60201,60203,60229],{"className":60202},[399],[238,60204,60206],{"className":60205,"style":1162},[403],[238,60207,60208,60211],{"style":13806},[238,60209],{"className":60210,"style":412},[411],[238,60212,60214],{"className":60213},[416,417,418,419],[238,60215,60217,60220,60223,60226],{"className":60216},[282,419],[238,60218,17558],{"className":60219},[282,419],[238,60221,1947],{"className":60222},[282,283,419],[238,60224,987],{"className":60225},[664,419],[238,60227,745],{"className":60228},[282,419],[238,60230,872],{"className":60231},[871],[238,60233,60235],{"className":60234},[399],[238,60236,60238],{"className":60237,"style":1193},[403],[238,60239],{},[238,60241,335],{"className":60242},[381],[238,60244,745],{"className":60245},[282],[238,60247,360],{"className":60248},[462],[238,60250],{"className":60251,"style":430},[429],[238,60253,992],{"className":60254},[434],[238,60256],{"className":60257,"style":430},[429],[238,60259,60261,60265,60332,60335,60338,60373,60376,60379,60382,60434,60437,60440,60443,60446,60449],{"className":60260},[273],[238,60262],{"className":60263,"style":60264},[277],"height:3.0652em;vertical-align:-1.4138em;",[238,60266,60268],{"className":60267},[1943,3493],[238,60269,60271,60324],{"className":60270},[395,848],[238,60272,60274,60321],{"className":60273},[399],[238,60275,60277,60297,60307],{"className":60276,"style":3503},[403],[238,60278,60279,60282],{"style":14125},[238,60280],{"className":60281,"style":3510},[411],[238,60283,60285],{"className":60284},[416,417,418,419],[238,60286,60288,60291,60294],{"className":60287},[282,419],[238,60289,13645],{"className":60290,"style":13930},[282,283,419],[238,60292,992],{"className":60293},[434,419],[238,60295,740],{"className":60296},[282,419],[238,60298,60299,60302],{"style":3529},[238,60300],{"className":60301,"style":3510},[411],[238,60303,60304],{},[238,60305,3390],{"className":60306},[1943,3538,3539],[238,60308,60309,60312],{"style":3542},[238,60310],{"className":60311,"style":3510},[411],[238,60313,60315],{"className":60314},[416,417,418,419],[238,60316,60318],{"className":60317},[282,419],[238,60319,1947],{"className":60320},[282,283,419],[238,60322,872],{"className":60323},[871],[238,60325,60327],{"className":60326},[399],[238,60328,60330],{"className":60329,"style":14157},[403],[238,60331],{},[238,60333],{"className":60334,"style":455},[429],[238,60336,29065],{"className":60337},[282],[238,60339,60341,60344],{"className":60340},[282],[238,60342,29072],{"className":60343},[282],[238,60345,60347],{"className":60346},[391],[238,60348,60350],{"className":60349},[395],[238,60351,60353],{"className":60352},[399],[238,60354,60356],{"className":60355,"style":55692},[403],[238,60357,60358,60361],{"style":1226},[238,60359],{"className":60360,"style":412},[411],[238,60362,60364],{"className":60363},[416,417,418,419],[238,60365,60367,60370],{"className":60366},[282,419],[238,60368,17558],{"className":60369},[282,419],[238,60371,13645],{"className":60372,"style":13930},[282,283,419],[238,60374,354],{"className":60375},[451],[238,60377],{"className":60378,"style":10986},[429],[238,60380],{"className":60381,"style":455},[429],[238,60383,60385,60388],{"className":60384},[282],[238,60386,4429],{"className":60387,"style":660},[282,283],[238,60389,60391],{"className":60390},[391],[238,60392,60394,60426],{"className":60393},[395,848],[238,60395,60397,60423],{"className":60396},[399],[238,60398,60400],{"className":60399,"style":1162},[403],[238,60401,60402,60405],{"style":13806},[238,60403],{"className":60404,"style":412},[411],[238,60406,60408],{"className":60407},[416,417,418,419],[238,60409,60411,60414,60417,60420],{"className":60410},[282,419],[238,60412,17558],{"className":60413},[282,419],[238,60415,1947],{"className":60416},[282,283,419],[238,60418,987],{"className":60419},[664,419],[238,60421,745],{"className":60422},[282,419],[238,60424,872],{"className":60425},[871],[238,60427,60429],{"className":60428},[399],[238,60430,60432],{"className":60431,"style":1193},[403],[238,60433],{},[238,60435,335],{"className":60436},[381],[238,60438,17558],{"className":60439},[282],[238,60441,360],{"className":60442},[462],[238,60444],{"className":60445,"style":430},[429],[238,60447,992],{"className":60448},[434],[238,60450],{"className":60451,"style":430},[429],[238,60453,60455,60459,60532,60535,60538,60579],{"className":60454},[273],[238,60456],{"className":60457,"style":60458},[277],"height:3.2149em;vertical-align:-1.4138em;",[238,60460,60462],{"className":60461},[1943,3493],[238,60463,60465,60524],{"className":60464},[395,848],[238,60466,60468,60521],{"className":60467},[399],[238,60469,60471,60491,60501],{"className":60470,"style":20473},[403],[238,60472,60473,60476],{"style":14125},[238,60474],{"className":60475,"style":3510},[411],[238,60477,60479],{"className":60478},[416,417,418,419],[238,60480,60482,60485,60488],{"className":60481},[282,419],[238,60483,13645],{"className":60484,"style":13930},[282,283,419],[238,60486,992],{"className":60487},[434,419],[238,60489,740],{"className":60490},[282,419],[238,60492,60493,60496],{"style":3529},[238,60494],{"className":60495,"style":3510},[411],[238,60497,60498],{},[238,60499,3390],{"className":60500},[1943,3538,3539],[238,60502,60503,60506],{"style":3542},[238,60504],{"className":60505,"style":3510},[411],[238,60507,60509],{"className":60508},[416,417,418,419],[238,60510,60512,60515,60518],{"className":60511},[282,419],[238,60513,1947],{"className":60514},[282,283,419],[238,60516,5577],{"className":60517},[664,419],[238,60519,745],{"className":60520},[282,419],[238,60522,872],{"className":60523},[871],[238,60525,60527],{"className":60526},[399],[238,60528,60530],{"className":60529,"style":14157},[403],[238,60531],{},[238,60533],{"className":60534,"style":455},[429],[238,60536,29065],{"className":60537},[282],[238,60539,60541,60544],{"className":60540},[282],[238,60542,29072],{"className":60543},[282],[238,60545,60547],{"className":60546},[391],[238,60548,60550],{"className":60549},[395],[238,60551,60553],{"className":60552},[399],[238,60554,60556],{"className":60555,"style":55692},[403],[238,60557,60558,60561],{"style":1226},[238,60559],{"className":60560,"style":412},[411],[238,60562,60564],{"className":60563},[416,417,418,419],[238,60565,60567,60570,60573,60576],{"className":60566},[282,419],[238,60568,17558],{"className":60569},[282,419],[238,60571,13645],{"className":60572,"style":13930},[282,283,419],[238,60574,987],{"className":60575},[664,419],[238,60577,745],{"className":60578},[282,419],[238,60580,3102],{"className":60581},[282],[11,60583,60584,60585,60636],{},"令 ",[238,60586,60588,60606],{"className":60587,"translate":242},[241],[238,60589,60591],{"className":60590},[246],[248,60592,60593],{"xmlns":250},[252,60594,60595,60603],{},[255,60596,60597,60599,60601],{},[258,60598,1947],{},[332,60600,617],{},[258,60602,1895],{"mathvariant":344},[262,60604,60605],{"encoding":264},"m\\to\\infty",[238,60607,60609,60627],{"className":60608,"ariaHidden":269},[268],[238,60610,60612,60615,60618,60621,60624],{"className":60611},[273],[238,60613],{"className":60614,"style":489},[277],[238,60616,1947],{"className":60617},[282,283],[238,60619],{"className":60620,"style":430},[429],[238,60622,617],{"className":60623},[434],[238,60625],{"className":60626,"style":430},[429],[238,60628,60630,60633],{"className":60629},[273],[238,60631],{"className":60632,"style":489},[277],[238,60634,1895],{"className":60635},[282],"，得到",[238,60638,60640],{"className":60639,"translate":242},[945],[238,60641,60643,60719],{"className":60642,"translate":242},[241],[238,60644,60646],{"className":60645},[246],[248,60647,60648],{"xmlns":250,"display":954},[252,60649,60650,60716],{},[255,60651,60652,60654,60656,60658,60660,60662,60678,60680,60682,60684,60686,60688,60690,60692,60694,60696,60712,60714],{},[258,60653,4429],{},[332,60655,335],{"stretchy":334},[738,60657,745],{},[332,60659,360],{"stretchy":334},[332,60661,992],{},[5561,60663,60664,60666],{},[738,60665,745],{},[255,60667,60668,60670,60672],{},[738,60669,745],{},[332,60671,5577],{},[337,60673,60674,60676],{},[738,60675,17938],{},[738,60677,17558],{},[332,60679,992],{},[738,60681,18351],{},[332,60683,354],{"separator":269},[429,60685],{"width":10450},[258,60687,4429],{},[332,60689,335],{"stretchy":334},[738,60691,17558],{},[332,60693,360],{"stretchy":334},[332,60695,992],{},[5561,60697,60698,60700],{},[738,60699,17938],{},[255,60701,60702,60704,60706],{},[738,60703,745],{},[332,60705,5577],{},[337,60707,60708,60710],{},[738,60709,17938],{},[738,60711,17558],{},[332,60713,992],{},[738,60715,18380],{},[262,60717,60718],{"encoding":264},"V(1)=\\frac{1}{1-0.9^2}=5.2632,\n\\qquad\nV(2)=\\frac{0.9}{1-0.9^2}=4.7368.",[238,60720,60722,60749,60869,60908,61026],{"className":60721,"ariaHidden":269},[268],[238,60723,60725,60728,60731,60734,60737,60740,60743,60746],{"className":60724},[273],[238,60726],{"className":60727,"style":444},[277],[238,60729,4429],{"className":60730,"style":660},[282,283],[238,60732,335],{"className":60733},[381],[238,60735,745],{"className":60736},[282],[238,60738,360],{"className":60739},[462],[238,60741],{"className":60742,"style":430},[429],[238,60744,992],{"className":60745},[434],[238,60747],{"className":60748,"style":430},[429],[238,60750,60752,60755,60860,60863,60866],{"className":60751},[273],[238,60753],{"className":60754,"style":18420},[277],[238,60756,60758,60761,60857],{"className":60757},[282],[238,60759],{"className":60760},[381,5662],[238,60762,60764],{"className":60763},[5561],[238,60765,60767,60849],{"className":60766},[395,848],[238,60768,60770,60846],{"className":60769},[399],[238,60771,60773,60827,60835],{"className":60772,"style":18439},[403],[238,60774,60775,60778],{"style":5678},[238,60776],{"className":60777,"style":1134},[411],[238,60779,60781,60784,60787,60790,60793,60796],{"className":60780},[282],[238,60782,745],{"className":60783},[282],[238,60785],{"className":60786,"style":660},[429],[238,60788,5577],{"className":60789},[664],[238,60791],{"className":60792,"style":660},[429],[238,60794,29065],{"className":60795},[282],[238,60797,60799,60802],{"className":60798},[282],[238,60800,29072],{"className":60801},[282],[238,60803,60805],{"className":60804},[391],[238,60806,60808],{"className":60807},[395],[238,60809,60811],{"className":60810},[399],[238,60812,60815],{"className":60813,"style":60814},[403],"height:0.7401em;",[238,60816,60818,60821],{"style":60817},"top:-2.989em;margin-right:0.05em;",[238,60819],{"className":60820,"style":412},[411],[238,60822,60824],{"className":60823},[416,417,418,419],[238,60825,17558],{"className":60826},[282,419],[238,60828,60829,60832],{"style":5702},[238,60830],{"className":60831,"style":1134},[411],[238,60833],{"className":60834,"style":5710},[5709],[238,60836,60837,60840],{"style":5713},[238,60838],{"className":60839,"style":1134},[411],[238,60841,60843],{"className":60842},[282],[238,60844,745],{"className":60845},[282],[238,60847,872],{"className":60848},[871],[238,60850,60852],{"className":60851},[399],[238,60853,60855],{"className":60854,"style":18491},[403],[238,60856],{},[238,60858],{"className":60859},[462,5662],[238,60861],{"className":60862,"style":430},[429],[238,60864,992],{"className":60865},[434],[238,60867],{"className":60868,"style":430},[429],[238,60870,60872,60875,60878,60881,60884,60887,60890,60893,60896,60899,60902,60905],{"className":60871},[273],[238,60873],{"className":60874,"style":444},[277],[238,60876,18351],{"className":60877},[282],[238,60879,354],{"className":60880},[451],[238,60882],{"className":60883,"style":10986},[429],[238,60885],{"className":60886,"style":455},[429],[238,60888,4429],{"className":60889,"style":660},[282,283],[238,60891,335],{"className":60892},[381],[238,60894,17558],{"className":60895},[282],[238,60897,360],{"className":60898},[462],[238,60900],{"className":60901,"style":430},[429],[238,60903,992],{"className":60904},[434],[238,60906],{"className":60907,"style":430},[429],[238,60909,60911,60914,61017,61020,61023],{"className":60910},[273],[238,60912],{"className":60913,"style":18420},[277],[238,60915,60917,60920,61014],{"className":60916},[282],[238,60918],{"className":60919},[381,5662],[238,60921,60923],{"className":60922},[5561],[238,60924,60926,61006],{"className":60925},[395,848],[238,60927,60929,61003],{"className":60928},[399],[238,60930,60932,60984,60992],{"className":60931,"style":18439},[403],[238,60933,60934,60937],{"style":5678},[238,60935],{"className":60936,"style":1134},[411],[238,60938,60940,60943,60946,60949,60952,60955],{"className":60939},[282],[238,60941,745],{"className":60942},[282],[238,60944],{"className":60945,"style":660},[429],[238,60947,5577],{"className":60948},[664],[238,60950],{"className":60951,"style":660},[429],[238,60953,29065],{"className":60954},[282],[238,60956,60958,60961],{"className":60957},[282],[238,60959,29072],{"className":60960},[282],[238,60962,60964],{"className":60963},[391],[238,60965,60967],{"className":60966},[395],[238,60968,60970],{"className":60969},[399],[238,60971,60973],{"className":60972,"style":60814},[403],[238,60974,60975,60978],{"style":60817},[238,60976],{"className":60977,"style":412},[411],[238,60979,60981],{"className":60980},[416,417,418,419],[238,60982,17558],{"className":60983},[282,419],[238,60985,60986,60989],{"style":5702},[238,60987],{"className":60988,"style":1134},[411],[238,60990],{"className":60991,"style":5710},[5709],[238,60993,60994,60997],{"style":5713},[238,60995],{"className":60996,"style":1134},[411],[238,60998,61000],{"className":60999},[282],[238,61001,17938],{"className":61002},[282],[238,61004,872],{"className":61005},[871],[238,61007,61009],{"className":61008},[399],[238,61010,61012],{"className":61011,"style":18491},[403],[238,61013],{},[238,61015],{"className":61016},[462,5662],[238,61018],{"className":61019,"style":430},[429],[238,61021,992],{"className":61022},[434],[238,61024],{"className":61025,"style":430},[429],[238,61027,61029,61032],{"className":61028},[273],[238,61030],{"className":61031,"style":2672},[277],[238,61033,18380],{"className":61034},[282],[11,61036,61037,61038,61122,61123,61151],{},"这个例子的结构值得注意。若只看奇数步或偶数步的子序列，误差按 ",[238,61039,61041,61063],{"className":61040,"translate":242},[241],[238,61042,61044],{"className":61043},[246],[248,61045,61046],{"xmlns":250},[252,61047,61048,61060],{},[255,61049,61050,61056,61058],{},[337,61051,61052,61054],{},[738,61053,17938],{},[738,61055,17558],{},[332,61057,992],{},[738,61059,18221],{},[262,61061,61062],{"encoding":264},"0.9^2=0.81",[238,61064,61066,61113],{"className":61065,"ariaHidden":269},[268],[238,61067,61069,61072,61075,61104,61107,61110],{"className":61068},[273],[238,61070],{"className":61071,"style":19909},[277],[238,61073,29065],{"className":61074},[282],[238,61076,61078,61081],{"className":61077},[282],[238,61079,29072],{"className":61080},[282],[238,61082,61084],{"className":61083},[391],[238,61085,61087],{"className":61086},[395],[238,61088,61090],{"className":61089},[399],[238,61091,61093],{"className":61092,"style":19909},[403],[238,61094,61095,61098],{"style":407},[238,61096],{"className":61097,"style":412},[411],[238,61099,61101],{"className":61100},[416,417,418,419],[238,61102,17558],{"className":61103},[282,419],[238,61105],{"className":61106,"style":430},[429],[238,61108,992],{"className":61109},[434],[238,61111],{"className":61112,"style":430},[429],[238,61114,61116,61119],{"className":61115},[273],[238,61117],{"className":61118,"style":2672},[277],[238,61120,18221],{"className":61121},[282]," 的比例衰减，因为状态每两步回到自身。但逐步的 sup 范数误差仍然按 ",[238,61124,61126,61139],{"className":61125,"translate":242},[241],[238,61127,61129],{"className":61128},[246],[248,61130,61131],{"xmlns":250},[252,61132,61133,61137],{},[255,61134,61135],{},[738,61136,17938],{},[262,61138,17938],{"encoding":264},[238,61140,61142],{"className":61141,"ariaHidden":269},[268],[238,61143,61145,61148],{"className":61144},[273],[238,61146],{"className":61147,"style":2672},[277],[238,61149,17938],{"className":61150},[282]," 衰减，只是最大误差所在的分量在两步之间交替。一般理论给出的是逐步最坏情形下的统一上界，具体问题的子序列可能具有更快的收敛速度。",[11,61153,16622,61154,61156,61157,49615],{},[357,61155,53572],{"href":53571},"，初始误差为 ",[238,61158,61160,61198],{"className":61159,"translate":242},[241],[238,61161,61163],{"className":61162},[246],[248,61164,61165],{"xmlns":250},[252,61166,61167,61195],{},[255,61168,61169,61171,61177,61179,61185,61191,61193],{},[258,61170,15588],{"mathvariant":344},[812,61172,61173,61175],{},[258,61174,4429],{},[738,61176,740],{},[332,61178,5577],{},[812,61180,61181,61183],{},[258,61182,4429],{},[258,61184,1895],{"mathvariant":344},[812,61186,61187,61189],{},[258,61188,15588],{"mathvariant":344},[258,61190,1895],{"mathvariant":344},[332,61192,992],{},[738,61194,18351],{},[262,61196,61197],{"encoding":264},"\\|V_0-V_\\infty\\|_\\infty=5.2632",[238,61199,61201,61259,61354],{"className":61200,"ariaHidden":269},[268],[238,61202,61204,61207,61210,61250,61253,61256],{"className":61203},[273],[238,61205],{"className":61206,"style":444},[277],[238,61208,15588],{"className":61209},[282],[238,61211,61213,61216],{"className":61212},[282],[238,61214,4429],{"className":61215,"style":660},[282,283],[238,61217,61219],{"className":61218},[391],[238,61220,61222,61242],{"className":61221},[395,848],[238,61223,61225,61239],{"className":61224},[399],[238,61226,61228],{"className":61227,"style":1162},[403],[238,61229,61230,61233],{"style":13806},[238,61231],{"className":61232,"style":412},[411],[238,61234,61236],{"className":61235},[416,417,418,419],[238,61237,740],{"className":61238},[282,419],[238,61240,872],{"className":61241},[871],[238,61243,61245],{"className":61244},[399],[238,61246,61248],{"className":61247,"style":879},[403],[238,61249],{},[238,61251],{"className":61252,"style":660},[429],[238,61254,5577],{"className":61255},[664],[238,61257],{"className":61258,"style":660},[429],[238,61260,61262,61265,61305,61345,61348,61351],{"className":61261},[273],[238,61263],{"className":61264,"style":444},[277],[238,61266,61268,61271],{"className":61267},[282],[238,61269,4429],{"className":61270,"style":660},[282,283],[238,61272,61274],{"className":61273},[391],[238,61275,61277,61297],{"className":61276},[395,848],[238,61278,61280,61294],{"className":61279},[399],[238,61281,61283],{"className":61282,"style":1927},[403],[238,61284,61285,61288],{"style":13806},[238,61286],{"className":61287,"style":412},[411],[238,61289,61291],{"className":61290},[416,417,418,419],[238,61292,1895],{"className":61293},[282,419],[238,61295,872],{"className":61296},[871],[238,61298,61300],{"className":61299},[399],[238,61301,61303],{"className":61302,"style":879},[403],[238,61304],{},[238,61306,61308,61311],{"className":61307},[282],[238,61309,15588],{"className":61310},[282],[238,61312,61314],{"className":61313},[391],[238,61315,61317,61337],{"className":61316},[395,848],[238,61318,61320,61334],{"className":61319},[399],[238,61321,61323],{"className":61322,"style":1927},[403],[238,61324,61325,61328],{"style":1165},[238,61326],{"className":61327,"style":412},[411],[238,61329,61331],{"className":61330},[416,417,418,419],[238,61332,1895],{"className":61333},[282,419],[238,61335,872],{"className":61336},[871],[238,61338,61340],{"className":61339},[399],[238,61341,61343],{"className":61342,"style":879},[403],[238,61344],{},[238,61346],{"className":61347,"style":430},[429],[238,61349,992],{"className":61350},[434],[238,61352],{"className":61353,"style":430},[429],[238,61355,61357,61360],{"className":61356},[273],[238,61358],{"className":61359,"style":2672},[277],[238,61361,18351],{"className":61362},[282],[238,61364,61366],{"className":61365,"translate":242},[945],[238,61367,61369,61417],{"className":61368,"translate":242},[241],[238,61370,61372],{"className":61371},[246],[248,61373,61374],{"xmlns":250,"display":954},[252,61375,61376,61414],{},[255,61377,61378,61380,61386,61388,61394,61400,61402,61404,61406,61412],{},[258,61379,15588],{"mathvariant":344},[812,61381,61382,61384],{},[258,61383,4429],{},[258,61385,3395],{},[332,61387,5577],{},[812,61389,61390,61392],{},[258,61391,4429],{},[258,61393,1895],{"mathvariant":344},[812,61395,61396,61398],{},[258,61397,15588],{"mathvariant":344},[258,61399,1895],{"mathvariant":344},[332,61401,2023],{},[738,61403,18351],{},[332,61405,608],{},[337,61407,61408,61410],{},[738,61409,17938],{},[258,61411,3395],{},[258,61413,3102],{"mathvariant":344},[262,61415,61416],{"encoding":264},"\\|V_k-V_\\infty\\|_\\infty\\le5.2632\\times0.9^k.",[238,61418,61420,61478,61573,61591],{"className":61419,"ariaHidden":269},[268],[238,61421,61423,61426,61429,61469,61472,61475],{"className":61422},[273],[238,61424],{"className":61425,"style":444},[277],[238,61427,15588],{"className":61428},[282],[238,61430,61432,61435],{"className":61431},[282],[238,61433,4429],{"className":61434,"style":660},[282,283],[238,61436,61438],{"className":61437},[391],[238,61439,61441,61461],{"className":61440},[395,848],[238,61442,61444,61458],{"className":61443},[399],[238,61445,61447],{"className":61446,"style":3618},[403],[238,61448,61449,61452],{"style":13806},[238,61450],{"className":61451,"style":412},[411],[238,61453,61455],{"className":61454},[416,417,418,419],[238,61456,3395],{"className":61457,"style":3520},[282,283,419],[238,61459,872],{"className":61460},[871],[238,61462,61464],{"className":61463},[399],[238,61465,61467],{"className":61466,"style":879},[403],[238,61468],{},[238,61470],{"className":61471,"style":660},[429],[238,61473,5577],{"className":61474},[664],[238,61476],{"className":61477,"style":660},[429],[238,61479,61481,61484,61524,61564,61567,61570],{"className":61480},[273],[238,61482],{"className":61483,"style":444},[277],[238,61485,61487,61490],{"className":61486},[282],[238,61488,4429],{"className":61489,"style":660},[282,283],[238,61491,61493],{"className":61492},[391],[238,61494,61496,61516],{"className":61495},[395,848],[238,61497,61499,61513],{"className":61498},[399],[238,61500,61502],{"className":61501,"style":1927},[403],[238,61503,61504,61507],{"style":13806},[238,61505],{"className":61506,"style":412},[411],[238,61508,61510],{"className":61509},[416,417,418,419],[238,61511,1895],{"className":61512},[282,419],[238,61514,872],{"className":61515},[871],[238,61517,61519],{"className":61518},[399],[238,61520,61522],{"className":61521,"style":879},[403],[238,61523],{},[238,61525,61527,61530],{"className":61526},[282],[238,61528,15588],{"className":61529},[282],[238,61531,61533],{"className":61532},[391],[238,61534,61536,61556],{"className":61535},[395,848],[238,61537,61539,61553],{"className":61538},[399],[238,61540,61542],{"className":61541,"style":1927},[403],[238,61543,61544,61547],{"style":1165},[238,61545],{"className":61546,"style":412},[411],[238,61548,61550],{"className":61549},[416,417,418,419],[238,61551,1895],{"className":61552},[282,419],[238,61554,872],{"className":61555},[871],[238,61557,61559],{"className":61558},[399],[238,61560,61562],{"className":61561,"style":879},[403],[238,61563],{},[238,61565],{"className":61566,"style":430},[429],[238,61568,2023],{"className":61569},[434],[238,61571],{"className":61572,"style":430},[429],[238,61574,61576,61579,61582,61585,61588],{"className":61575},[273],[238,61577],{"className":61578,"style":18081},[277],[238,61580,18351],{"className":61581},[282],[238,61583],{"className":61584,"style":660},[429],[238,61586,608],{"className":61587},[664],[238,61589],{"className":61590,"style":660},[429],[238,61592,61594,61597,61600,61629],{"className":61593},[273],[238,61595],{"className":61596,"style":3588},[277],[238,61598,29065],{"className":61599},[282],[238,61601,61603,61606],{"className":61602},[282],[238,61604,29072],{"className":61605},[282],[238,61607,61609],{"className":61608},[391],[238,61610,61612],{"className":61611},[395],[238,61613,61615],{"className":61614},[399],[238,61616,61618],{"className":61617,"style":3588},[403],[238,61619,61620,61623],{"style":1226},[238,61621],{"className":61622,"style":412},[411],[238,61624,61626],{"className":61625},[416,417,418,419],[238,61627,3395],{"className":61628,"style":3520},[282,283,419],[238,61630,3102],{"className":61631},[282],[11,61633,13131,61634,61686,61687,29168,61716,61686,61768,61797],{},[238,61635,61637,61656],{"className":61636,"translate":242},[241],[238,61638,61640],{"className":61639},[246],[248,61641,61642],{"xmlns":250},[252,61643,61644,61653],{},[255,61645,61646,61648,61650],{},[258,61647,3395],{},[332,61649,992],{},[738,61651,61652],{},"20",[262,61654,61655],{"encoding":264},"k=20",[238,61657,61659,61677],{"className":61658,"ariaHidden":269},[268],[238,61660,61662,61665,61668,61671,61674],{"className":61661},[273],[238,61663],{"className":61664,"style":29023},[277],[238,61666,3395],{"className":61667,"style":3520},[282,283],[238,61669],{"className":61670,"style":430},[429],[238,61672,992],{"className":61673},[434],[238,61675],{"className":61676,"style":430},[429],[238,61678,61680,61683],{"className":61679},[273],[238,61681],{"className":61682,"style":2672},[277],[238,61684,61652],{"className":61685},[282]," 时，右端约为 ",[238,61688,61690,61704],{"className":61689,"translate":242},[241],[238,61691,61693],{"className":61692},[246],[248,61694,61695],{"xmlns":250},[252,61696,61697,61702],{},[255,61698,61699],{},[738,61700,61701],{},"0.64",[262,61703,61701],{"encoding":264},[238,61705,61707],{"className":61706,"ariaHidden":269},[268],[238,61708,61710,61713],{"className":61709},[273],[238,61711],{"className":61712,"style":2672},[277],[238,61714,61701],{"className":61715},[282],[238,61717,61719,61738],{"className":61718,"translate":242},[241],[238,61720,61722],{"className":61721},[246],[248,61723,61724],{"xmlns":250},[252,61725,61726,61735],{},[255,61727,61728,61730,61732],{},[258,61729,3395],{},[332,61731,992],{},[738,61733,61734],{},"50",[262,61736,61737],{"encoding":264},"k=50",[238,61739,61741,61759],{"className":61740,"ariaHidden":269},[268],[238,61742,61744,61747,61750,61753,61756],{"className":61743},[273],[238,61745],{"className":61746,"style":29023},[277],[238,61748,3395],{"className":61749,"style":3520},[282,283],[238,61751],{"className":61752,"style":430},[429],[238,61754,992],{"className":61755},[434],[238,61757],{"className":61758,"style":430},[429],[238,61760,61762,61765],{"className":61761},[273],[238,61763],{"className":61764,"style":2672},[277],[238,61766,61734],{"className":61767},[282],[238,61769,61771,61785],{"className":61770,"translate":242},[241],[238,61772,61774],{"className":61773},[246],[248,61775,61776],{"xmlns":250},[252,61777,61778,61783],{},[255,61779,61780],{},[738,61781,61782],{},"0.027",[262,61784,61782],{"encoding":264},[238,61786,61788],{"className":61787,"ariaHidden":269},[268],[238,61789,61791,61794],{"className":61790},[273],[238,61792],{"className":61793,"style":2672},[277],[238,61795,61782],{"className":61796},[282],"。这与几何收敛的预期一致。",[225,61799,61801],{"id":61800},"_54-折扣因子的必要性","5.4 折扣因子的必要性",[11,61803,61804,61805,35185,61855,61905],{},"压缩性依赖 ",[238,61806,61808,61825],{"className":61807,"translate":242},[241],[238,61809,61811],{"className":61810},[246],[248,61812,61813],{"xmlns":250},[252,61814,61815,61823],{},[255,61816,61817,61819,61821],{},[258,61818,730],{},[332,61820,1892],{},[738,61822,745],{},[262,61824,17495],{"encoding":264},[238,61826,61828,61846],{"className":61827,"ariaHidden":269},[268],[238,61829,61831,61834,61837,61840,61843],{"className":61830},[273],[238,61832],{"className":61833,"style":760},[277],[238,61835,730],{"className":61836,"style":764},[282,283],[238,61838],{"className":61839,"style":430},[429],[238,61841,1892],{"className":61842},[434],[238,61844],{"className":61845,"style":430},[429],[238,61847,61849,61852],{"className":61848},[273],[238,61850],{"className":61851,"style":2672},[277],[238,61853,745],{"className":61854},[282],[238,61856,61858,61875],{"className":61857,"translate":242},[241],[238,61859,61861],{"className":61860},[246],[248,61862,61863],{"xmlns":250},[252,61864,61865,61873],{},[255,61866,61867,61869,61871],{},[258,61868,730],{},[332,61870,992],{},[738,61872,745],{},[262,61874,29189],{"encoding":264},[238,61876,61878,61896],{"className":61877,"ariaHidden":269},[268],[238,61879,61881,61884,61887,61890,61893],{"className":61880},[273],[238,61882],{"className":61883,"style":2853},[277],[238,61885,730],{"className":61886,"style":764},[282,283],[238,61888],{"className":61889,"style":430},[429],[238,61891,992],{"className":61892},[434],[238,61894],{"className":61895,"style":430},[429],[238,61897,61899,61902],{"className":61898},[273],[238,61900],{"className":61901,"style":2672},[277],[238,61903,745],{"className":61904},[282],"，贝尔曼算子一般不再是压缩映射，贝尔曼方程可能没有有限解，也可能有多个解。以下两个例子说明这两种失效方式。",[11,61907,61908,61909,61951,61952,62025,62026,17972],{},"第一个例子是零奖励的确定性循环。设两个状态 ",[238,61910,61912,61930],{"className":61911,"translate":242},[241],[238,61913,61915],{"className":61914},[246],[248,61916,61917],{"xmlns":250},[252,61918,61919,61927],{},[255,61920,61921,61923,61925],{},[738,61922,745],{},[332,61924,354],{"separator":269},[738,61926,17558],{},[262,61928,61929],{"encoding":264},"1,2",[238,61931,61933],{"className":61932,"ariaHidden":269},[268],[238,61934,61936,61939,61942,61945,61948],{"className":61935},[273],[238,61937],{"className":61938,"style":17777},[277],[238,61940,745],{"className":61941},[282],[238,61943,354],{"className":61944},[451],[238,61946],{"className":61947,"style":455},[429],[238,61949,17558],{"className":61950},[282],"，转移为 ",[238,61953,61955,61977],{"className":61954,"translate":242},[241],[238,61956,61958],{"className":61957},[246],[248,61959,61960],{"xmlns":250},[252,61961,61962,61974],{},[255,61963,61964,61966,61968,61970,61972],{},[738,61965,745],{},[332,61967,617],{},[738,61969,17558],{},[332,61971,617],{},[738,61973,745],{},[262,61975,61976],{"encoding":264},"1\\to2\\to1",[238,61978,61980,61998,62016],{"className":61979,"ariaHidden":269},[268],[238,61981,61983,61986,61989,61992,61995],{"className":61982},[273],[238,61984],{"className":61985,"style":2672},[277],[238,61987,745],{"className":61988},[282],[238,61990],{"className":61991,"style":430},[429],[238,61993,617],{"className":61994},[434],[238,61996],{"className":61997,"style":430},[429],[238,61999,62001,62004,62007,62010,62013],{"className":62000},[273],[238,62002],{"className":62003,"style":2672},[277],[238,62005,17558],{"className":62006},[282],[238,62008],{"className":62009,"style":430},[429],[238,62011,617],{"className":62012},[434],[238,62014],{"className":62015,"style":430},[429],[238,62017,62019,62022],{"className":62018},[273],[238,62020],{"className":62021,"style":2672},[277],[238,62023,745],{"className":62024},[282],"，奖励恒为零，",[238,62027,62029,62046],{"className":62028,"translate":242},[241],[238,62030,62032],{"className":62031},[246],[248,62033,62034],{"xmlns":250},[252,62035,62036,62044],{},[255,62037,62038,62040,62042],{},[258,62039,730],{},[332,62041,992],{},[738,62043,745],{},[262,62045,29189],{"encoding":264},[238,62047,62049,62067],{"className":62048,"ariaHidden":269},[268],[238,62050,62052,62055,62058,62061,62064],{"className":62051},[273],[238,62053],{"className":62054,"style":2853},[277],[238,62056,730],{"className":62057,"style":764},[282,283],[238,62059],{"className":62060,"style":430},[429],[238,62062,992],{"className":62063},[434],[238,62065],{"className":62066,"style":430},[429],[238,62068,62070,62073],{"className":62069},[273],[238,62071],{"className":62072,"style":2672},[277],[238,62074,745],{"className":62075},[282],[238,62077,62079],{"className":62078,"translate":242},[945],[238,62080,62082,62136],{"className":62081,"translate":242},[241],[238,62083,62085],{"className":62084},[246],[248,62086,62087],{"xmlns":250,"display":954},[252,62088,62089,62133],{},[255,62090,62091,62093,62095,62097,62099,62101,62103,62105,62107,62109,62111,62113,62115,62117,62119,62121,62123,62125,62127,62129,62131],{},[258,62092,4429],{},[332,62094,335],{"stretchy":334},[738,62096,745],{},[332,62098,360],{"stretchy":334},[332,62100,992],{},[258,62102,4429],{},[332,62104,335],{"stretchy":334},[738,62106,17558],{},[332,62108,360],{"stretchy":334},[332,62110,354],{"separator":269},[429,62112],{"width":10450},[258,62114,4429],{},[332,62116,335],{"stretchy":334},[738,62118,17558],{},[332,62120,360],{"stretchy":334},[332,62122,992],{},[258,62124,4429],{},[332,62126,335],{"stretchy":334},[738,62128,745],{},[332,62130,360],{"stretchy":334},[258,62132,3102],{"mathvariant":344},[262,62134,62135],{"encoding":264},"V(1)=V(2),\\qquad V(2)=V(1).",[238,62137,62139,62166,62214],{"className":62138,"ariaHidden":269},[268],[238,62140,62142,62145,62148,62151,62154,62157,62160,62163],{"className":62141},[273],[238,62143],{"className":62144,"style":444},[277],[238,62146,4429],{"className":62147,"style":660},[282,283],[238,62149,335],{"className":62150},[381],[238,62152,745],{"className":62153},[282],[238,62155,360],{"className":62156},[462],[238,62158],{"className":62159,"style":430},[429],[238,62161,992],{"className":62162},[434],[238,62164],{"className":62165,"style":430},[429],[238,62167,62169,62172,62175,62178,62181,62184,62187,62190,62193,62196,62199,62202,62205,62208,62211],{"className":62168},[273],[238,62170],{"className":62171,"style":444},[277],[238,62173,4429],{"className":62174,"style":660},[282,283],[238,62176,335],{"className":62177},[381],[238,62179,17558],{"className":62180},[282],[238,62182,360],{"className":62183},[462],[238,62185,354],{"className":62186},[451],[238,62188],{"className":62189,"style":10986},[429],[238,62191],{"className":62192,"style":455},[429],[238,62194,4429],{"className":62195,"style":660},[282,283],[238,62197,335],{"className":62198},[381],[238,62200,17558],{"className":62201},[282],[238,62203,360],{"className":62204},[462],[238,62206],{"className":62207,"style":430},[429],[238,62209,992],{"className":62210},[434],[238,62212],{"className":62213,"style":430},[429],[238,62215,62217,62220,62223,62226,62229,62232],{"className":62216},[273],[238,62218],{"className":62219,"style":444},[277],[238,62221,4429],{"className":62222,"style":660},[282,283],[238,62224,335],{"className":62225},[381],[238,62227,745],{"className":62228},[282],[238,62230,360],{"className":62231},[462],[238,62233,3102],{"className":62234},[282],[11,62236,62237,62238,62341],{},"任意常数函数 ",[238,62239,62241,62275],{"className":62240,"translate":242},[241],[238,62242,62244],{"className":62243},[246],[248,62245,62246],{"xmlns":250},[252,62247,62248,62272],{},[255,62249,62250,62252,62254,62256,62258,62260,62262,62264,62266,62268,62270],{},[258,62251,4429],{},[332,62253,335],{"stretchy":334},[738,62255,745],{},[332,62257,360],{"stretchy":334},[332,62259,992],{},[258,62261,4429],{},[332,62263,335],{"stretchy":334},[738,62265,17558],{},[332,62267,360],{"stretchy":334},[332,62269,992],{},[258,62271,24276],{},[262,62273,62274],{"encoding":264},"V(1)=V(2)=c",[238,62276,62278,62305,62332],{"className":62277,"ariaHidden":269},[268],[238,62279,62281,62284,62287,62290,62293,62296,62299,62302],{"className":62280},[273],[238,62282],{"className":62283,"style":444},[277],[238,62285,4429],{"className":62286,"style":660},[282,283],[238,62288,335],{"className":62289},[381],[238,62291,745],{"className":62292},[282],[238,62294,360],{"className":62295},[462],[238,62297],{"className":62298,"style":430},[429],[238,62300,992],{"className":62301},[434],[238,62303],{"className":62304,"style":430},[429],[238,62306,62308,62311,62314,62317,62320,62323,62326,62329],{"className":62307},[273],[238,62309],{"className":62310,"style":444},[277],[238,62312,4429],{"className":62313,"style":660},[282,283],[238,62315,335],{"className":62316},[381],[238,62318,17558],{"className":62319},[282],[238,62321,360],{"className":62322},[462],[238,62324],{"className":62325,"style":430},[429],[238,62327,992],{"className":62328},[434],[238,62330],{"className":62331,"style":430},[429],[238,62333,62335,62338],{"className":62334},[273],[238,62336],{"className":62337,"style":489},[277],[238,62339,24276],{"className":62340},[282,283]," 都是解，因此不动点不唯一。Banach 定理的唯一性结论失效。",[11,62343,62344,62345,62373],{},"第二个例子是每步奖励为 ",[238,62346,62348,62361],{"className":62347,"translate":242},[241],[238,62349,62351],{"className":62350},[246],[248,62352,62353],{"xmlns":250},[252,62354,62355,62359],{},[255,62356,62357],{},[738,62358,745],{},[262,62360,745],{"encoding":264},[238,62362,62364],{"className":62363,"ariaHidden":269},[268],[238,62365,62367,62370],{"className":62366},[273],[238,62368],{"className":62369,"style":2672},[277],[238,62371,745],{"className":62372},[282]," 的同一循环。此时",[238,62375,62377],{"className":62376,"translate":242},[945],[238,62378,62380,62422],{"className":62379,"translate":242},[241],[238,62381,62383],{"className":62382},[246],[248,62384,62385],{"xmlns":250,"display":954},[252,62386,62387,62419],{},[255,62388,62389,62395,62397,62411,62413,62415,62417],{},[812,62390,62391,62393],{},[258,62392,3380],{},[258,62394,820],{},[332,62396,992],{},[3386,62398,62399,62401,62409],{},[332,62400,3390],{},[255,62402,62403,62405,62407],{},[258,62404,3395],{},[332,62406,992],{},[738,62408,740],{},[258,62410,1895],{"mathvariant":344},[738,62412,745],{},[332,62414,992],{},[258,62416,1895],{"mathvariant":344},[332,62418,354],{"separator":269},[262,62420,62421],{"encoding":264},"G_t=\\sum_{k=0}^{\\infty}1=\\infty,",[238,62423,62425,62480,62568],{"className":62424,"ariaHidden":269},[268],[238,62426,62428,62431,62471,62474,62477],{"className":62427},[273],[238,62429],{"className":62430,"style":833},[277],[238,62432,62434,62437],{"className":62433},[282],[238,62435,3380],{"className":62436},[282,283],[238,62438,62440],{"className":62439},[391],[238,62441,62443,62463],{"className":62442},[395,848],[238,62444,62446,62460],{"className":62445},[399],[238,62447,62449],{"className":62448,"style":855},[403],[238,62450,62451,62454],{"style":1165},[238,62452],{"className":62453,"style":412},[411],[238,62455,62457],{"className":62456},[416,417,418,419],[238,62458,820],{"className":62459},[282,283,419],[238,62461,872],{"className":62462},[871],[238,62464,62466],{"className":62465},[399],[238,62467,62469],{"className":62468,"style":879},[403],[238,62470],{},[238,62472],{"className":62473,"style":430},[429],[238,62475,992],{"className":62476},[434],[238,62478],{"className":62479,"style":430},[429],[238,62481,62483,62486,62553,62556,62559,62562,62565],{"className":62482},[273],[238,62484],{"className":62485,"style":3489},[277],[238,62487,62489],{"className":62488},[1943,3493],[238,62490,62492,62545],{"className":62491},[395,848],[238,62493,62495,62542],{"className":62494},[399],[238,62496,62498,62518,62528],{"className":62497,"style":3503},[403],[238,62499,62500,62503],{"style":3506},[238,62501],{"className":62502,"style":3510},[411],[238,62504,62506],{"className":62505},[416,417,418,419],[238,62507,62509,62512,62515],{"className":62508},[282,419],[238,62510,3395],{"className":62511,"style":3520},[282,283,419],[238,62513,992],{"className":62514},[434,419],[238,62516,740],{"className":62517},[282,419],[238,62519,62520,62523],{"style":3529},[238,62521],{"className":62522,"style":3510},[411],[238,62524,62525],{},[238,62526,3390],{"className":62527},[1943,3538,3539],[238,62529,62530,62533],{"style":3542},[238,62531],{"className":62532,"style":3510},[411],[238,62534,62536],{"className":62535},[416,417,418,419],[238,62537,62539],{"className":62538},[282,419],[238,62540,1895],{"className":62541},[282,419],[238,62543,872],{"className":62544},[871],[238,62546,62548],{"className":62547},[399],[238,62549,62551],{"className":62550,"style":3564},[403],[238,62552],{},[238,62554],{"className":62555,"style":455},[429],[238,62557,745],{"className":62558},[282],[238,62560],{"className":62561,"style":430},[429],[238,62563,992],{"className":62564},[434],[238,62566],{"className":62567,"style":430},[429],[238,62569,62571,62574,62577],{"className":62570},[273],[238,62572],{"className":62573,"style":2853},[277],[238,62575,1895],{"className":62576},[282],[238,62578,354],{"className":62579},[451],[11,62581,62582,62583,62626],{},"折扣回报不再是有限实数，值函数不属于 ",[238,62584,62586,62605],{"className":62585,"translate":242},[241],[238,62587,62589],{"className":62588},[246],[248,62590,62591],{"xmlns":250},[252,62592,62593,62603],{},[255,62594,62595,62597,62599,62601],{},[258,62596,19971],{},[332,62598,335],{"stretchy":334},[258,62600,260],{},[332,62602,360],{"stretchy":334},[262,62604,21579],{"encoding":264},[238,62606,62608],{"className":62607,"ariaHidden":269},[268],[238,62609,62611,62614,62617,62620,62623],{"className":62610},[273],[238,62612],{"className":62613,"style":444},[277],[238,62615,19971],{"className":62616,"style":20011},[282,283],[238,62618,335],{"className":62619},[381],[238,62621,260],{"className":62622,"style":284},[282,283],[238,62624,360],{"className":62625},[462],"。对应的贝尔曼方程为",[238,62628,62630],{"className":62629,"translate":242},[945],[238,62631,62633,62695],{"className":62632,"translate":242},[241],[238,62634,62636],{"className":62635},[246],[248,62637,62638],{"xmlns":250,"display":954},[252,62639,62640,62692],{},[255,62641,62642,62644,62646,62648,62650,62652,62654,62656,62658,62660,62662,62664,62666,62668,62670,62672,62674,62676,62678,62680,62682,62684,62686,62688,62690],{},[258,62643,4429],{},[332,62645,335],{"stretchy":334},[738,62647,745],{},[332,62649,360],{"stretchy":334},[332,62651,992],{},[738,62653,745],{},[332,62655,987],{},[258,62657,4429],{},[332,62659,335],{"stretchy":334},[738,62661,17558],{},[332,62663,360],{"stretchy":334},[332,62665,354],{"separator":269},[429,62667],{"width":10450},[258,62669,4429],{},[332,62671,335],{"stretchy":334},[738,62673,17558],{},[332,62675,360],{"stretchy":334},[332,62677,992],{},[738,62679,745],{},[332,62681,987],{},[258,62683,4429],{},[332,62685,335],{"stretchy":334},[738,62687,745],{},[332,62689,360],{"stretchy":334},[332,62691,354],{"separator":269},[262,62693,62694],{"encoding":264},"V(1)=1+V(2),\\qquad V(2)=1+V(1),",[238,62696,62698,62725,62743,62791,62809],{"className":62697,"ariaHidden":269},[268],[238,62699,62701,62704,62707,62710,62713,62716,62719,62722],{"className":62700},[273],[238,62702],{"className":62703,"style":444},[277],[238,62705,4429],{"className":62706,"style":660},[282,283],[238,62708,335],{"className":62709},[381],[238,62711,745],{"className":62712},[282],[238,62714,360],{"className":62715},[462],[238,62717],{"className":62718,"style":430},[429],[238,62720,992],{"className":62721},[434],[238,62723],{"className":62724,"style":430},[429],[238,62726,62728,62731,62734,62737,62740],{"className":62727},[273],[238,62729],{"className":62730,"style":18081},[277],[238,62732,745],{"className":62733},[282],[238,62735],{"className":62736,"style":660},[429],[238,62738,987],{"className":62739},[664],[238,62741],{"className":62742,"style":660},[429],[238,62744,62746,62749,62752,62755,62758,62761,62764,62767,62770,62773,62776,62779,62782,62785,62788],{"className":62745},[273],[238,62747],{"className":62748,"style":444},[277],[238,62750,4429],{"className":62751,"style":660},[282,283],[238,62753,335],{"className":62754},[381],[238,62756,17558],{"className":62757},[282],[238,62759,360],{"className":62760},[462],[238,62762,354],{"className":62763},[451],[238,62765],{"className":62766,"style":10986},[429],[238,62768],{"className":62769,"style":455},[429],[238,62771,4429],{"className":62772,"style":660},[282,283],[238,62774,335],{"className":62775},[381],[238,62777,17558],{"className":62778},[282],[238,62780,360],{"className":62781},[462],[238,62783],{"className":62784,"style":430},[429],[238,62786,992],{"className":62787},[434],[238,62789],{"className":62790,"style":430},[429],[238,62792,62794,62797,62800,62803,62806],{"className":62793},[273],[238,62795],{"className":62796,"style":18081},[277],[238,62798,745],{"className":62799},[282],[238,62801],{"className":62802,"style":660},[429],[238,62804,987],{"className":62805},[664],[238,62807],{"className":62808,"style":660},[429],[238,62810,62812,62815,62818,62821,62824,62827],{"className":62811},[273],[238,62813],{"className":62814,"style":444},[277],[238,62816,4429],{"className":62817,"style":660},[282,283],[238,62819,335],{"className":62820},[381],[238,62822,745],{"className":62823},[282],[238,62825,360],{"className":62826},[462],[238,62828,354],{"className":62829},[451],[11,62831,62832,62833,62883],{},"它也没有有限解。这说明 ",[238,62834,62836,62853],{"className":62835,"translate":242},[241],[238,62837,62839],{"className":62838},[246],[248,62840,62841],{"xmlns":250},[252,62842,62843,62851],{},[255,62844,62845,62847,62849],{},[258,62846,730],{},[332,62848,1892],{},[738,62850,745],{},[262,62852,17495],{"encoding":264},[238,62854,62856,62874],{"className":62855,"ariaHidden":269},[268],[238,62857,62859,62862,62865,62868,62871],{"className":62858},[273],[238,62860],{"className":62861,"style":760},[277],[238,62863,730],{"className":62864,"style":764},[282,283],[238,62866],{"className":62867,"style":430},[429],[238,62869,1892],{"className":62870},[434],[238,62872],{"className":62873,"style":430},[429],[238,62875,62877,62880],{"className":62876},[273],[238,62878],{"className":62879,"style":2672},[277],[238,62881,745],{"className":62882},[282]," 不是技术上的方便假设，而是保证回报有界、算子压缩、解唯一的结构性条件。",[220,62885,62887],{"id":62886},"六最优子结构与最优性原理","六、最优子结构与最优性原理",[225,62889,62891],{"id":62890},"_61-有限期情形","6.1 有限期情形",[11,62893,62894,62895,62966,62967,2382,63155,63225,63226,63254],{},"最优子结构最清晰的表述出现在有限期问题中。设时间范围为 ",[238,62896,62898,62924],{"className":62897,"translate":242},[241],[238,62899,62901],{"className":62900},[246],[248,62902,62903],{"xmlns":250},[252,62904,62905,62921],{},[255,62906,62907,62909,62911,62913,62915,62917,62919],{},[738,62908,740],{},[332,62910,354],{"separator":269},[738,62912,745],{},[332,62914,354],{"separator":269},[332,62916,2840],{},[332,62918,354],{"separator":269},[258,62920,11971],{},[262,62922,62923],{"encoding":264},"0,1,\\dots,T",[238,62925,62927],{"className":62926,"ariaHidden":269},[268],[238,62928,62930,62933,62936,62939,62942,62945,62948,62951,62954,62957,62960,62963],{"className":62929},[273],[238,62931],{"className":62932,"style":6651},[277],[238,62934,740],{"className":62935},[282],[238,62937,354],{"className":62938},[451],[238,62940],{"className":62941,"style":455},[429],[238,62943,745],{"className":62944},[282],[238,62946,354],{"className":62947},[451],[238,62949],{"className":62950,"style":455},[429],[238,62952,2840],{"className":62953},[3041],[238,62955],{"className":62956,"style":455},[429],[238,62958,354],{"className":62959},[451],[238,62961],{"className":62962,"style":455},[429],[238,62964,11971],{"className":62965,"style":377},[282,283],"，策略为一个序列 ",[238,62968,62970,63014],{"className":62969,"translate":242},[241],[238,62971,62973],{"className":62972},[246],[248,62974,62975],{"xmlns":250},[252,62976,62977,63011],{},[255,62978,62979,62981,62983,62985,62991,62993,62995,62997,63009],{},[258,62980,2297],{},[332,62982,992],{},[332,62984,335],{"stretchy":334},[812,62986,62987,62989],{},[258,62988,2297],{},[738,62990,740],{},[332,62992,354],{"separator":269},[332,62994,2840],{},[332,62996,354],{"separator":269},[812,62998,62999,63001],{},[258,63000,2297],{},[255,63002,63003,63005,63007],{},[258,63004,11971],{},[332,63006,5577],{},[738,63008,745],{},[332,63010,360],{"stretchy":334},[262,63012,63013],{"encoding":264},"\\pi=(\\pi_0,\\dots,\\pi_{T-1})",[238,63015,63017,63035],{"className":63016,"ariaHidden":269},[268],[238,63018,63020,63023,63026,63029,63032],{"className":63019},[273],[238,63021],{"className":63022,"style":489},[277],[238,63024,2297],{"className":63025,"style":2313},[282,283],[238,63027],{"className":63028,"style":430},[429],[238,63030,992],{"className":63031},[434],[238,63033],{"className":63034,"style":430},[429],[238,63036,63038,63041,63044,63084,63087,63090,63093,63096,63099,63102,63152],{"className":63037},[273],[238,63039],{"className":63040,"style":444},[277],[238,63042,335],{"className":63043},[381],[238,63045,63047,63050],{"className":63046},[282],[238,63048,2297],{"className":63049,"style":2313},[282,283],[238,63051,63053],{"className":63052},[391],[238,63054,63056,63076],{"className":63055},[395,848],[238,63057,63059,63073],{"className":63058},[399],[238,63060,63062],{"className":63061,"style":1162},[403],[238,63063,63064,63067],{"style":41515},[238,63065],{"className":63066,"style":412},[411],[238,63068,63070],{"className":63069},[416,417,418,419],[238,63071,740],{"className":63072},[282,419],[238,63074,872],{"className":63075},[871],[238,63077,63079],{"className":63078},[399],[238,63080,63082],{"className":63081,"style":879},[403],[238,63083],{},[238,63085,354],{"className":63086},[451],[238,63088],{"className":63089,"style":455},[429],[238,63091,2840],{"className":63092},[3041],[238,63094],{"className":63095,"style":455},[429],[238,63097,354],{"className":63098},[451],[238,63100],{"className":63101,"style":455},[429],[238,63103,63105,63108],{"className":63104},[282],[238,63106,2297],{"className":63107,"style":2313},[282,283],[238,63109,63111],{"className":63110},[391],[238,63112,63114,63144],{"className":63113},[395,848],[238,63115,63117,63141],{"className":63116},[399],[238,63118,63121],{"className":63119,"style":63120},[403],"height:0.3283em;",[238,63122,63123,63126],{"style":41515},[238,63124],{"className":63125,"style":412},[411],[238,63127,63129],{"className":63128},[416,417,418,419],[238,63130,63132,63135,63138],{"className":63131},[282,419],[238,63133,11971],{"className":63134,"style":377},[282,283,419],[238,63136,5577],{"className":63137},[664,419],[238,63139,745],{"className":63140},[282,419],[238,63142,872],{"className":63143},[871],[238,63145,63147],{"className":63146},[399],[238,63148,63150],{"className":63149,"style":1193},[403],[238,63151],{},[238,63153,360],{"className":63154},[462],[238,63156,63158,63176],{"className":63157,"translate":242},[241],[238,63159,63161],{"className":63160},[246],[248,63162,63163],{"xmlns":250},[252,63164,63165,63173],{},[255,63166,63167],{},[812,63168,63169,63171],{},[258,63170,2297],{},[258,63172,820],{},[262,63174,63175],{"encoding":264},"\\pi_t",[238,63177,63179],{"className":63178,"ariaHidden":269},[268],[238,63180,63182,63185],{"className":63181},[273],[238,63183],{"className":63184,"style":3115},[277],[238,63186,63188,63191],{"className":63187},[282],[238,63189,2297],{"className":63190,"style":2313},[282,283],[238,63192,63194],{"className":63193},[391],[238,63195,63197,63217],{"className":63196},[395,848],[238,63198,63200,63214],{"className":63199},[399],[238,63201,63203],{"className":63202,"style":855},[403],[238,63204,63205,63208],{"style":41515},[238,63206],{"className":63207,"style":412},[411],[238,63209,63211],{"className":63210},[416,417,418,419],[238,63212,820],{"className":63213},[282,283,419],[238,63215,872],{"className":63216},[871],[238,63218,63220],{"className":63219},[399],[238,63221,63223],{"className":63222,"style":879},[403],[238,63224],{}," 是时刻 ",[238,63227,63229,63242],{"className":63228,"translate":242},[241],[238,63230,63232],{"className":63231},[246],[248,63233,63234],{"xmlns":250},[252,63235,63236,63240],{},[255,63237,63238],{},[258,63239,820],{},[262,63241,820],{"encoding":264},[238,63243,63245],{"className":63244,"ariaHidden":269},[268],[238,63246,63248,63251],{"className":63247},[273],[238,63249],{"className":63250,"style":908},[277],[238,63252,820],{"className":63253},[282,283]," 的策略。定义",[238,63256,63258],{"className":63257,"translate":242},[945],[238,63259,63261,63420],{"className":63260,"translate":242},[241],[238,63262,63264],{"className":63263},[246],[248,63265,63266],{"xmlns":250,"display":954},[252,63267,63268,63417],{},[958,63269,63270,63349,63383],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,63271,63272,63292],{},[966,63273,63274],{},[969,63275,63276],{"scriptlevel":740,"displaystyle":269},[255,63277,63278,63286,63288,63290],{},[10131,63279,63280,63282,63284],{},[258,63281,4429],{},[258,63283,820],{},[258,63285,2297],{},[332,63287,335],{"stretchy":334},[258,63289,341],{},[332,63291,360],{"stretchy":334},[966,63293,63294],{},[969,63295,63296],{"scriptlevel":740,"displaystyle":269},[255,63297,63298,63300,63302,63308,63311,63331,63343],{},[255,63299],{},[332,63301,992],{},[812,63303,63304,63306],{},[258,63305,4452],{"mathvariant":620},[258,63307,2297],{},[332,63309,736],{"fence":334,"stretchy":269,"minsize":63310,"maxsize":63310},"3em",[3386,63312,63313,63315,63323],{},[332,63314,3390],{},[255,63316,63317,63319,63321],{},[258,63318,3395],{},[332,63320,992],{},[258,63322,820],{},[255,63324,63325,63327,63329],{},[258,63326,11971],{},[332,63328,5577],{},[738,63330,745],{},[337,63332,63333,63335],{},[258,63334,730],{},[255,63336,63337,63339,63341],{},[258,63338,3395],{},[332,63340,5577],{},[258,63342,820],{},[812,63344,63345,63347],{},[258,63346,600],{},[258,63348,3395],{},[963,63350,63351,63357],{},[966,63352,63353],{},[969,63354,63355],{"scriptlevel":740,"displaystyle":269},[255,63356],{},[966,63358,63359],{},[969,63360,63361],{"scriptlevel":740,"displaystyle":269},[255,63362,63363,63365,63367,63369,63375,63377,63379,63381],{},[255,63364],{},[429,63366],{"width":10450},[332,63368,349],{},[812,63370,63371,63373],{},[258,63372,341],{},[258,63374,820],{},[332,63376,992],{},[258,63378,341],{},[332,63380,4479],{"fence":334,"stretchy":269,"minsize":63310,"maxsize":63310},[332,63382,354],{"separator":269},[963,63384,63385,63405],{},[966,63386,63387],{},[969,63388,63389],{"scriptlevel":740,"displaystyle":269},[255,63390,63391,63399,63401,63403],{},[10131,63392,63393,63395,63397],{},[258,63394,4429],{},[258,63396,11971],{},[258,63398,2297],{},[332,63400,335],{"stretchy":334},[258,63402,341],{},[332,63404,360],{"stretchy":334},[966,63406,63407],{},[969,63408,63409],{"scriptlevel":740,"displaystyle":269},[255,63410,63411,63413,63415],{},[255,63412],{},[332,63414,992],{},[738,63416,29065],{},[262,63418,63419],{"encoding":264},"\\begin{aligned}\nV_t^\\pi(s)\n&=\\mathbb{E}_\\pi\\Bigg[\\sum_{k=t}^{T-1}\\gamma^{k-t}r_k\\\\\n&\\qquad\\mid s_t=s\\Bigg],\\\\\nV_T^\\pi(s)&=0.\n\\end{aligned}",[238,63421,63423],{"className":63422,"ariaHidden":269},[268],[238,63424,63426,63430],{"className":63425},[273],[238,63427],{"className":63428,"style":63429},[277],"height:7.9305em;vertical-align:-3.7152em;",[238,63431,63433],{"className":63432},[282],[238,63434,63436,63609],{"className":63435},[958],[238,63437,63439],{"className":63438},[1117],[238,63440,63442,63600],{"className":63441},[395,848],[238,63443,63445,63597],{"className":63444},[399],[238,63446,63449,63519,63528],{"className":63447,"style":63448},[403],"height:4.2152em;",[238,63450,63452,63456],{"style":63451},"top:-6.2152em;",[238,63453],{"className":63454,"style":63455},[411],"height:3.8283em;",[238,63457,63459,63510,63513,63516],{"className":63458},[282],[238,63460,63462,63465],{"className":63461},[282],[238,63463,4429],{"className":63464,"style":660},[282,283],[238,63466,63468],{"className":63467},[391],[238,63469,63471,63502],{"className":63470},[395,848],[238,63472,63474,63499],{"className":63473},[399],[238,63475,63477,63488],{"className":63476,"style":4620},[403],[238,63478,63479,63482],{"style":10172},[238,63480],{"className":63481,"style":412},[411],[238,63483,63485],{"className":63484},[416,417,418,419],[238,63486,820],{"className":63487},[282,283,419],[238,63489,63490,63493],{"style":1226},[238,63491],{"className":63492,"style":412},[411],[238,63494,63496],{"className":63495},[416,417,418,419],[238,63497,2297],{"className":63498,"style":2313},[282,283,419],[238,63500,872],{"className":63501},[871],[238,63503,63505],{"className":63504},[399],[238,63506,63508],{"className":63507,"style":10202},[403],[238,63509],{},[238,63511,335],{"className":63512},[381],[238,63514,341],{"className":63515},[282,283],[238,63517,360],{"className":63518},[462],[238,63520,63522,63525],{"style":63521},"top:-2.8631em;",[238,63523],{"className":63524,"style":63455},[411],[238,63526],{"className":63527},[282],[238,63529,63531,63534],{"style":63530},"top:-0.4731em;",[238,63532],{"className":63533,"style":63455},[411],[238,63535,63537,63588,63591,63594],{"className":63536},[282],[238,63538,63540,63543],{"className":63539},[282],[238,63541,4429],{"className":63542,"style":660},[282,283],[238,63544,63546],{"className":63545},[391],[238,63547,63549,63580],{"className":63548},[395,848],[238,63550,63552,63577],{"className":63551},[399],[238,63553,63555,63566],{"className":63554,"style":4620},[403],[238,63556,63557,63560],{"style":10172},[238,63558],{"className":63559,"style":412},[411],[238,63561,63563],{"className":63562},[416,417,418,419],[238,63564,11971],{"className":63565,"style":377},[282,283,419],[238,63567,63568,63571],{"style":1226},[238,63569],{"className":63570,"style":412},[411],[238,63572,63574],{"className":63573},[416,417,418,419],[238,63575,2297],{"className":63576,"style":2313},[282,283,419],[238,63578,872],{"className":63579},[871],[238,63581,63583],{"className":63582},[399],[238,63584,63586],{"className":63585,"style":10202},[403],[238,63587],{},[238,63589,335],{"className":63590},[381],[238,63592,341],{"className":63593},[282,283],[238,63595,360],{"className":63596},[462],[238,63598,872],{"className":63599},[871],[238,63601,63603],{"className":63602},[399],[238,63604,63607],{"className":63605,"style":63606},[403],"height:3.7152em;",[238,63608],{},[238,63610,63612],{"className":63611},[1422],[238,63613,63615,63956],{"className":63614},[395,848],[238,63616,63618,63953],{"className":63617},[399],[238,63619,63621,63846,63930],{"className":63620,"style":63448},[403],[238,63622,63623,63626],{"style":63451},[238,63624],{"className":63625,"style":63455},[411],[238,63627,63629,63632,63635,63638,63641,63681,63688,63691,63765,63768,63806],{"className":63628},[282],[238,63630],{"className":63631},[282],[238,63633],{"className":63634,"style":430},[429],[238,63636,992],{"className":63637},[434],[238,63639],{"className":63640,"style":430},[429],[238,63642,63644,63647],{"className":63643},[282],[238,63645,4452],{"className":63646},[282,714],[238,63648,63650],{"className":63649},[391],[238,63651,63653,63673],{"className":63652},[395,848],[238,63654,63656,63670],{"className":63655},[399],[238,63657,63659],{"className":63658,"style":1927},[403],[238,63660,63661,63664],{"style":1165},[238,63662],{"className":63663,"style":412},[411],[238,63665,63667],{"className":63666},[416,417,418,419],[238,63668,2297],{"className":63669,"style":2313},[282,283,419],[238,63671,872],{"className":63672},[871],[238,63674,63676],{"className":63675},[399],[238,63677,63679],{"className":63678,"style":879},[403],[238,63680],{},[238,63682,63684],{"className":63683},[282],[238,63685,736],{"className":63686},[18719,63687],"size4",[238,63689],{"className":63690,"style":455},[429],[238,63692,63694],{"className":63693},[1943,3493],[238,63695,63697,63757],{"className":63696},[395,848],[238,63698,63700,63754],{"className":63699},[399],[238,63701,63704,63724,63734],{"className":63702,"style":63703},[403],"height:1.8283em;",[238,63705,63706,63709]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V_t^\\pi(s)",[238,64023,64025,64100],{"className":64024,"ariaHidden":269},[268],[238,64026,64028,64031,64082,64085,64088,64091,64094,64097],{"className":64027},[273],[238,64029],{"className":64030,"style":444},[277],[238,64032,64034,64037],{"className":64033},[282],[238,64035,4429],{"className":64036,"style":660},[282,283],[238,64038,64040],{"className":64039},[391],[238,64041,64043,64074],{"className":64042},[395,848],[238,64044,64046,64071],{"className":64045},[399],[238,64047,64049,64060],{"className":64048,"style":22096},[403],[238,64050,64051,64054],{"style":10172},[238,64052],{"className":64053,"style":412},[411],[238,64055,64057],{"className":64056},[416,417,418,419],[238,64058,820],{"className":64059},[282,283,419],[238,64061,64062,64065],{"style":407},[238,64063],{"className":64064,"style":412},[411],[238,64066,64068],{"className":64067},[416,417,418,419],[238,64069,22083],{"className":64070},[664,419],[238,64072,872],{"className":64073},[871],[238,64075,64077],{"className":64076},[399],[238,64078,64080],{"className":64079,"style":10202},[403],[238,64081],{},[238,64083,335],{"className":64084},[381],[238,64086,341],{"className":64087},[282,283],[238,64089,360],{"className":64090},[462],[238,64092],{"className":64093,"style":430},[429],[238,64095,992],{"className":64096},[434],[238,64098],{"className":64099,"style":430},[429],[238,64101,64103,64106,64149,64152,64203,64206,64209],{"className":64102},[273],[238,64104],{"className":64105,"style":444},[277],[238,64107,64109,64112],{"className":64108},[1943],[238,64110,21664],{"className":64111},[1943],[238,64113,64115],{"className":64114},[391],[238,64116,64118,64140],{"className":64117},[395,848],[238,64119,64121,64137],{"className":64120},[399],[238,64122,64125],{"className":64123,"style":64124},[403],"height:0.0573em;",[238,64126,64128,64131],{"style":64127},"top:-2.4559em;margin-right:0.05em;",[238,64129],{"className":64130,"style":412},[411],[238,64132,64134],{"className":64133},[416,417,418,419],[238,64135,2297],{"className":64136,"style":2313},[282,283,419],[238,64138,872],{"className":64139},[871],[238,64141,64143],{"className":64142},[399],[238,64144,64147],{"className":64145,"style":64146},[403],"height:0.2441em;",[238,64148],{},[238,64150],{"className":64151,"style":455},[429],[238,64153,64155,64158],{"className":64154},[282],[238,64156,4429],{"className":64157,"style":660},[282,283],[238,64159,64161],{"className":64160},[391],[238,64162,64164,64195],{"className":64163},[395,848],[238,64165,64167,64192],{"className":64166},[399],[238,64168,64170,64181],{"className":64169,"style":6609},[403],[238,64171,64172,64175],{"style":10172},[238,64173],{"className":64174,"style":412},[411],[238,64176,64178],{"className":64177},[416,417,418,419],[238,64179,820],{"className":64180},[282,283,419],[238,64182,64183,64186],{"style":407},[238,64184],{"className":64185,"style":412},[411],[238,64187,64189],{"className":64188},[416,417,418,419],[238,64190,2297],{"className":64191,"style":2313},[282,283,419],[238,64193,872],{"className":64194},[871],[238,64196,64198],{"className":64197},[399],[238,64199,64201],{"className":64200,"style":10202},[403],[238,64202],{},[238,64204,335],{"className":64205},[381],[238,64207,341],{"className":64208},[282,283],[238,64210,360],{"className":64211},[462],[230,64213,64216,64355],{"id":64214,"title":64215},"theorem-finite-horizon","定理 6.1（有限期最优性原理）",[11,64217,26642,64218,6724],{},[238,64219,64221,64259],{"className":64220,"translate":242},[241],[238,64222,64224],{"className":64223},[246],[248,64225,64226],{"xmlns":250},[252,64227,64228,64256],{},[255,64229,64230,64232,64234,64236,64238,64240,64242,64244,64246,64248,64250,64252,64254],{},[258,64231,820],{},[332,64233,992],{},[258,64235,11971],{},[332,64237,5577],{},[738,64239,745],{},[332,64241,354],{"separator":269},[258,64243,11971],{},[332,64245,5577],{},[738,64247,17558],{},[332,64249,354],{"separator":269},[332,64251,2840],{},[332,64253,354],{"separator":269},[738,64255,740],{},[262,64257,64258],{"encoding":264},"t=T-1,T-2,\\dots,0",[238,64260,64262,64280,64298,64325],{"className":64261,"ariaHidden":269},[268],[238,64263,64265,64268,64271,64274,64277],{"className":64264},[273],[238,64266],{"className":64267,"style":908},[277],[238,64269,820],{"className":64270},[282,283],[238,64272],{"className":64273,"style":430},[429],[238,64275,992],{"className":64276},[434],[238,64278],{"className":64279,"style":430},[429],[238,64281,64283,64286,64289,64292,64295],{"className":64282},[273],[238,64284],{"className":64285,"style":653},[277],[238,64287,11971],{"className":64288,"style":377},[282,283],[238,64290],{"className":64291,"style":660},[429],[238,64293,5577],{"className":64294},[664],[238,64296],{"className":64297,"style":660},[429],[238,64299,64301,64304,64307,64310,64313,64316,64319,64322],{"className":64300},[273],[238,64302],{"className":64303,"style":6651},[277],[238,64305,745],{"className":64306},[282],[238,64308,354],{"className":64309},[451],[238,64311],{"className":64312,"style":455},[429],[238,64314,11971],{"className":64315,"style":377},[282,283],[238,64317],{"className":64318,"style":660},[429],[238,64320,5577],{"className":64321},[664],[238,64323],{"className":64324,"style":660},[429],[238,64326,64328,64331,64334,64337,64340,64343,64346,64349,64352],{"className":64327},[273],[238,64329],{"className":64330,"style":17777},[277],[238,64332,17558],{"className":64333},[282],[238,64335,354],{"className":64336},[451],[238,64338],{"className":64339,"style":455},[429],[238,64341,2840],{"className":64342},[3041],[238,64344],{"className":64345,"style":455},[429],[238,64347,354],{"className":64348},[451],[238,64350],{"className":64351,"style":455},[429],[238,64353,740],{"className":64354},[282],[238,64356,64358],{"className":64357,"translate":242},[945],[238,64359,64361,64443],{"className":64360,"translate":242},[241],[238,64362,64364],{"className":64363},[246],[248,64365,64366],{"xmlns":250,"display":954},[252,64367,64368,64440],{},[255,64369,64370,64378,64380,64390,64404,64406,64408,64416,64418,64424,64438],{},[10131,64371,64372,64374,64376],{},[258,64373,4429],{},[258,64375,820],{},[258,64377,2297],{},[332,64379,992],{},[337,64381,64382,64384],{},[258,64383,11971],{"mathvariant":816},[812,64385,64386,64388],{},[258,64387,2297],{},[258,64389,820],{},[10131,64391,64392,64394,64402],{},[258,64393,4429],{},[255,64395,64396,64398,64400],{},[258,64397,820],{},[332,64399,987],{},[738,64401,745],{},[258,64403,2297],{},[332,64405,354],{"separator":269},[429,64407],{"width":10450},[10131,64409,64410,64412,64414],{},[258,64411,4429],{},[258,64413,820],{},[332,64415,22083],{},[332,64417,992],{},[337,64419,64420,64422],{},[258,64421,11971],{"mathvariant":816},[332,64423,22083],{},[10131,64425,64426,64428,64436],{},[258,64427,4429],{},[255,64429,64430,64432,64434],{},[258,64431,820],{},[332,64433,987],{},[738,64435,745],{},[332,64437,22083],{},[258,64439,3102],{"mathvariant":344},[262,64441,64442],{"encoding":264},"V_t^\\pi=\\mathcal{T}^{\\pi_t} V_{t+1}^\\pi,\n\\qquad\nV_t^*=\\mathcal{T}^* V_{t+1}^*.",[238,64444,64446,64513,64721],{"className":64445,"ariaHidden":269},[268],[238,64447,64449,64453,64504,64507,64510],{"className":64448},[273],[238,64450],{"className":64451,"style":64452},[277],"height:0.9614em;vertical-align:-0.247em;",[238,64454,64456,64459],{"className":64455},[282],[238,64457,4429],{"className":64458,"style":660},[282,283],[238,64460,64462],{"className":64461},[391],[238,64463,64465,64496],{"className":64464},[395,848],[238,64466,64468,64493],{"className":64467},[399],[238,64469,64471,64482],{"className":64470,"style":4620},[403],[238,64472,64473,64476],{"style":10172},[238,64474],{"className":64475,"style":412},[411],[238,64477,64479],{"className":64478},[416,417,418,419],[238,64480,820],{"className":64481},[282,283,419],[238,64483,64484,64487],{"style":1226},[238,64485],{"className":64486,"style":412},[411],[238,64488,64490],{"className":64489},[416,417,418,419],[238,64491,2297],{"className":64492,"style":2313},[282,283,419],[238,64494,872],{"className":64495},[871],[238,64497,64499],{"className":64498},[399],[238,64500,64502],{"className":64501,"style":10202},[403],[238,64503],{},[238,64505],{"className":64506,"style":430},[429],[238,64508,992],{"className":64509},[434],[238,64511],{"className":64512,"style":430},[429],[238,64514,64516,64520,64591,64652,64655,64658,64661,64712,64715,64718],{"className":64515},[273],[238,64517],{"className":64518,"style":64519},[277],"height:1.044em;vertical-align:-0.3053em;",[238,64521,64523,64526],{"className":64522},[282],[238,64524,11971],{"className":64525,"style":11992},[282,840],[238,64527,64529],{"className":64528},[391],[238,64530,64532],{"className":64531},[395],[238,64533,64535],{"className":64534},[399],[238,64536,64538],{"className":64537,"style":4620},[403],[238,64539,64540,64543],{"style":1226},[238,64541],{"className":64542,"style":412},[411],[238,64544,64546],{"className":64545},[416,417,418,419],[238,64547,64549],{"className":64548},[282,419],[238,64550,64552,64555],{"className":64551},[282,419],[238,64553,2297],{"className":64554,"style":2313},[282,283,419],[238,64556,64558],{"className":64557},[391],[238,64559,64561,64582],{"className":64560},[395,848],[238,64562,64564,64579],{"className":64563},[399],[238,64565,64568],{"className":64566,"style":64567},[403],"height:0.2963em;",[238,64569,64570,64573],{"style":42307},[238,64571],{"className":64572,"style":4203},[411],[238,64574,64576],{"className":64575},[416,4207,4208,419],[238,64577,820],{"className":64578},[282,283,419],[238,64580,872],{"className":64581},[871],[238,64583,64585],{"className":64584},[399],[238,64586,64589],{"className":64587,"style":64588},[403],"height:0.143em;",[238,64590],{},[238,64592,64594,64597],{"className":64593},[282],[238,64595,4429],{"className":64596,"style":660},[282,283],[238,64598,64600],{"className":64599},[391],[238,64601,64603,64643],{"className":64602},[395,848],[238,64604,64606,64640],{"className":64605},[399],[238,64607,64609,64629],{"className":64608,"style":4620},[403],[238,64610,64611,64614],{"style":10172},[238,64612],{"className":64613,"style":412},[411],[238,64615,64617],{"className":64616},[416,417,418,419],[238,64618,64620,64623,64626],{"className":64619},[282,419],[238,64621,820],{"className":64622},[282,283,419],[238,64624,987],{"className":64625},[664,419],[238,64627,745],{"className":64628},[282,419],[238,64630,64631,64634],{"style":1226},[238,64632],{"className":64633,"style":412},[411],[238,64635,64637],{"className":64636},[416,417,418,419],[238,64638,2297],{"className":64639,"style":2313},[282,283,419],[238,64641,872],{"className":64642},[871],[238,64644,64646],{"className":64645},[399],[238,64647,64650],{"className":64648,"style":64649},[403],"height:0.3053em;",[238,64651],{},[238,64653,354],{"className":64654},[451],[238,64656],{"className":64657,"style":10986},[429],[238,64659],{"className":64660,"style":455},[429],[238,64662,64664,64667],{"className":64663},[282],[238,64665,4429],{"className":64666,"style":660},[282,283],[238,64668,64670],{"className":64669},[391],[238,64671,64673,64704],{"className":64672},[395,848],[238,64674,64676,64701],{"className":64675},[399],[238,64677,64679,64690],{"className":64678,"style":22441},[403],[238,64680,64681,64684],{"style":10172},[238,64682],{"className":64683,"style":412},[411],[238,64685,64687],{"className":64686},[416,417,418,419],[238,64688,820],{"className":64689},[282,283,419],[238,64691,64692,64695],{"style":1226},[238,64693],{"className":64694,"style":412},[411],[238,64696,64698],{"className":64697},[416,417,418,419],[238,64699,22083],{"className":64700},[664,419],[238,64702,872],{"className":64703},[871],[238,64705,64707],{"className":64706},[399],[238,64708,64710],{"className":64709,"style":10202},[403],[238,64711],{},[238,64713],{"className":64714,"style":430},[429],[238,64716,992],{"className":64717},[434],[238,64719],{"className":64720,"style":430},[429],[238,64722,64724,64727,64756,64816],{"className":64723},[273],[238,64725],{"className":64726,"style":64519},[2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对 ",[238,64825,64827,64840],{"className":64826,"translate":242},[241],[238,64828,64830],{"className":64829},[246],[248,64831,64832],{"xmlns":250},[252,64833,64834,64838],{},[255,64835,64836],{},[258,64837,820],{},[262,64839,820],{"encoding":264},[238,64841,64843],{"className":64842,"ariaHidden":269},[268],[238,64844,64846,64849],{"className":64845},[273],[238,64847],{"className":64848,"style":908},[277],[238,64850,820],{"className":64851},[282,283]," 从 ",[238,64854,64856,64874],{"className":64855,"translate":242},[241],[238,64857,64859],{"className":64858},[246],[248,64860,64861],{"xmlns":250},[252,64862,64863,64871],{},[255,64864,64865,64867,64869],{},[258,64866,11971],{},[332,64868,5577],{},[738,64870,745],{},[262,64872,64873],{"encoding":264},"T-1",[238,64875,64877,64895],{"className":64876,"ariaHidden":269},[268],[238,64878,64880,64883,64886,64889,64892],{"className":64879},[273],[238,64881],{"className":64882,"style":653},[277],[238,64884,11971],{"className":64885,"style":377},[282,283],[238,64887],{"className":64888,"style":660},[429],[238,64890,5577],{"className":64891},[664],[238,64893],{"className":64894,"style":660},[429],[238,64896,64898,64901],{"className":64897},[273],[238,64899],{"className":64900,"style":2672},[277],[238,64902,745],{"className":64903},[282]," 到 ",[238,64906,64908,64921],{"className":64907,"translate":242},[241],[238,64909,64911],{"className":64910},[246],[248,64912,64913],{"xmlns":250},[252,64914,64915,64919],{},[255,64916,64917],{},[738,64918,740],{},[262,64920,740],{"encoding":264},[238,64922,64924],{"className":64923,"ariaHidden":269},[268],[238,64925,64927,64930],{"className":64926},[273],[238,64928],{"className":64929,"style":2672},[277],[238,64931,740],{"className":64932},[282]," 反向归纳。",[238,64935,64937,64955],{"className":64936,"translate":242},[241],[238,64938,64940],{"className":64939},[246],[248,64941,64942],{"xmlns":250},[252,64943,64944,64952],{},[255,64945,64946,64948,64950],{},[258,64947,820],{},[332,64949,992],{},[258,64951,11971],{},[262,64953,64954],{"encoding":264},"t=T",[238,64956,64958,64976],{"className":64957,"ariaHidden":269},[268],[238,64959,64961,64964,64967,64970,64973],{"className":64960},[273],[238,64962],{"className":64963,"style":908},[277],[238,64965,820],{"className":64966},[282,283],[238,64968],{"className":64969,"style":430},[429],[238,64971,992],{"className":64972},[434],[238,64974],{"className":64975,"style":430},[429],[238,64977,64979,64982],{"className":64978},[273],[238,64980],{"className":64981,"style":278},[277],[238,64983,11971],{"className":64984,"style":377},[282,283]," 时两端均为零。设结论对 ",[238,64987,64989,65007],{"className":64988,"translate":242},[241],[238,64990,64992],{"className":64991},[246],[248,64993,64994],{"xmlns":250},[252,64995,64996,65004],{},[255,64997,64998,65000,65002],{},[258,64999,820],{},[332,65001,987],{},[738,65003,745],{},[262,65005,65006],{"encoding":264},"t+1",[238,65008,65010,65029],{"className":65009,"ariaHidden":269},[268],[238,65011,65013,65017,65020,65023,65026],{"className":65012},[273],[238,65014],{"className":65015,"style":65016},[277],"height:0.6984em;vertical-align:-0.0833em;",[238,65018,820],{"className":65019},[282,283],[238,65021],{"className":65022,"style":660},[429],[238,65024,987],{"className":65025},[664],[238,65027],{"className":65028,"style":660},[429],[238,65030,65032,65035],{"className":65031},[273],[238,65033],{"className":65034,"style":2672},[277],[238,65036,745],{"className":65037},[282]," 成立。对任意策略 ",[238,65040,65042,65055],{"className":65041,"translate":242},[241],[238,65043,65045],{"className":65044},[246],[248,65046,65047],{"xmlns":250},[252,65048,65049,65053],{},[255,65050,65051],{},[258,65052,2297],{},[262,65054,2300],{"encoding":264},[238,65056,65058],{"className":65057,"ariaHidden":269},[268],[238,65059,65061,65064],{"className":65060},[273],[238,65062],{"className":65063,"style":489},[277],[238,65065,2297],{"className":65066,"style":2313},[282,283],"，在 ",[238,65069,65071,65093],{"className":65070,"translate":242},[241],[238,65072,65074],{"className":65073},[246],[248,65075,65076],{"xmlns":250},[252,65077,65078,65090],{},[255,65079,65080,65086,65088],{},[812,65081,65082,65084],{},[258,65083,341],{},[258,65085,820],{},[332,65087,992],{},[258,65089,341],{},[262,65091,65092],{"encoding":264},"s_t=s",[238,65094,65096,65151],{"className":65095,"ariaHidden":269},[268],[238,65097,65099,65102,65142,65145,65148],{"className":65098},[273],[238,65100],{"className":65101,"style":3115},[277],[238,65103,65105,65108],{"className":65104},[282],[238,65106,341],{"className":65107},[282,283],[238,65109,65111],{"className":65110},[391],[238,65112,65114,65134],{"className":65113},[395,848],[238,65115,65117,65131],{"className":65116},[399],[238,65118,65120],{"className":65119,"style":855},[403],[238,65121,65122,65125],{"style":1165},[238,65123],{"className":65124,"style":412},[411],[238,65126,65128],{"className":65127},[416,417,418,419],[238,65129,820],{"className":65130},[282,283,419],[238,65132,872],{"className":65133},[871],[238,65135,65137],{"className":65136},[399],[238,65138,65140],{"className":65139,"style":879},[403],[238,65141],{},[238,65143],{"className":65144,"style":430},[429],[238,65146,992],{"className":65147},[434],[238,65149],{"className":65150,"style":430},[429],[238,65152,65154,65157],{"className":65153},[273],[238,65155],{"className":65156,"style":489},[277],[238,65158,341],{"className":65159},[282,283]," 处分解回报：",[238,65162,65164],{"className":65163,"translate":242},[945],[238,65165,65167,65294],{"className":65166,"translate":242},[241],[238,65168,65170],{"className":65169},[246],[248,65171,65172],{"xmlns":250,"display":954},[252,65173,65174,65291],{},[958,65175,65176,65257],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,65177,65178,65198],{},[966,65179,65180],{},[969,65181,65182],{"scriptlevel":740,"displaystyle":269},[255,65183,65184,65192,65194,65196],{},[10131,65185,65186,65188,65190],{},[258,65187,4429],{},[258,65189,820],{},[258,65191,2297],{},[332,65193,335],{"stretchy":334},[258,65195,341],{},[332,65197,360],{"stretchy":334},[966,65199,65200],{},[969,65201,65202],{"scriptlevel":740,"displaystyle":269},[255,65203,65204,65206,65208,65214,65217,65223,65225,65227,65241,65243,65255],{},[255,65205],{},[332,65207,992],{},[812,65209,65210,65212],{},[258,65211,4452],{"mathvariant":620},[258,65213,2297],{},[332,65215,736],{"fence":334,"stretchy":269,"minsize":65216,"maxsize":65216},"1.8em",[812,65218,65219,65221],{},[258,65220,600],{},[258,65222,820],{},[332,65224,987],{},[258,65226,730],{},[10131,65228,65229,65231,65239],{},[258,65230,4429],{},[255,65232,65233,65235,65237],{},[258,65234,820],{},[332,65236,987],{},[738,65238,745],{},[258,65240,2297],{},[332,65242,335],{"stretchy":334},[812,65244,65245,65247],{},[258,65246,341],{},[255,65248,65249,65251,65253],{},[258,65250,820],{},[332,65252,987],{},[738,65254,745],{},[332,65256,360],{"stretchy":334},[963,65258,65259,65265],{},[966,65260,65261],{},[969,65262,65263],{"scriptlevel":740,"displaystyle":269},[255,65264],{},[966,65266,65267],{},[969,65268,65269],{"scriptlevel":740,"displaystyle":269},[255,65270,65271,65273,65275,65277,65283,65285,65287,65289],{},[255,65272],{},[429,65274],{"width":10450},[332,65276,349],{},[812,65278,65279,65281],{},[258,65280,341],{},[258,65282,820],{},[332,65284,992],{},[258,65286,341],{},[332,65288,4479],{"fence":334,"stretchy":269,"minsize":65216,"maxsize":65216},[258,65290,3102],{"mathvariant":344},[262,65292,65293],{"encoding":264},"\\begin{aligned}\nV_t^\\pi(s)\n&=\\mathbb{E}_\\pi\\Big[r_t+\\gamma V_{t+1}^\\pi(s_{t+1})\\\\\n&\\qquad\\mid s_t=s\\Big].\n\\end{aligned}",[238,65295,65297],{"className":65296,"ariaHidden":269},[268],[238,65298,65300,65304],{"className":65299},[273],[238,65301],{"className":65302,"style":65303},[277],"height:3.9em;vertical-align:-1.7em;",[238,65305,65307],{"className":65306},[282],[238,65308,65310,65412],{"className":65309},[958],[238,65311,65313],{"className":65312},[1117],[238,65314,65316,65403],{"className":65315},[395,848],[238,65317,65319,65400],{"className":65318},[399],[238,65320,65323,65392],{"className":65321,"style":65322},[403],"height:2.2em;",[238,65324,65326,65329],{"style":65325},"top:-4.2em;",[238,65327],{"className":65328,"style":33299},[411],[238,65330,65332,65383,65386,65389],{"className":65331},[282],[238,65333,65335,65338],{"className":65334},[282],[238,65336,4429],{"className":65337,"style":660},[282,283],[238,65339,65341],{"className":65340},[391],[238,65342,65344,65375],{"className":65343},[395,848],[238,65345,65347,65372],{"className":65346},[399],[238,65348,65350,65361],{"className":65349,"style":4620},[403],[238,65351,65352,65355],{"style":10172},[238,65353],{"className":65354,"style":412},[411],[238,65356,65358],{"className":65357},[416,417,418,419],[238,65359,820],{"className":65360},[282,283,419],[238,65362,65363,65366],{"style":1226},[238,65364],{"className":65365,"style":412},[411],[238,65367,65369],{"className":65368},[416,417,418,419],[238,65370,2297],{"className":65371,"style":2313},[282,283,419],[238,65373,872],{"className":65374},[871],[238,65376,65378],{"className":65377},[399],[238,65379,65381],{"className":65380,"style":10202},[403],[238,65382],{},[238,65384,335],{"className":65385},[381],[238,65387,341],{"className":65388},[282,283],[238,65390,360],{"className":65391},[462],[238,65393,65394,65397],{"style":6033},[238,65395],{"className":65396,"style":33299},[411],[238,65398],{"className":65399},[282],[238,65401,872],{"className":65402},[871],[238,65404,65406],{"className":65405},[399],[238,65407,65410],{"className":65408,"style":65409},[403],"height:1.7em;",[238,65411],{},[238,65413,65415],{"className":65414},[1422],[238,65416,65418,65744],{"className":65417},[395,848],[238,65419,65421,65741],{"className":65420},[399],[238,65422,65424,65657],{"className":65423,"style":65322},[403],[238,65425,65426,65429],{"style":65325},[238,65427],{"className":65428,"style":33299},[411],[238,65430,65432,65435,65438,65441,65444,65484,65490,65530,65533,65536,65539,65542,65602,65605,65654],{"className":65431},[282],[238,65433],{"className":65434},[282],[238,65436],{"className":65437,"style":430},[429],[238,65439,992],{"className":65440},[434],[238,65442],{"className":65443,"style":430},[429],[238,65445,65447,65450],{"className":65446},[282],[238,65448,4452],{"className":65449},[282,714],[238,65451,65453],{"className":65452},[391],[238,65454,65456,65476],{"className":65455},[395,848],[238,65457,65459,65473],{"className":65458},[399],[238,65460,65462],{"className":65461,"style":1927},[403],[238,65463,65464,65467],{"style":1165},[238,65465],{"className":65466,"style":412},[411],[238,65468,65470],{"className":65469},[416,417,418,419],[238,65471,2297],{"className":65472,"style":2313},[282,283,419],[238,65474,872],{"className":65475},[871],[238,65477,65479],{"className":65478},[399],[238,65480,65482],{"className":65481,"style":879},[403],[238,65483],{},[238,65485,65487],{"className":65486},[282],[238,65488,736],{"className":65489},[18719,33403],[238,65491,65493,65496],{"className":65492},[282],[238,65494,600],{"className":65495,"style":637},[282,283],[238,65497,65499],{"className":65498},[391],[238,65500,65502,65522],{"className":65501},[395,848],[238,65503,65505,65519],{"className":65504},[399],[238,65506,65508],{"className":65507,"style":855},[403],[238,65509,65510,65513],{"style":3136},[238,65511],{"className":65512,"style":412},[411],[238,65514,65516],{"className":65515},[416,417,418,419],[238,65517,820],{"className":65518},[282,283,419],[238,65520,872],{"className":65521},[871],[238,65523,65525],{"className":65524},[399],[238,65526,65528],{"className":65527,"style":879},[403],[238,65529],{},[238,65531],{"className":65532,"style":660},[429],[238,65534,987],{"className":65535},[664],[238,65537],{"className":65538,"style":660},[429],[238,65540,730],{"className":65541,"style":764},[282,283],[238,65543,65545,65548],{"className":65544},[282],[238,65546,4429],{"className":65547,"style":660},[282,283],[238,65549,65551],{"className":65550},[391],[238,65552,65554,65594],{"className":65553},[395,848],[238,65555,65557,65591],{"className":65556},[399],[238,65558,65560,65580],{"className":65559,"style":4620},[403],[238,65561,65562,65565],{"style":10172},[238,65563],{"className":65564,"style":412},[411],[238,65566,65568],{"className":65567},[416,417,418,419],[238,65569,65571,65574,65577],{"className":65570},[282,419],[238,65572,820],{"className":65573},[282,283,419],[238,65575,987],{"className":65576},[664,419],[238,65578,745],{"className":65579},[282,419],[238,65581,65582,65585],{"style":1226},[238,65583],{"className":65584,"style":412},[411],[238,65586,65588],{"className":65587},[416,417,418,419],[238,65589,2297],{"className":65590,"style":2313},[282,283,419],[238,65592,872],{"className":65593},[871],[238,65595,65597],{"className":65596},[399],[238,65598,65600],{"className":65599,"style":64649},[403],[238,65601],{},[238,65603,335],{"className":65604},[381],[238,65606,65608,65611],{"className":65607},[282],[238,65609,341],{"className":65610},[282,283],[238,65612,65614],{"className":65613},[391],[238,65615,65617,65646],{"className":65616},[395,848],[238,65618,65620,65643],{"className":65619},[399],[238,65621,65623],{"className":65622,"style":1162},[403],[238,65624,65625,65628],{"style":1165},[238,65626],{"className":65627,"style":412},[411],[238,65629,65631],{"className":65630},[416,417,418,419],[238,65632,65634,65637,65640],{"className":65633},[282,419],[238,65635,820],{"className":65636},[282,283,419],[238,65638,987],{"className":65639},[664,419],[238,65641,745],{"className":65642},[282,419],[238,65644,872],{"className":65645},[871],[238,65647,65649],{"className":65648},[399],[238,65650,65652],{"className":65651,"style":1193},[403],[238,65653],{},[238,65655,360],{"className":65656},[462],[238,65658,65659,65662],{"style":6033},[238,65660],{"className":65661,"style":33299},[411],[238,65663,65665,65668,65671,65674,65677,65680,65720,65723,65726,65729,65732,65738],{"className":65664},[282],[238,65666],{"className":65667},[282],[238,65669],{"className":65670,"style":10986},[429],[238,65672],{"className":65673,"style":430},[429],[238,65675,349],{"className":65676},[434],[238,65678],{"className":65679,"style":430},[429],[238,65681,65683,65686],{"className":65682},[282],[238,65684,341],{"className":65685},[282,283],[238,65687,65689],{"className":65688},[391],[238,65690,65692,65712],{"className":65691},[395,848],[238,65693,65695,65709],{"className":65694},[399],[238,65696,65698],{"className":65697,"style":855},[403],[238,65699,65700,65703],{"style":1165},[238,65701],{"className":65702,"style":412},[411],[238,65704,65706],{"className":65705},[416,417,418,419],[238,65707,820],{"className":65708},[282,283,419],[238,65710,872],{"className":65711},[871],[238,65713,65715],{"className":65714},[399],[238,65716,65718],{"className":65717,"style":879},[403],[238,65719],{},[238,65721],{"className":65722,"style":430},[429],[238,65724,992],{"className":65725},[434],[238,65727],{"className":65728,"style":430},[429],[238,65730,341],{"className":65731},[282,283],[238,65733,65735],{"className":65734},[282],[238,65736,4479],{"className":65737},[18719,33403],[238,65739,3102],{"className":65740},[282],[238,65742,872],{"className":65743},[871],[238,65745,65747],{"className":65746},[399],[238,65748,65750],{"className":65749,"style":65409},[403],[238,65751],{},[11,65753,65754,65755,65856,65857,49615],{},"由 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的定义，",[238,65858,65860,65902],{"className":65859,"translate":242},[241],[238,65861,65863],{"className":65862},[246],[248,65864,65865],{"xmlns":250},[252,65866,65867,65899],{},[255,65868,65869,65883,65885],{},[10131,65870,65871,65873,65881],{},[258,65872,4429],{},[255,65874,65875,65877,65879],{},[258,65876,820],{},[332,65878,987],{},[738,65880,745],{},[258,65882,2297],{},[332,65884,2023],{},[10131,65886,65887,65889,65897],{},[258,65888,4429],{},[255,65890,65891,65893,65895],{},[258,65892,820],{},[332,65894,987],{},[738,65896,745],{},[332,65898,22083],{},[262,65900,65901],{"encoding":264},"V_{t+1}^\\pi\\le 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时刻可达的最优值。取 ",[238,67476,67478,67496],{"className":67477,"translate":242},[241],[238,67479,67481],{"className":67480},[246],[248,67482,67483],{"xmlns":250},[252,67484,67485,67493],{},[255,67486,67487],{},[812,67488,67489,67491],{},[258,67490,357],{},[258,67492,820],{},[262,67494,67495],{"encoding":264},"a_t",[238,67497,67499],{"className":67498,"ariaHidden":269},[268],[238,67500,67502,67505],{"className":67501},[273],[238,67503],{"className":67504,"style":3115},[277],[238,67506,67508,67511],{"className":67507},[282],[238,67509,357],{"className":67510},[282,283],[238,67512,67514],{"className":67513},[391],[238,67515,67517,67537],{"className":67516},[395,848],[238,67518,67520,67534],{"className":67519},[399],[238,67521,67523],{"className":67522,"style":855},[403],[238,67524,67525,67528],{"style":1165},[238,67526],{"className":67527,"style":412},[411],[238,67529,67531],{"className":67530},[416,417,418,419],[238,67532,820],{"className":67533},[282,283,419],[238,67535,872],{"className":67536},[871],[238,67538,67540],{"className":67539},[399],[238,67541,67543],{"className":67542,"style":879},[403],[238,67544],{}," 为使上式右端最大的动作，并从 ",[238,67547,67549,67566],{"className":67548,"translate":242},[241],[238,67550,67552],{"className":67551},[246],[248,67553,67554],{"xmlns":250},[252,67555,67556,67564],{},[255,67557,67558,67560,67562],{},[258,67559,820],{},[332,67561,987],{},[738,67563,745],{},[262,67565,65006],{"encoding":264},[238,67567,67569,67587],{"className":67568,"ariaHidden":269},[268],[238,67570,67572,67575,67578,67581,67584],{"className":67571},[273],[238,67573],{"className":67574,"style":65016},[277],[238,67576,820],{"className":67577},[282,283],[238,67579],{"className":67580,"style":660},[429],[238,67582,987],{"className":67583},[664],[238,67585],{"className":67586,"style":660},[429],[238,67588,67590,67593],{"className":67589},[273],[238,67591],{"className":67592,"style":2672},[277],[238,67594,745],{"className":67595},[282]," 开始采用达到 ",[238,67598,67600,67625],{"className":67599,"translate":242},[241],[238,67601,67603],{"className":67602},[246],[248,67604,67605],{"xmlns":250},[252,67606,67607,67623],{},[255,67608,67609],{},[10131,67610,67611,67613,67621],{},[258,67612,4429],{},[255,67614,67615,67617,67619],{},[258,67616,820],{},[332,67618,987],{},[738,67620,745],{},[332,67622,22083],{},[262,67624,65783],{"encoding":264},[238,67626,67628],{"className":67627,"ariaHidden":269},[268],[238,67629,67631,67634],{"className":67630},[273],[238,67632],{"className":67633,"style":65793},[277],[238,67635,67637,67640],{"className":67636},[282],[238,67638,4429],{"className":67639,"style":660},[282,283],[238,67641,67643],{"className":67642},[391],[238,67644,67646,67686],{"className":67645},[395,848],[238,67647,67649,67683],{"className":67648},[399],[238,67650,67652,67672],{"className":67651,"style":22096},[403],[238,67653,67654,67657],{"style":65814},[238,67655],{"className":67656,"style":412},[411],[238,67658,67660],{"className":67659},[416,417,418,419],[238,67661,67663,67666,67669],{"className":67662},[282,419],[238,67664,820],{"className":67665},[282,283,419],[238,67667,987],{"className":67668},[664,419],[238,67670,745],{"className":67671},[282,419],[238,67673,67674,67677],{"style":407},[238,67675],{"className":67676,"style":412},[411],[238,67678,67680],{"className":67679},[416,417,418,419],[238,67681,22083],{"className":67682},[664,419],[238,67684,872],{"className":67685},[871],[238,67687,67689],{"className":67688},[399],[238,67690,67692],{"className":67691,"style":65853},[403],[238,67693],{}," 的最优续策略，则等号可以达到。在有限状态与有限动作下，可以逐状态选取最优续策略，因此不需要额外的可测选择条件。故 ",[238,67696,67698,67740],{"className":67697,"translate":242},[241],[238,67699,67701],{"className":67700},[246],[248,67702,67703],{"xmlns":250},[252,67704,67705,67737],{},[255,67706,67707,67715,67717,67723],{},[10131,67708,67709,67711,67713],{},[258,67710,4429],{},[258,67712,820],{},[332,67714,22083],{},[332,67716,992],{},[337,67718,67719,67721],{},[258,67720,11971],{"mathvariant":816},[332,67722,22083],{},[10131,67724,67725,67727,67735],{},[258,67726,4429],{},[255,67728,67729,67731,67733],{},[258,67730,820],{},[332,67732,987],{},[738,67734,745],{},[332,67736,22083],{},[262,67738,67739],{"encoding":264},"V_t^*=\\mathcal{T}^*V_{t+1}^*",[238,67741,67743,67810],{"className":67742,"ariaHidden":269},[268],[238,67744,67746,67750,67801,67804,67807],{"className":67745},[273],[238,67747],{"className":67748,"style":67749},[277],"height:0.9357em;vertical-align:-0.247em;",[238,67751,67753,67756],{"className":67752},[282],[238,67754,4429],{"className":67755,"style":660},[282,283],[238,67757,67759],{"className":67758},[391],[238,67760,67762,67793],{"className":67761},[395,848],[238,67763,67765,67790],{"className":67764},[399],[238,67766,67768,67779],{"className":67767,"style":22096},[403],[238,67769,67770,67773],{"style":10172},[238,67771],{"className":67772,"style":412},[411],[238,67774,67776],{"className":67775},[416,417,418,419],[238,67777,820],{"className":67778},[282,283,419],[238,67780,67781,67784],{"style":407},[238,67782],{"className":67783,"style":412},[411],[238,67785,67787],{"className":67786},[416,417,418,419],[238,67788,22083],{"className":67789},[664,419],[238,67791,872],{"className":67792},[871],[238,67794,67796],{"className":67795},[399],[238,67797,67799],{"className":67798,"style":10202},[403],[238,67800],{},[238,67802],{"className":67803,"style":430},[429],[238,67805,992],{"className":67806},[434],[238,67808],{"className":67809,"style":430},[429],[238,67811,67813,67816,67845],{"className":67812},[273],[238,67814],{"className":67815,"style":65793},[277],[238,67817,67819,67822],{"className":67818},[282],[238,67820,11971],{"className":67821,"style":11992},[282,840],[238,67823,67825],{"className":67824},[391],[238,67826,67828],{"className":67827},[395],[238,67829,67831],{"className":67830},[399],[238,67832,67834],{"className":67833,"style":22096},[403],[238,67835,67836,67839],{"style":407},[238,67837],{"className":67838,"style":412},[411],[238,67840,67842],{"className":67841},[416,417,418,419],[238,67843,22083],{"className":67844},[664,419],[238,67846,67848,67851],{"className":67847},[282],[238,67849,4429],{"className":67850,"style":660},[282,283],[238,67852,67854],{"className":67853},[391],[238,67855,67857,67897],{"className":67856},[395,848],[238,67858,67860,67894],{"className":67859},[399],[238,67861,67863,67883],{"className":67862,"style":22096},[403],[238,67864,67865,67868],{"style":65814},[238,67866],{"className":67867,"style":412},[411],[238,67869,67871],{"className":67870},[416,417,418,419],[238,67872,67874,67877,67880],{"className":67873},[282,419],[238,67875,820],{"className":67876},[282,283,419],[238,67878,987],{"className":67879},[664,419],[238,67881,745],{"className":67882},[282,419],[238,67884,67885,67888],{"style":407},[238,67886],{"className":67887,"style":412},[411],[238,67889,67891],{"className":67890},[416,417,418,419],[238,67892,22083],{"className":67893},[664,419],[238,67895,872],{"className":67896},[871],[238,67898,67900],{"className":67899},[399],[238,67901,67903],{"className":67902,"style":65853},[403],[238,67904],{},"。策略评估式同理。证毕。",[11,67907,67908,67909,67937,67938,68035,68036,68120,68121,68205],{},"这个定理就是反向归纳的数学形式。它说明，在时刻 ",[238,67910,67912,67925],{"className":67911,"translate":242},[241],[238,67913,67915],{"className":67914},[246],[248,67916,67917],{"xmlns":250},[252,67918,67919,67923],{},[255,67920,67921],{},[258,67922,820],{},[262,67924,820],{"encoding":264},[238,67926,67928],{"className":67927,"ariaHidden":269},[268],[238,67929,67931,67934],{"className":67930},[273],[238,67932],{"className":67933,"style":908},[277],[238,67935,820],{"className":67936},[282,283],"，最优决策只需要知道当前状态与下一时刻的最优值函数 ",[238,67939,67941,67966],{"className":67940,"translate":242},[241],[238,67942,67944],{"className":67943},[246],[248,67945,67946],{"xmlns":250},[252,67947,67948,67964],{},[255,67949,67950],{},[10131,67951,67952,67954,67962],{},[258,67953,4429],{},[255,67955,67956,67958,67960],{},[258,67957,820],{},[332,67959,987],{},[738,67961,745],{},[332,67963,22083],{},[262,67965,65783],{"encoding":264},[238,67967,67969],{"className":67968,"ariaHidden":269},[268],[238,67970,67972,67975],{"className":67971},[273],[238,67973],{"className":67974,"style":65793},[277],[238,67976,67978,67981],{"className":67977},[282],[238,67979,4429],{"className":67980,"style":660},[282,283],[238,67982,67984],{"className":67983},[391],[238,67985,67987,68027],{"className":67986},[395,848],[238,67988,67990,68024],{"className":67989},[399],[238,67991,67993,68013],{"className":67992,"style":22096},[403],[238,67994,67995,67998],{"style":65814},[238,67996],{"className":67997,"style":412},[411],[238,67999,68001],{"className":68000},[416,417,418,419],[238,68002,68004,68007,68010],{"className":68003},[282,419],[238,68005,820],{"className":68006},[282,283,419],[238,68008,987],{"className":68009},[664,419],[238,68011,745],{"className":68012},[282,419],[238,68014,68015,68018],{"style":407},[238,68016],{"className":68017,"style":412},[411],[238,68019,68021],{"className":68020},[416,417,418,419],[238,68022,22083],{"className":68023},[664,419],[238,68025,872],{"className":68026},[871],[238,68028,68030],{"className":68029},[399],[238,68031,68033],{"className":68032,"style":65853},[403],[238,68034],{},"。一旦后继状态 ",[238,68037,68039,68062],{"className":68038,"translate":242},[241],[238,68040,68042],{"className":68041},[246],[248,68043,68044],{"xmlns":250},[252,68045,68046,68060],{},[255,68047,68048],{},[812,68049,68050,68052],{},[258,68051,341],{},[255,68053,68054,68056,68058],{},[258,68055,820],{},[332,68057,987],{},[738,68059,745],{},[262,68061,3824],{"encoding":264},[238,68063,68065],{"className":68064,"ariaHidden":269},[268],[238,68066,68068,68071],{"className":68067},[273],[238,68069],{"className":68070,"style":3834},[277],[238,68072,68074,68077],{"className":68073},[282],[238,68075,341],{"className":68076},[282,283],[238,68078,68080],{"className":68079},[391],[238,68081,68083,68112],{"className":68082},[395,848],[238,68084,68086,68109],{"className":68085},[399],[238,68087,68089],{"className":68088,"style":1162},[403],[238,68090,68091,68094],{"style":1165},[238,68092],{"className":68093,"style":412},[411],[238,68095,68097],{"className":68096},[416,417,418,419],[238,68098,68100,68103,68106],{"className":68099},[282,419],[238,68101,820],{"className":68102},[282,283,419],[238,68104,987],{"className":68105},[664,419],[238,68107,745],{"className":68108},[282,419],[238,68110,872],{"className":68111},[871],[238,68113,68115],{"className":68114},[399],[238,68116,68118],{"className":68117,"style":1193},[403],[238,68119],{}," 确定，从 ",[238,68122,68124,68147],{"className":68123,"translate":242},[241],[238,68125,68127],{"className":68126},[246],[248,68128,68129],{"xmlns":250},[252,68130,68131,68145],{},[255,68132,68133],{},[812,68134,68135,68137],{},[258,68136,341],{},[255,68138,68139,68141,68143],{},[258,68140,820],{},[332,68142,987],{},[738,68144,745],{},[262,68146,3824],{"encoding":264},[238,68148,68150],{"className":68149,"ariaHidden":269},[268],[238,68151,68153,68156],{"className":68152},[273],[238,68154],{"className":68155,"style":3834},[277],[238,68157,68159,68162],{"className":68158},[282],[238,68160,341],{"className":68161},[282,283],[238,68163,68165],{"className":68164},[391],[238,68166,68168,68197],{"className":68167},[395,848],[238,68169,68171,68194],{"className":68170},[399],[238,68172,68174],{"className":68173,"style":1162},[403],[238,68175,68176,68179],{"style":1165},[238,68177],{"className":68178,"style":412},[411],[238,68180,68182],{"className":68181},[416,417,418,419],[238,68183,68185,68188,68191],{"className":68184},[282,419],[238,68186,820],{"className":68187},[282,283,419],[238,68189,987],{"className":68190},[664,419],[238,68192,745],{"className":68193},[282,419],[238,68195,872],{"className":68196},[871],[238,68198,68200],{"className":68199},[399],[238,68201,68203],{"className":68202,"style":1193},[403],[238,68204],{}," 开始的剩余决策必须构成该子问题的最优策略。否则，替换该续策略就可以严格改进原策略，与整体最优性矛盾。这就是贝尔曼最优性原理在最简单情形下的完整证明。",[225,68207,68209],{"id":68208},"_62-无限期情形","6.2 无限期情形",[11,68211,68212,68213,68215,68216,68274],{},"无限期问题不能直接使用反向归纳，因为不存在一个最后的时刻。但第 4 节的压缩映射论证给出了同样的结论。",[357,68214,37451],{"href":37450}," 表明，无限期最优贝尔曼方程就是 ",[238,68217,68219,68236],{"className":68218,"translate":242},[241],[238,68220,68222],{"className":68221},[246],[248,68223,68224],{"xmlns":250},[252,68225,68226,68234],{},[255,68227,68228],{},[337,68229,68230,68232],{},[258,68231,11971],{"mathvariant":816},[332,68233,22083],{},[262,68235,22086],{"encoding":264},[238,68237,68239],{"className":68238,"ariaHidden":269},[268],[238,68240,68242,68245],{"className":68241},[273],[238,68243],{"className":68244,"style":22096},[277],[238,68246,68248,68251],{"className":68247},[282],[238,68249,11971],{"className":68250,"style":11992},[282,840],[238,68252,68254],{"className":68253},[391],[238,68255,68257],{"className":68256},[395],[238,68258,68260],{"className":68259},[399],[238,68261,68263],{"className":68262,"style":22096},[403],[238,68264,68265,68268],{"style":407},[238,68266],{"className":68267,"style":412},[411],[238,68269,68271],{"className":68270},[416,417,418,419],[238,68272,22083],{"className":68273},[664,419]," 的不动点方程，而最优值函数正是该方程的唯一有界解。",[11,68276,68277,68278,68493],{},"因此，最优子结构在无限期情形下不是一条需要额外假设的公理，而是折扣、马尔可夫性与可加效用三者共同作用的结果。折扣保证了算子的压缩性，马尔可夫性保证后继状态足以概括历史，可加效用保证了回报可以分解为 ",[238,68279,68281,68322],{"className":68280,"translate":242},[241],[238,68282,68284],{"className":68283},[246],[248,68285,68286],{"xmlns":250},[252,68287,68288,68320],{},[255,68289,68290,68296,68298,68304,68306,68308],{},[812,68291,68292,68294],{},[258,68293,3380],{},[258,68295,820],{},[332,68297,992],{},[812,68299,68300,68302],{},[258,68301,600],{},[258,68303,820],{},[332,68305,987],{},[258,68307,730],{},[812,68309,68310,68312],{},[258,68311,3380],{},[255,68313,68314,68316,68318],{},[258,68315,820],{},[332,68317,987],{},[738,68319,745],{},[262,68321,7662],{"encoding":264},[238,68323,68325,68380,68435],{"className":68324,"ariaHidden":269},[268],[238,68326,68328,68331,68371,68374,68377],{"className":68327},[273],[238,68329],{"className":68330,"style":833},[277],[238,68332,68334,68337],{"className":68333},[282],[238,68335,3380],{"className":68336},[282,283],[238,68338,68340],{"className":68339},[391],[238,68341,68343,68363],{"className":68342},[395,848],[238,68344,68346,68360],{"className":68345},[399],[238,68347,68349],{"className":68348,"style":855},[403],[238,68350,68351,68354],{"style":1165},[238,68352],{"className":68353,"style":412},[411],[238,68355,68357],{"className":68356},[416,417,418,419],[238,68358,820],{"className":68359},[282,283,419],[238,68361,872],{"className":68362},[871],[238,68364,68366],{"className":68365},[399],[238,68367,68369],{"className":68368,"style":879},[403],[238,68370],{},[238,68372],{"className":68373,"style":430},[429],[238,68375,992],{"className":68376},[434],[238,68378],{"className":68379,"style":430},[429],[238,68381,68383,68386,68426,68429,68432],{"className":68382},[273],[238,68384],{"className":68385,"style":7727},[277],[238,68387,68389,68392],{"className":68388},[282],[238,68390,600],{"className":68391,"style":637},[282,283],[238,68393,68395],{"className":68394},[391],[238,68396,68398,68418],{"className":68397},[395,848],[238,68399,68401,68415],{"className":68400},[399],[238,68402,68404],{"className":68403,"style":855},[403],[238,68405,68406,68409],{"style":3136},[238,68407],{"className":68408,"style":412},[411],[238,68410,68412],{"className":68411},[416,417,418,419],[238,68413,820],{"className":68414},[282,283,419],[238,68416,872],{"className":68417},[871],[238,68419,68421],{"className":68420},[399],[238,68422,68424],{"className":68423,"style":879},[403],[238,68425],{},[238,68427],{"className":68428,"style":660},[429],[238,68430,987],{"className":68431},[664],[238,68433],{"className":68434,"style":660},[429],[238,68436,68438,68441,68444],{"className":68437},[273],[238,68439],{"className":68440,"style":7783},[277],[238,68442,730],{"className":68443,"style":764},[282,283],[238,68445,68447,68450],{"className":68446},[282],[238,68448,3380],{"className":68449},[282,283],[238,68451,68453],{"className":68452},[391],[238,68454,68456,68485],{"className":68455},[395,848],[238,68457,68459,68482],{"className":68458},[399],[238,68460,68462],{"className":68461,"style":1162},[403],[238,68463,68464,68467],{"style":1165},[238,68465],{"className":68466,"style":412},[411],[238,68468,68470],{"className":68469},[416,417,418,419],[238,68471,68473,68476,68479],{"className":68472},[282,419],[238,68474,820],{"className":68475},[282,283,419],[238,68477,987],{"className":68478},[664,419],[238,68480,745],{"className":68481},[282,419],[238,68483,872],{"className":68484},[871],[238,68486,68488],{"className":68487},[399],[238,68489,68491],{"className":68490,"style":1193},[403],[238,68492],{},"。缺少其中任何一个条件，贝尔曼方程的标准形式都可能失效。",[225,68495,68497],{"id":68496},"_63-成立条件","6.3 成立条件",[11,68499,68500],{},"最优子结构的成立依赖以下三个条件。",[11,68502,68503,68504,68704,68705,68920],{},"第一，效用必须是可加且按常数折扣的。也就是说，回报必须能够写成 ",[238,68505,68507,68551],{"className":68506,"translate":242},[241],[238,68508,68510],{"className":68509},[246],[248,68511,68512],{"xmlns":250},[252,68513,68514,68548],{},[255,68515,68516,68530,68536],{},[10131,68517,68518,68520,68528],{},[332,68519,3390],{},[255,68521,68522,68524,68526],{},[258,68523,3395],{},[332,68525,992],{},[738,68527,740],{},[258,68529,1895],{"mathvariant":344},[337,68531,68532,68534],{},[258,68533,730],{},[258,68535,3395],{},[812,68537,68538,68540],{},[258,68539,600],{},[255,68541,68542,68544,68546],{},[258,68543,820],{},[332,68545,987],{},[258,68547,3395],{},[262,68549,68550],{"encoding":264},"\\sum_{k=0}^{\\infty}\\gamma^k r_{t+k}",[238,68552,68554],{"className":68553,"ariaHidden":269},[268],[238,68555,68557,68560,68623,68626,68655],{"className":68556},[273],[238,68558],{"className":68559,"style":15918},[277],[238,68561,68563,68566],{"className":68562},[1943],[238,68564,3390],{"className":68565,"style":4152},[1943,3538,4151],[238,68567,68569],{"className":68568},[391],[238,68570,68572,68615],{"className":68571},[395,848],[238,68573,68575,68612],{"className":68574},[399],[238,68576,68578,68598],{"className":68577,"style":15937},[403],[238,68579,68580,68583],{"style":4168},[238,68581],{"className":68582,"style":412},[411],[238,68584,68586],{"className":68585},[416,417,418,419],[238,68587,68589,68592,68595],{"className":68588},[282,419],[238,68590,3395],{"className":68591,"style":3520},[282,283,419],[238,68593,992],{"className":68594},[434,419],[238,68596,740],{"className":68597},[282,419],[238,68599,68600,68603],{"style":15960},[238,68601],{"className":68602,"style":412},[411],[238,68604,68606],{"className":68605},[416,417,418,419],[238,68607,68609],{"className":68608},[282,419],[238,68610,1895],{"className":68611},[282,419],[238,68613,872],{"className":68614},[871],[238,68616,68618],{"className":68617},[399],[238,68619,68621],{"className":68620,"style":4224},[403],[238,68622],{},[238,68624],{"className":68625,"style":455},[429],[238,68627,68629,68632],{"className":68628},[282],[238,68630,730],{"className":68631,"style":764},[282,283],[238,68633,68635],{"className":68634},[391],[238,68636,68638],{"className":68637},[395],[238,68639,68641],{"className":68640},[399],[238,68642,68644],{"className":68643,"style":15776},[403],[238,68645,68646,68649],{"style":407},[238,68647],{"className":68648,"style":412},[411],[238,68650,68652],{"className":68651},[416,417,418,419],[238,68653,3395],{"className":68654,"style":3520},[282,283,419],[238,68656,68658,68661],{"className":68657},[282],[238,68659,600],{"className":68660,"style":637},[282,283],[238,68662,68664],{"className":68663},[391],[238,68665,68667,68696],{"className":68666},[395,848],[238,68668,68670,68693],{"className":68669},[399],[238,68671,68673],{"className":68672,"style":3618},[403],[238,68674,68675,68678],{"style":3136},[238,68676],{"className":68677,"style":412},[411],[238,68679,68681],{"className":68680},[416,417,418,419],[238,68682,68684,68687,68690],{"className":68683},[282,419],[238,68685,820],{"className":68686},[282,283,419],[238,68688,987],{"className":68689},[664,419],[238,68691,3395],{"className":68692,"style":3520},[282,283,419],[238,68694,872],{"className":68695},[871],[238,68697,68699],{"className":68698},[399],[238,68700,68702],{"className":68701,"style":1193},[403],[238,68703],{},"。这个形式保证了 ",[238,68706,68708,68749],{"className":68707,"translate":242},[241],[238,68709,68711],{"className":68710},[246],[248,68712,68713],{"xmlns":250},[252,68714,68715,68747],{},[255,68716,68717,68723,68725,68731,68733,68735],{},[812,68718,68719,68721],{},[258,68720,3380],{},[258,68722,820],{},[332,68724,992],{},[812,68726,68727,68729],{},[258,68728,600],{},[258,68730,820],{},[332,68732,987],{},[258,68734,730],{},[812,68736,68737,68739],{},[258,68738,3380],{},[255,68740,68741,68743,68745],{},[258,68742,820],{},[332,68744,987],{},[738,68746,745],{},[262,68748,7662],{"encoding":264},[238,68750,68752,68807,68862],{"className":68751,"ariaHidden":269},[268],[238,68753,68755,68758,68798,68801,68804],{"className":68754},[273],[238,68756],{"className":68757,"style":833},[277],[238,68759,68761,68764],{"className":68760},[282],[238,68762,3380],{"className":68763},[282,283],[238,68765,68767],{"className":68766},[391],[238,68768,68770,68790],{"className":68769},[395,848],[238,68771,68773,68787],{"className":68772},[399],[238,68774,68776],{"className":68775,"style":855},[403],[238,68777,68778,68781],{"style":1165},[238,68779],{"className":68780,"style":412},[411],[238,68782,68784],{"className":68783},[416,417,418,419],[238,68785,820],{"className":68786},[282,283,419],[238,68788,872],{"className":68789},[871],[238,68791,68793],{"className":68792},[399],[238,68794,68796],{"className":68795,"style":879},[403],[238,68797],{},[238,68799],{"className":68800,"style":430},[429],[238,68802,992],{"className":68803},[434],[238,68805],{"className":68806,"style":430},[429],[238,68808,68810,68813,68853,68856,68859],{"className":68809},[273],[238,68811],{"className":68812,"style":7727},[277],[238,68814,68816,68819],{"className":68815},[282],[238,68817,600],{"className":68818,"style":637},[282,283],[238,68820,68822],{"className":68821},[391],[238,68823,68825,68845],{"className":68824},[395,848],[238,68826,68828,68842],{"className":68827},[399],[238,68829,68831],{"className":68830,"style":855},[403],[238,68832,68833,68836],{"style":3136},[238,68834],{"className":68835,"style":412},[411],[238,68837,68839],{"className":68838},[416,417,418,419],[238,68840,820],{"className":68841},[282,283,419],[238,68843,872],{"className":68844},[871],[238,68846,68848],{"className":68847},[399],[238,68849,68851],{"className":68850,"style":879},[403],[238,68852],{},[238,68854],{"className":68855,"style":660},[429],[238,68857,987],{"className":68858},[664],[238,68860],{"className":68861,"style":660},[429],[238,68863,68865,68868,68871],{"className":68864},[273],[238,68866],{"className":68867,"style":7783},[277],[238,68869,730],{"className":68870,"style":764},[282,283],[238,68872,68874,68877],{"className":68873},[282],[238,68875,3380],{"className":68876},[282,283],[238,68878,68880],{"className":68879},[391],[238,68881,68883,68912],{"className":68882},[395,848],[238,68884,68886,68909],{"className":68885},[399],[238,68887,68889],{"className":68888,"style":1162},[403],[238,68890,68891,68894],{"style":1165},[238,68892],{"className":68893,"style":412},[411],[238,68895,68897],{"className":68896},[416,417,418,419],[238,68898,68900,68903,68906],{"className":68899},[282,419],[238,68901,820],{"className":68902},[282,283,419],[238,68904,987],{"className":68905},[664,419],[238,68907,745],{"className":68908},[282,419],[238,68910,872],{"className":68911},[871],[238,68913,68915],{"className":68914},[399],[238,68916,68918],{"className":68917,"style":1193},[403],[238,68919],{},"。若效用是历史整体的非线性函数，例如某些非可加偏好，那么未来就不能被压缩成一个标量值函数。双曲折扣是一个典型例子。在双曲折扣下，今天的偏好与明天的偏好可能不一致，因而不存在一个与时间无关的递归值函数。",[11,68922,68923,68924,68993],{},"第二，状态必须满足马尔可夫性。若未来的转移依赖于更早的历史，当前状态 ",[238,68925,68927,68944],{"className":68926,"translate":242},[241],[238,68928,68930],{"className":68929},[246],[248,68931,68932],{"xmlns":250},[252,68933,68934,68942],{},[255,68935,68936],{},[812,68937,68938,68940],{},[258,68939,341],{},[258,68941,820],{},[262,68943,3747],{"encoding":264},[238,68945,68947],{"className":68946,"ariaHidden":269},[268],[238,68948,68950,68953],{"className":68949},[273],[238,68951],{"className":68952,"style":3115},[277],[238,68954,68956,68959],{"className":68955},[282],[238,68957,341],{"className":68958},[282,283],[238,68960,68962],{"className":68961},[391],[238,68963,68965,68985],{"className":68964},[395,848],[238,68966,68968,68982],{"className":68967},[399],[238,68969,68971],{"className":68970,"style":855},[403],[238,68972,68973,68976],{"style":1165},[238,68974],{"className":68975,"style":412},[411],[238,68977,68979],{"className":68978},[416,417,418,419],[238,68980,820],{"className":68981},[282,283,419],[238,68983,872],{"className":68984},[871],[238,68986,68988],{"className":68987},[399],[238,68989,68991],{"className":68990,"style":879},[403],[238,68992],{}," 就不是充分的统计量。此时可以把状态扩充为包含必要历史信息的新状态，例如部分可观测问题中的信念状态。在扩充后的状态空间上，贝尔曼方程仍然可以保持相同的形式。",[11,68995,68996,68997,69047,69048,69098],{},"第三，回报必须有界，或者折扣必须足够强。有限期问题即使 ",[238,68998,69000,69017],{"className":68999,"translate":242},[241],[238,69001,69003],{"className":69002},[246],[248,69004,69005],{"xmlns":250},[252,69006,69007,69015],{},[255,69008,69009,69011,69013],{},[258,69010,730],{},[332,69012,992],{},[738,69014,745],{},[262,69016,29189],{"encoding":264},[238,69018,69020,69038],{"className":69019,"ariaHidden":269},[268],[238,69021,69023,69026,69029,69032,69035],{"className":69022},[273],[238,69024],{"className":69025,"style":2853},[277],[238,69027,730],{"className":69028,"style":764},[282,283],[238,69030],{"className":69031,"style":430},[429],[238,69033,992],{"className":69034},[434],[238,69036],{"className":69037,"style":430},[429],[238,69039,69041,69044],{"className":69040},[273],[238,69042],{"className":69043,"style":2672},[277],[238,69045,745],{"className":69046},[282]," 也可以有有界回报，因为求和的项数是有限的。无限期问题则需要 ",[238,69049,69051,69068],{"className":69050,"translate":242},[241],[238,69052,69054],{"className":69053},[246],[248,69055,69056],{"xmlns":250},[252,69057,69058,69066],{},[255,69059,69060,69062,69064],{},[258,69061,730],{},[332,69063,1892],{},[738,69065,745],{},[262,69067,17495],{"encoding":264},[238,69069,69071,69089],{"className":69070,"ariaHidden":269},[268],[238,69072,69074,69077,69080,69083,69086],{"className":69073},[273],[238,69075],{"className":69076,"style":760},[277],[238,69078,730],{"className":69079,"style":764},[282,283],[238,69081],{"className":69082,"style":430},[429],[238,69084,1892],{"className":69085},[434],[238,69087],{"className":69088,"style":430},[429],[238,69090,69092,69095],{"className":69091},[273],[238,69093],{"className":69094,"style":2672},[277],[238,69096,745],{"className":69097},[282],"，或者采用平均奖励、吸收态等替代准则。否则，值函数可能发散，或者贝尔曼方程的解不唯一。",[11,69100,69101],{},"这三个条件共同说明，最优子结构是一个有明确适用范围的定理，而不是一条无条件成立的哲学原则。",[220,69103,69105],{"id":69104},"七边界与常见误区","七、边界与常见误区",[225,69107,69109],{"id":69108},"_71-连续状态空间","7.1 连续状态空间",[11,69111,69112,69113,69156],{},"在标准的可测性假设下，连续状态空间中的 ",[238,69114,69116,69135],{"className":69115,"translate":242},[241],[238,69117,69119],{"className":69118},[246],[248,69120,69121],{"xmlns":250},[252,69122,69123,69133],{},[255,69124,69125,69127,69129,69131],{},[258,69126,19971],{},[332,69128,335],{"stretchy":334},[258,69130,260],{},[332,69132,360],{"stretchy":334},[262,69134,21579],{"encoding":264},[238,69136,69138],{"className":69137,"ariaHidden":269},[268],[238,69139,69141,69144,69147,69150,69153],{"className":69140},[273],[238,69142],{"className":69143,"style":444},[277],[238,69145,19971],{"className":69146,"style":20011},[282,283],[238,69148,335],{"className":69149},[381],[238,69151,260],{"className":69152,"style":284},[282,283],[238,69154,360],{"className":69155},[462]," 仍然是 Banach 空间，压缩映射的论证也仍然成立。因此，贝尔曼期望方程的唯一有界解依然存在，贝尔曼最优算子也依然有唯一有界不动点，值迭代的几何收敛性同样成立。在适当的可测选择条件下，该不动点等于最优值函数。",[11,69158,69159,69160,69218],{},"变化主要发生在 ",[238,69161,69163,69180],{"className":69162,"translate":242},[241],[238,69164,69166],{"className":69165},[246],[248,69167,69168],{"xmlns":250},[252,69169,69170,69178],{},[255,69171,69172],{},[337,69173,69174,69176],{},[258,69175,11971],{"mathvariant":816},[332,69177,22083],{},[262,69179,22086],{"encoding":264},[238,69181,69183],{"className":69182,"ariaHidden":269},[268],[238,69184,69186,69189],{"className":69185},[273],[238,69187],{"className":69188,"style":22096},[277],[238,69190,69192,69195],{"className":69191},[282],[238,69193,11971],{"className":69194,"style":11992},[282,840],[238,69196,69198],{"className":69197},[391],[238,69199,69201],{"className":69200},[395],[238,69202,69204],{"className":69203},[399],[238,69205,69207],{"className":69206,"style":22096},[403],[238,69208,69209,69212],{"style":407},[238,69210],{"className":69211,"style":412},[411],[238,69213,69215],{"className":69214},[416,417,418,419],[238,69216,22083],{"className":69217},[664,419]," 的定义上。此时最大值未必取到，应将其写成",[238,69220,69222],{"className":69221,"translate":242},[945],[238,69223,69225,69376],{"className":69224,"translate":242},[241],[238,69226,69228],{"className":69227},[246],[248,69229,69230],{"xmlns":250,"display":954},[252,69231,69232,69373],{},[958,69233,69234,69311],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,69235,69236,69260],{},[966,69237,69238],{},[969,69239,69240],{"scriptlevel":740,"displaystyle":269},[255,69241,69242,69244,69250,69252,69254,69256,69258],{},[332,69243,335],{"stretchy":334},[337,69245,69246,69248],{},[258,69247,11971],{"mathvariant":816},[332,69249,22083],{},[258,69251,4429],{},[332,69253,360],{"stretchy":334},[332,69255,335],{"stretchy":334},[258,69257,341],{},[332,69259,360],{"stretchy":334},[966,69261,69262],{},[969,69263,69264],{"scriptlevel":740,"displaystyle":269},[255,69265,69266,69268,69270,69280,69282,69289,69291,69293,69295,69297,69299,69301,69303,69309],{},[255,69267],{},[332,69269,992],{},[6755,69271,69272,69278],{},[255,69273,69274,69276],{},[258,69275,21664],{},[332,69277,1889],{},[258,69279,357],{},[332,69281,13150],{"fence":334,"stretchy":269,"minsize":63310,"maxsize":63310},[812,69283,69284,69287],{},[332,69285,69286],{},"∫",[258,69288,260],{},[332,69290,736],{"fence":334,"stretchy":269,"minsize":65216,"maxsize":65216},[258,69292,600],{},[332,69294,335],{"stretchy":334},[258,69296,341],{},[332,69298,354],{"separator":269},[258,69300,357],{},[332,69302,354],{"separator":269},[337,69304,69305,69307],{},[258,69306,341],{},[332,69308,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,69310,360],{"stretchy":334},[963,69312,69313,69319],{},[966,69314,69315],{},[969,69316,69317],{"scriptlevel":740,"displaystyle":269},[255,69318],{},[966,69320,69321],{},[969,69322,69323],{"scriptlevel":740,"displaystyle":269},[255,69324,69325,69327,69329,69331,69333,69335,69337,69343,69345,69347,69349,69351,69353,69359,69361,69363,69365,69367,69369,69371],{},[255,69326],{},[429,69328],{"width":10450},[332,69330,987],{},[258,69332,730],{},[258,69334,4429],{},[332,69336,335],{"stretchy":334},[337,69338,69339,69341],{},[258,69340,341],{},[332,69342,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,69344,360],{"stretchy":334},[332,69346,4479],{"fence":334,"stretchy":269,"minsize":65216,"maxsize":65216},[258,69348,330],{},[332,69350,335],{"stretchy":334},[258,69352,29624],{"mathvariant":344},[337,69354,69355,69357],{},[258,69356,341],{},[332,69358,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,69360,349],{},[258,69362,341],{},[332,69364,354],{"separator":269},[258,69366,357],{},[332,69368,360],{"stretchy":334},[332,69370,13164],{"fence":334,"stretchy":269,"minsize":63310,"maxsize":63310},[258,69372,3102],{"mathvariant":344},[262,69374,69375],{"encoding":264},"\\begin{aligned}\n(\\mathcal{T}^*V)(s)\n&=\\sup_{a}\\Bigg\\{\\int_S \\Big[r(s,a,s')\\\\\n&\\qquad+\\gamma V(s')\\Big]P(\\mathrm{d}s'\\mid s,a)\\Bigg\\}.\n\\end{aligned}",[238,69377,69379],{"className":69378,"ariaHidden":269},[268],[238,69380,69382,69386],{"className":69381},[273],[238,69383],{"className":69384,"style":69385},[277],"height:6.3001em;vertical-align:-2.9em;",[238,69387,69389],{"className":69388},[282],[238,69390,69392,69482],{"className":69391},[958],[238,69393,69395],{"className":69394},[1117],[238,69396,69398,69473],{"className":69397},[395,848],[238,69399,69401,69470],{"className":69400},[399],[238,69402,69405,69462],{"className":69403,"style":69404},[403],"height:3.4em;",[238,69406,69408,69412],{"style":69407},"top:-5.4em;",[238,69409],{"className":69410,"style":69411},[411],"height:3.75em;",[238,69413,69415,69418,69447,69450,69453,69456,69459],{"className":69414},[282],[238,69416,335],{"className":69417},[381],[238,69419,69421,69424],{"className":69420},[282],[238,69422,11971],{"className":69423,"style":11992},[282,840],[238,69425,69427],{"className":69426},[391],[238,69428,69430],{"className":69429},[395],[238,69431,69433],{"className":69432},[399],[238,69434,69436],{"className":69435,"style":22441},[403],[238,69437,69438,69441],{"style":1226},[238,69439],{"className":69440,"style":412},[411],[238,69442,69444],{"className":69443},[416,417,418,419],[238,69445,22083],{"className":69446},[664,419],[238,69448,4429],{"className":69449,"style":660},[282,283],[238,69451,360],{"className":69452},[462],[238,69454,335],{"className":69455},[381],[238,69457,341],{"className":69458},[282,283],[238,69460,360],{"className":69461},[462],[238,69463,69464,69467],{"style":6033},[238,69465],{"className":69466,"style":69411},[411],[238,69468],{"className":69469},[282],[238,69471,872],{"className":69472},[871],[238,69474,69476],{"className":69475},[399],[238,69477,69480],{"className":69478,"style":69479},[403],"height:2.9em;",[238,69481],{},[238,69483,69485],{"className":69484},[1422],[238,69486,69488,69835],{"className":69487},[395,848],[238,69489,69491,69832],{"className":69490},[399],[238,69492,69494,69685],{"className":69493,"style":69404},[403],[238,69495,69496,69499],{"style":69407},[238,69497],{"className":69498,"style":69411},[411],[238,69500,69502,69505,69508,69511,69514,69561,69564,69570,69573,69617,69620,69626,69629,69632,69635,69638,69641,69644,69647,69650,69682],{"className":69501},[282],[238,69503],{"className":69504},[282],[238,69506],{"className":69507,"style":430},[429],[238,69509,992],{"className":69510},[434],[238,69512],{"className":69513,"style":430},[429],[238,69515,69517],{"className":69516},[1943,3493],[238,69518,69520,69553],{"className":69519},[395,848],[238,69521,69523,69550],{"className":69522},[399],[238,69524,69526,69540],{"className":69525,"style":489},[403],[238,69527,69528,69531],{"style":36928},[238,69529],{"className":69530,"style":1134},[411],[238,69532,69534],{"className":69533},[416,417,418,419],[238,69535,69537],{"className":69536},[282,419],[238,69538,357],{"className":69539},[282,283,419],[238,69541,69542,69545],{"style":21798},[238,69543],{"className":69544,"style":1134},[411],[238,69546,69547],{},[238,69548,21664],{"className":69549},[1943],[238,69551,872],{"className":69552},[871],[238,69554,69556],{"className":69555},[399],[238,69557,69559],{"className":69558,"style":36960},[403],[238,69560],{},[238,69562],{"className":69563,"style":455},[429],[238,69565,69567],{"className":69566},[282],[238,69568,13150],{"className":69569},[18719,63687],[238,69571],{"className":69572,"style":455},[429],[238,69574,69576,69580],{"className":69575},[1943],[238,69577,69286],{"className":69578,"style":69579},[1943,3538,3539],"margin-right:0.4445em;position:relative;top:-0.0011em;",[238,69581,69583],{"className":69582},[391],[238,69584,69586,69608],{"className":69585},[395,848],[238,69587,69589,69605],{"className":69588},[399],[238,69590,69593],{"className":69591,"style":69592},[403],"height:-0.4336em;",[238,69594,69596,69599],{"style":69595},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[238,69597],{"className":69598,"style":412},[411],[238,69600,69602],{"className":69601},[416,417,418,419],[238,69603,260],{"className":69604,"style":284},[282,283,419],[238,69606,872],{"className":69607},[871],[238,69609,69611],{"className":69610},[399],[238,69612,69615],{"className":69613,"style":69614},[403],"height:0.9119em;",[238,69616],{},[238,69618],{"className":69619,"style":455},[429],[238,69621,69623],{"className":69622},[282],[238,69624,736],{"className":69625},[18719,33403],[238,69627,600],{"className":69628,"style":637},[282,283],[238,69630,335],{"className":69631},[381],[238,69633,341],{"className":69634},[282,283],[238,69636,354],{"className":69637},[451],[238,69639],{"className":69640,"style":455},[429],[238,69642,357],{"className":69643},[282,283],[238,69645,354],{"className":69646},[451],[238,69648],{"className":69649,"style":455},[429],[238,69651,69653,69656],{"className":69652},[282],[238,69654,341],{"className":69655},[282,283],[238,69657,69659],{"className":69658},[391],[238,69660,69662],{"className":69661},[395],[238,69663,69665],{"className":69664},[399],[238,69666,69668],{"className":69667,"style":1223},[403],[238,69669,69670,69673],{"style":1226},[238,69671],{"className":69672,"style":412},[411],[238,69674,69676],{"className":69675},[416,417,418,419],[238,69677,69679],{"className":69678},[282,419],[238,69680,346],{"className":69681},[282,419],[238,69683,360],{"className":69684},[462],[238,69686,69687,69690],{"style":6033},[238,69688],{"className":69689,"style":69411},[411],[238,69691,69693,69696,69699,69702,69705,69708,69711,69714,69717,69749,69752,69758,69761,69764,69767,69799,69802,69805,69808,69811,69814,69817,69820,69823,69829],{"className":69692},[282],[238,69694],{"className":69695},[282],[238,69697],{"className":69698,"style":10986},[429],[238,69700],{"className":69701,"style":660},[429],[238,69703,987],{"className":69704},[664],[238,69706],{"className":69707,"style":660},[429],[238,69709,730],{"className":69710,"style":764},[282,283],[238,69712,4429],{"className":69713,"style":660},[282,283],[238,69715,335],{"className":69716},[381],[238,69718,69720,69723],{"className":69719},[282],[238,69721,341],{"className":69722},[282,283],[238,69724,69726],{"className":69725},[391],[238,69727,69729],{"className":69728},[395],[238,69730,69732],{"className":69731},[399],[238,69733,69735],{"className":69734,"style":1223},[403],[238,69736,69737,69740],{"style":1226},[238,69738],{"className":69739,"style":412},[411],[238,69741,69743],{"className":69742},[416,417,418,419],[238,69744,69746],{"className":69745},[282,419],[238,69747,346],{"className":69748},[282,419],[238,69750,360],{"className":69751},[462],[238,69753,69755],{"className":69754},[282],[238,69756,4479],{"className":69757},[18719,33403],[238,69759,330],{"className":69760,"style":377},[282,283],[238,69762,335],{"className":69763},[381],[238,69765,29624],{"className":69766},[282,35276],[238,69768,69770,69773],{"className":69769},[282],[238,69771,341],{"className":69772},[282,283],[238,69774,69776],{"className":69775},[391],[238,69777,69779],{"className":69778},[395],[238,69780,69782],{"className":69781},[399],[238,69783,69785],{"className":69784,"style":1223},[403],[238,69786,69787,69790],{"style":1226},[238,69788],{"className":69789,"style":412},[411],[238,69791,69793],{"className":69792},[416,417,418,419],[238,69794,69796],{"className":69795},[282,419],[238,69797,346],{"className":69798},[282,419],[238,69800],{"className":69801,"style":430},[429],[238,69803,349],{"className":69804},[434],[238,69806],{"className":69807,"style":430},[429],[238,69809,341],{"className":69810},[282,283],[238,69812,354],{"className":69813},[451],[238,69815],{"className":69816,"style":455},[429],[238,69818,357],{"className":69819},[282,283],[238,69821,360],{"className":69822},[462],[238,69824,69826],{"className":69825},[282],[238,69827,13164],{"className":69828},[18719,63687],[238,69830,3102],{"className":69831},[282],[238,69833,872],{"className":69834},[871],[238,69836,69838],{"className":69837},[399],[238,69839,69841],{"className":69840,"style":69479},[403],[238,69842],{},[11,69844,69845,69846,69904,69905,69933],{},"压缩性证明中的「最大值保持逐点序」可以替换为「上确界保持逐点序」，因此 ",[238,69847,69849,69866],{"className":69848,"translate":242},[241],[238,69850,69852],{"className":69851},[246],[248,69853,69854],{"xmlns":250},[252,69855,69856,69864],{},[255,69857,69858],{},[337,69859,69860,69862],{},[258,69861,11971],{"mathvariant":816},[332,69863,22083],{},[262,69865,22086],{"encoding":264},[238,69867,69869],{"className":69868,"ariaHidden":269},[268],[238,69870,69872,69875],{"className":69871},[273],[238,69873],{"className":69874,"style":22096},[277],[238,69876,69878,69881],{"className":69877},[282],[238,69879,11971],{"className":69880,"style":11992},[282,840],[238,69882,69884],{"className":69883},[391],[238,69885,69887],{"className":69886},[395],[238,69888,69890],{"className":69889},[399],[238,69891,69893],{"className":69892,"style":22096},[403],[238,69894,69895,69898],{"style":407},[238,69896],{"className":69897,"style":412},[411],[238,69899,69901],{"className":69900},[416,417,418,419],[238,69902,22083],{"className":69903},[664,419]," 仍然是 ",[238,69906,69908,69921],{"className":69907,"translate":242},[241],[238,69909,69911],{"className":69910},[246],[248,69912,69913],{"xmlns":250},[252,69914,69915,69919],{},[255,69916,69917],{},[258,69918,730],{},[262,69920,26785],{"encoding":264},[238,69922,69924],{"className":69923,"ariaHidden":269},[268],[238,69925,69927,69930],{"className":69926},[273],[238,69928],{"className":69929,"style":2853},[277],[238,69931,730],{"className":69932,"style":764},[282,283],"-压缩映射。唯一不动点仍然存在。",[11,69935,69936,69937,69995,69996,70094],{},"但是，最优策略的存在性需要额外条件。即使 ",[238,69938,69940,69957],{"className":69939,"translate":242},[241],[238,69941,69943],{"className":69942},[246],[248,69944,69945],{"xmlns":250},[252,69946,69947,69955],{},[255,69948,69949],{},[337,69950,69951,69953],{},[258,69952,11971],{"mathvariant":816},[332,69954,22083],{},[262,69956,22086],{"encoding":264},[238,69958,69960],{"className":69959,"ariaHidden":269},[268],[238,69961,69963,69966],{"className":69962},[273],[238,69964],{"className":69965,"style":22096},[277],[238,69967,69969,69972],{"className":69968},[282],[238,69970,11971],{"className":69971,"style":11992},[282,840],[238,69973,69975],{"className":69974},[391],[238,69976,69978],{"className":69977},[395],[238,69979,69981],{"className":69980},[399],[238,69982,69984],{"className":69983,"style":22096},[403],[238,69985,69986,69989],{"style":407},[238,69987],{"className":69988,"style":412},[411],[238,69990,69992],{"className":69991},[416,417,418,419],[238,69993,22083],{"className":69994},[664,419]," 有唯一不动点，也不一定存在一个可测的映射 ",[238,69997,69999,70028],{"className":69998,"translate":242},[241],[238,70000,70002],{"className":70001},[246],[248,70003,70004],{"xmlns":250},[252,70005,70006,70025],{},[255,70007,70008,70010,70013,70019,70021,70023],{},[258,70009,341],{},[332,70011,70012],{},"↦",[337,70014,70015,70017],{},[258,70016,2297],{},[332,70018,22083],{},[332,70020,335],{"stretchy":334},[258,70022,341],{},[332,70024,360],{"stretchy":334},[262,70026,70027],{"encoding":264},"s\\mapsto\\pi^*(s)",[238,70029,70031,70050],{"className":70030,"ariaHidden":269},[268],[238,70032,70034,70038,70041,70044,70047],{"className":70033},[273],[238,70035],{"className":70036,"style":70037},[277],"height:0.522em;vertical-align:-0.011em;",[238,70039,341],{"className":70040},[282,283],[238,70042],{"className":70043,"style":430},[429],[238,70045,70012],{"className":70046},[434],[238,70048],{"className":70049,"style":430},[429],[238,70051,70053,70056,70085,70088,70091],{"className":70052},[273],[238,70054],{"className":70055,"style":444},[277],[238,70057,70059,70062],{"className":70058},[282],[238,70060,2297],{"className":70061,"style":2313},[282,283],[238,70063,70065],{"className":70064},[391],[238,70066,70068],{"className":70067},[395],[238,70069,70071],{"className":70070},[399],[238,70072,70074],{"className":70073,"style":22096},[403],[238,70075,70076,70079],{"style":407},[238,70077],{"className":70078,"style":412},[411],[238,70080,70082],{"className":70081},[416,417,418,419],[238,70083,22083],{"className":70084},[664,419],[238,70086,335],{"className":70087},[381],[238,70089,341],{"className":70090},[282,283],[238,70092,360],{"className":70093},[462]," 在每个状态都取到上确界。通常需要动作集紧、奖励与转移核连续，或者使用可测选择定理。本文的有限状态与有限动作假设避开了这些技术问题。这也是为什么在一般情形下，贝尔曼最优方程常写成上确界形式，而不是最大值形式。",[225,70096,70098],{"id":70097},"_72-平均奖励与无折扣","7.2 平均奖励与无折扣",[11,70100,13131,70101,70151,70152,70182,70183,70212],{},[238,70102,70104,70121],{"className":70103,"translate":242},[241],[238,70105,70107],{"className":70106},[246],[248,70108,70109],{"xmlns":250},[252,70110,70111,70119],{},[255,70112,70113,70115,70117],{},[258,70114,730],{},[332,70116,992],{},[738,70118,745],{},[262,70120,29189],{"encoding":264},[238,70122,70124,70142],{"className":70123,"ariaHidden":269},[268],[238,70125,70127,70130,70133,70136,70139],{"className":70126},[273],[238,70128],{"className":70129,"style":2853},[277],[238,70131,730],{"className":70132,"style":764},[282,283],[238,70134],{"className":70135,"style":430},[429],[238,70137,992],{"className":70138},[434],[238,70140],{"className":70141,"style":430},[429],[238,70143,70145,70148],{"className":70144},[273],[238,70146],{"className":70147,"style":2672},[277],[238,70149,745],{"className":70150},[282]," 且时间范围为无限时，折扣回报可能发散，贝尔曼方程的标准形式不再适用。此时常用的替代准则是平均奖励准则。它引入增益 ",[238,70153,70155,70170],{"className":70154,"translate":242},[241],[238,70156,70158],{"className":70157},[246],[248,70159,70160],{"xmlns":250},[252,70161,70162,70167],{},[255,70163,70164],{},[258,70165,70166],{},"ρ",[262,70168,70169],{"encoding":264},"\\rho",[238,70171,70173],{"className":70172,"ariaHidden":269},[268],[238,70174,70176,70179],{"className":70175},[273],[238,70177],{"className":70178,"style":2853},[277],[238,70180,70166],{"className":70181},[282,283]," 与偏差函数 ",[238,70184,70186,70200],{"className":70185,"translate":242},[241],[238,70187,70189],{"className":70188},[246],[248,70190,70191],{"xmlns":250},[252,70192,70193,70198],{},[255,70194,70195],{},[258,70196,70197],{},"h",[262,70199,70197],{"encoding":264},[238,70201,70203],{"className":70202,"ariaHidden":269},[268],[238,70204,70206,70209],{"className":70205},[273],[238,70207],{"className":70208,"style":29023},[277],[238,70210,70197],{"className":70211},[282,283],"，其最优方程形如",[238,70214,70216],{"className":70215,"translate":242},[945],[238,70217,70219,70355],{"className":70218,"translate":242},[241],[238,70220,70222],{"className":70221},[246],[248,70223,70224],{"xmlns":250,"display":954},[252,70225,70226,70352],{},[958,70227,70228,70286],{"rowspacing":960,"columnalign":961,"columnspacing":345},[963,70229,70230,70248],{},[966,70231,70232],{},[969,70233,70234],{"scriptlevel":740,"displaystyle":269},[255,70235,70236,70238,70240,70242,70244,70246],{},[258,70237,70166],{},[332,70239,987],{},[258,70241,70197],{},[332,70243,335],{"stretchy":334},[258,70245,341],{},[332,70247,360],{"stretchy":334},[966,70249,70250],{},[969,70251,70252],{"scriptlevel":740,"displaystyle":269},[255,70253,70254,70256,70258,70268,70270,70276,70278,70280,70282,70284],{},[255,70255],{},[332,70257,992],{},[6755,70259,70260,70266],{},[255,70261,70262,70264],{},[258,70263,1886],{},[332,70265,1889],{},[258,70267,357],{},[332,70269,13150],{"fence":334,"stretchy":269,"minsize":63310,"maxsize":63310},[39888,70271,70272,70274],{"accent":269},[258,70273,600],{},[332,70275,39894],{},[332,70277,335],{"stretchy":334},[258,70279,341],{},[332,70281,354],{"separator":269},[258,70283,357],{},[332,70285,360],{"stretchy":334},[963,70287,70288,70294],{},[966,70289,70290],{},[969,70291,70292],{"scriptlevel":740,"displaystyle":269},[255,70293],{},[966,70295,70296],{},[969,70297,70298],{"scriptlevel":740,"displaystyle":269},[255,70299,70300,70302,70304,70306,70316,70318,70320,70326,70328,70330,70332,70334,70336,70338,70340,70346,70348,70350],{},[255,70301],{},[429,70303],{"width":10450},[332,70305,987],{},[6755,70307,70308,70310],{},[332,70309,3390],{},[337,70311,70312,70314],{},[258,70313,341],{},[332,70315,346],{"mathvariant":344,"lspace":345,"rspace":345},[258,70317,330],{},[332,70319,335],{"stretchy":334},[337,70321,70322,70324],{},[258,70323,341],{},[332,70325,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,70327,349],{},[258,70329,341],{},[332,70331,354],{"separator":269},[258,70333,357],{},[332,70335,360],{"stretchy":334},[258,70337,70197],{},[332,70339,335],{"stretchy":334},[337,70341,70342,70344],{},[258,70343,341],{},[332,70345,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,70347,360],{"stretchy":334},[332,70349,13164],{"fence":334,"stretchy":269,"minsize":63310,"maxsize":63310},[332,70351,354],{"separator":269},[262,70353,70354],{"encoding":264},"\\begin{aligned}\n\\rho+h(s)&=\\max_{a}\\Bigg\\{\\bar r(s,a)\\\\\n&\\qquad+\\sum_{s'}P(s'\\mid s,a)h(s')\\Bigg\\},\n\\end{aligned}",[238,70356,70358],{"className":70357,"ariaHidden":269},[268],[238,70359,70361,70365],{"className":70360},[273],[238,70362],{"className":70363,"style":70364},[277],"height:6.344em;vertical-align:-2.922em;",[238,70366,70368],{"className":70367},[282],[238,70369,70371,70437],{"className":70370},[958],[238,70372,70374],{"className":70373},[1117],[238,70375,70377,70428],{"className":70376},[395,848],[238,70378,70380,70425],{"className":70379},[399],[238,70381,70384,70417],{"className":70382,"style":70383},[403],"height:3.422em;",[238,70385,70387,70390],{"style":70386},"top:-5.422em;",[238,70388],{"className":70389,"style":69411},[411],[238,70391,70393,70396,70399,70402,70405,70408,70411,70414],{"className":70392},[282],[238,70394,70166],{"className":70395},[282,283],[238,70397],{"className":70398,"style":660},[429],[238,70400,987],{"className":70401},[664],[238,70403],{"className":70404,"style":660},[429],[238,70406,70197],{"className":70407},[282,283],[238,70409,335],{"className":70410},[381],[238,70412,341],{"className":70413},[282,283],[238,70415,360],{"className":70416},[462],[238,70418,70419,70422],{"style":11303},[238,70420],{"className":70421,"style":69411},[411],[238,70423],{"className":70424},[282],[238,70426,872],{"className":70427},[871],[238,70429,70431],{"className":70430},[399],[238,70432,70435],{"className":70433,"style":70434},[403],"height:2.922em;",[238,70436],{},[238,70438,70440],{"className":70439},[1422],[238,70441,70443,70793],{"className":70442},[395,848],[238,70444,70446,70790],{"className":70445},[399],[238,70447,70449,70576],{"className":70448,"style":70383},[403],[238,70450,70451,70454],{"style":70386},[238,70452],{"className":70453,"style":69411},[411],[238,70455,70457,70460,70463,70466,70469,70516,70519,70525,70558,70561,70564,70567,70570,70573],{"className":70456},[282],[238,70458],{"className":70459},[282],[238,70461],{"className":70462,"style":430},[429],[238,70464,992],{"className":70465},[434],[238,70467],{"className":70468,"style":430},[429],[238,70470,70472],{"className":70471},[1943,3493],[238,70473,70475,70508],{"className":70474},[395,848],[238,70476,70478,70505],{"className":70477},[399],[238,70479,70481,70495],{"className":70480,"style":489},[403],[238,70482,70483,70486],{"style":22498},[238,70484],{"className":70485,"style":1134},[411],[238,70487,70489],{"className":70488},[416,417,418,419],[238,70490,70492],{"className":70491},[282,419],[238,70493,357],{"className":70494},[282,283,419],[238,70496,70497,70500],{"style":21798},[238,70498],{"className":70499,"style":1134},[411],[238,70501,70502],{},[238,70503,1886],{"className":70504},[1943],[238,70506,872],{"className":70507},[871],[238,70509,70511],{"className":70510},[399],[238,70512,70514],{"className":70513,"style":22530},[403],[238,70515],{},[238,70517],{"className":70518,"style":455},[429],[238,70520,70522],{"className":70521},[282],[238,70523,13150],{"className":70524},[18719,63687],[238,70526,70528],{"className":70527},[282,39911],[238,70529,70531],{"className":70530},[395],[238,70532,70534],{"className":70533},[399],[238,70535,70538,70546],{"className":70536,"style":70537},[403],"height:0.5678em;",[238,70539,70540,70543],{"style":21798},[238,70541],{"className":70542,"style":1134},[411],[238,70544,600],{"className":70545,"style":637},[282,283],[238,70547,70548,70551],{"style":21798},[238,70549],{"className":70550,"style":1134},[411],[238,70552,70555],{"className":70553,"style":70554},[39938],"left:-0.1944em;",[238,70556,39894],{"className":70557},[282],[238,70559,335],{"className":70560},[381],[238,70562,341],{"className":70563},[282,283],[238,70565,354],{"className":70566},[451],[238,70568],{"className":70569,"style":455},[429],[238,70571,357],{"className":70572},[282,283],[238,70574,360],{"className":70575},[462],[238,70577,70578,70581],{"style":11303},[238,70579],{"className":70580,"style":69411},[411],[238,70582,70584,70587,70590,70593,70596,70599,70675,70678,70681,70684,70716,70719,70722,70725,70728,70731,70734,70737,70740,70743,70746,70778,70781,70787],{"className":70583},[282],[238,70585],{"className":70586},[282],[238,70588],{"className":70589,"style":10986},[429],[238,70591],{"className":70592,"style":660},[429],[238,70594,987],{"className":70595},[664],[238,70597],{"className":70598,"style":660},[429],[238,70600,70602],{"className":70601},[1943,3493],[238,70603,70605,70667],{"className":70604},[395,848],[238,70606,70608,70664],{"className":70607},[399],[238,70609,70611,70654],{"className":70610,"style":6869},[403],[238,70612,70613,70616],{"style":7296},[238,70614],{"className":70615,"style":3510},[411],[238,70617,70619],{"className":70618},[416,417,418,419],[238,70620,70622],{"className":70621},[282,419],[238,70623,70625,70628],{"className":70624},[282,419],[238,70626,341],{"className":70627},[282,283,419],[238,70629,70631],{"className":70630},[391],[238,70632,70634],{"className":70633},[395],[238,70635,70637],{"className":70636},[399],[238,70638,70640],{"className":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r(s,a)=\\sum_{s'}P(s'\\mid s,a)r(s,a,s')",[238,70887,70889,70953,71081],{"className":70888,"ariaHidden":269},[268],[238,70890,70892,70895,70926,70929,70932,70935,70938,70941,70944,70947,70950],{"className":70891},[273],[238,70893],{"className":70894,"style":444},[277],[238,70896,70898],{"className":70897},[282,39911],[238,70899,70901],{"className":70900},[395],[238,70902,70904],{"className":70903},[399],[238,70905,70907,70915],{"className":70906,"style":70537},[403],[238,70908,70909,70912],{"style":21798},[238,70910],{"className":70911,"style":1134},[411],[238,70913,600],{"className":70914,"style":637},[282,283],[238,70916,70917,70920],{"style":21798},[238,70918],{"className":70919,"style":1134},[411],[238,70921,70923],{"className":70922,"style":70554},[39938],[238,70924,39894],{"className":70925},[282],[238,70927,335],{"className":70928},[381],[238,70930,341],{"className":70931},[282,283],[238,70933,354],{"className":70934},[451],[238,70936],{"className":70937,"style":455},[429],[238,70939,357],{"className":70940},[282,283],[238,70942,360],{"className":70943},[462],[238,70945],{"className":70946,"style":430},[429],[238,70948,992],{"className":70949},[434],[238,70951],{"className":70952,"style":430},[429],[238,70954,70956,70959,71031,71034,71037,71040,71072,71075,71078],{"className":70955},[273],[238,70957],{"className":70958,"style":4144},[277],[238,70960,70962,70965],{"className":70961},[1943],[238,70963,3390],{"className":70964,"style":4152},[1943,3538,4151],[238,70966,70968],{"className":70967},[391],[238,70969,70971,71023],{"className":70970},[395,848],[238,70972,70974,71020],{"className":70973},[399],[238,70975,70977],{"className":70976,"style":4165},[403],[238,70978,70979,70982],{"style":4168},[238,70980],{"className":70981,"style":412},[411],[238,70983,70985],{"className":70984},[416,417,418,419],[238,70986,70988],{"className":70987},[282,419],[238,70989,70991,70994],{"className":70990},[282,419],[238,70992,341],{"className":70993},[282,283,419],[238,70995,70997],{"className":70996},[391],[238,70998,71000],{"className":70999},[395],[238,71001,71003],{"className":71002},[399],[238,71004,71006],{"className":71005,"style":4196},[403],[238,71007,71008,71011],{"style":4199},[238,71009],{"className":71010,"style":4203},[411],[238,71012,71014],{"className":71013},[416,4207,4208,419],[238,71015,71017],{"className":71016},[282,419],[238,71018,346],{"className":71019},[282,419],[238,71021,872],{"className":71022},[871],[238,71024,71026],{"className":71025},[399],[238,71027,71029],{"className":71028,"style":4224},[403],[238,71030],{},[238,71032],{"className":71033,"style":455},[429],[238,71035,330],{"className":71036,"style":377},[282,283],[238,71038,335],{"className":71039},[381],[238,71041,71043,71046],{"className":71042},[282],[238,71044,341],{"className":71045},[282,283],[238,71047,71049],{"className":71048},[391],[238,71050,71052],{"className":71051},[395],[238,71053,71055],{"className":71054},[399],[238,71056,71058],{"className":71057,"style":404},[403],[238,71059,71060,71063],{"style":407},[238,71061],{"className":71062,"style":412},[411],[238,71064,71066],{"className":71065},[416,417,418,419],[238,71067,71069],{"className":71068},[282,419],[238,71070,346],{"className":71071},[282,419],[238,71073],{"className":71074,"style":430},[429],[238,71076,349],{"className":71077},[434],[238,71079],{"className":71080,"style":430},[429],[238,71082,71084,71087,71090,71093,71096,71099,71102,71105,71108,71111,71114,71117,71120,71123,71126,71158],{"className":71083},[273],[238,71085],{"className":71086,"style":373},[277],[238,71088,341],{"className":71089},[282,283],[238,71091,354],{"className":71092},[451],[238,71094],{"className":71095,"style":455},[429],[238,71097,357],{"className":71098},[282,283],[238,71100,360],{"className":71101},[462],[238,71103,600],{"className":71104,"style":637},[282,283],[238,71106,335],{"className":71107},[381],[238,71109,341],{"className":71110},[282,283],[238,71112,354],{"className":71113},[451],[238,71115],{"className":71116,"style":455},[429],[238,71118,357],{"className":71119},[282,283],[238,71121,354],{"className":71122},[451],[238,71124],{"className":71125,"style":455},[429],[238,71127,71129,71132],{"className":71128},[282],[238,71130,341],{"className":71131},[282,283],[238,71133,71135],{"className":71134},[391],[238,71136,71138],{"className":71137},[395],[238,71139,71141],{"className":71140},[399],[238,71142,71144],{"className":71143,"style":404},[403],[238,71145,71146,71149],{"style":407},[238,71147],{"className":71148,"style":412},[411],[238,71150,71152],{"className":71151},[416,417,418,419],[238,71153,71155],{"className":71154},[282,419],[238,71156,346],{"className":71157},[282,419],[238,71159,360],{"className":71160},[462]," 是期望即时奖励。这个方程与折扣贝尔曼方程在结构上相似，但数学性质不同。平均奖励问题通常只保证偏差函数在相差一个常数的意义下唯一，因为若 ",[238,71163,71165,71178],{"className":71164,"translate":242},[241],[238,71166,71168],{"className":71167},[246],[248,71169,71170],{"xmlns":250},[252,71171,71172,71176],{},[255,71173,71174],{},[258,71175,70197],{},[262,71177,70197],{"encoding":264},[238,71179,71181],{"className":71180,"ariaHidden":269},[268],[238,71182,71184,71187],{"className":71183},[273],[238,71185],{"className":71186,"style":29023},[277],[238,71188,70197],{"className":71189},[282,283]," 是解，则 ",[238,71192,71194,71212],{"className":71193,"translate":242},[241],[238,71195,71197],{"className":71196},[246],[248,71198,71199],{"xmlns":250},[252,71200,71201,71209],{},[255,71202,71203,71205,71207],{},[258,71204,70197],{},[332,71206,987],{},[258,71208,24276],{},[262,71210,71211],{"encoding":264},"h+c",[238,71213,71215,71234],{"className":71214,"ariaHidden":269},[268],[238,71216,71218,71222,71225,71228,71231],{"className":71217},[273],[238,71219],{"className":71220,"style":71221},[277],"height:0.7778em;vertical-align:-0.0833em;",[238,71223,70197],{"className":71224},[282,283],[238,71226],{"className":71227,"style":660},[429],[238,71229,987],{"className":71230},[664],[238,71232],{"className":71233,"style":660},[429],[238,71235,71237,71240],{"className":71236},[273],[238,71238],{"className":71239,"style":489},[277],[238,71241,24276],{"className":71242},[282,283]," 也是解。折扣因子 ",[238,71245,71247,71264],{"className":71246,"translate":242},[241],[238,71248,71250],{"className":71249},[246],[248,71251,71252],{"xmlns":250},[252,71253,71254,71262],{},[255,71255,71256,71258,71260],{},[258,71257,730],{},[332,71259,1892],{},[738,71261,745],{},[262,71263,17495],{"encoding":264},[238,71265,71267,71285],{"className":71266,"ariaHidden":269},[268],[238,71268,71270,71273,71276,71279,71282],{"className":71269},[273],[238,71271],{"className":71272,"style":760},[277],[238,71274,730],{"className":71275,"style":764},[282,283],[238,71277],{"className":71278,"style":430},[429],[238,71280,1892],{"className":71281},[434],[238,71283],{"className":71284,"style":430},[429],[238,71286,71288,71291],{"className":71287},[273],[238,71289],{"className":71290,"style":2672},[277],[238,71292,745],{"className":71293},[282]," 消除了这种平移自由度，因此折扣情形的不动点是真正唯一的。",[11,71296,71297,71298,71348,71349,71399],{},"有限期问题是一个例外。即使 ",[238,71299,71301,71318],{"className":71300,"translate":242},[241],[238,71302,71304],{"className":71303},[246],[248,71305,71306],{"xmlns":250},[252,71307,71308,71316],{},[255,71309,71310,71312,71314],{},[258,71311,730],{},[332,71313,992],{},[738,71315,745],{},[262,71317,29189],{"encoding":264},[238,71319,71321,71339],{"className":71320,"ariaHidden":269},[268],[238,71322,71324,71327,71330,71333,71336],{"className":71323},[273],[238,71325],{"className":71326,"style":2853},[277],[238,71328,730],{"className":71329,"style":764},[282,283],[238,71331],{"className":71332,"style":430},[429],[238,71334,992],{"className":71335},[434],[238,71337],{"className":71338,"style":430},[429],[238,71340,71342,71345],{"className":71341},[273],[238,71343],{"className":71344,"style":2672},[277],[238,71346,745],{"className":71347},[282],"，只要时间范围有限，回报就是有限项之和，值函数仍然有界，反向归纳仍然成立。因此，",[238,71350,71352,71369],{"className":71351,"translate":242},[241],[238,71353,71355],{"className":71354},[246],[248,71356,71357],{"xmlns":250},[252,71358,71359,71367],{},[255,71360,71361,71363,71365],{},[258,71362,730],{},[332,71364,1892],{},[738,71366,745],{},[262,71368,17495],{"encoding":264},[238,71370,71372,71390],{"className":71371,"ariaHidden":269},[268],[238,71373,71375,71378,71381,71384,71387],{"className":71374},[273],[238,71376],{"className":71377,"style":760},[277],[238,71379,730],{"className":71380,"style":764},[282,283],[238,71382],{"className":71383,"style":430},[429],[238,71385,1892],{"className":71386},[434],[238,71388],{"className":71389,"style":430},[429],[238,71391,71393,71396],{"className":71392},[273],[238,71394],{"className":71395,"style":2672},[277],[238,71397,745],{"className":71398},[282]," 的必要性主要针对无限期问题。",[225,71401,71403],{"id":71402},"_73-注意要点","7.3 注意要点",[11,71405,71406,71409,71410,71579],{},[1688,71407,71408],{},"把贝尔曼方程写成没有期望的形式。"," 在确定性转移的情形下，",[238,71411,71413,71463],{"className":71412,"translate":242},[241],[238,71414,71416],{"className":71415},[246],[248,71417,71418],{"xmlns":250},[252,71419,71420,71460],{},[255,71421,71422,71424,71426,71428,71430,71432,71434,71436,71438,71440,71442,71444,71446,71448,71450,71452,71458],{},[258,71423,4429],{},[332,71425,335],{"stretchy":334},[258,71427,341],{},[332,71429,360],{"stretchy":334},[332,71431,992],{},[258,71433,600],{},[332,71435,335],{"stretchy":334},[258,71437,341],{},[332,71439,354],{"separator":269},[258,71441,357],{},[332,71443,360],{"stretchy":334},[332,71445,987],{},[258,71447,730],{},[258,71449,4429],{},[332,71451,335],{"stretchy":334},[337,71453,71454,71456],{},[258,71455,341],{},[332,71457,346],{"mathvariant":344,"lspace":345,"rspace":345},[332,71459,360],{"stretchy":334},[262,71461,71462],{"encoding":264},"V(s)=r(s,a)+\\gamma V(s')",[238,71464,71466,71493,71529],{"className":71465,"ariaHidden":269},[268],[238,71467,71469,71472,71475,71478,71481,71484,71487,71490],{"className":71468},[273],[238,71470],{"className":71471,"style":444},[277],[238,71473,4429],{"className":71474,"style":660},[282,283],[238,71476,335],{"className":71477},[381],[238,71479,341],{"className":71480},[282,283],[238,71482,360],{"className":71483},[462],[238,71485],{"className":71486,"style":430},[429],[238,71488,992],{"className":71489},[434],[238,71491],{"className":71492,"style":430},[429],[238,71494,71496,71499,71502,71505,71508,71511,71514,71517,71520,71523,71526],{"className":71495},[273],[238,71497],{"className":71498,"style":444},[277],[238,71500,600],{"className":71501,"style":637},[282,283],[238,71503,335],{"className":71504},[381],[238,71506,341],{"className":71507},[282,283],[238,71509,354],{"className":71510},[451],[238,71512],{"className":71513,"style":455},[429],[238,71515,357],{"className":71516},[282,283],[238,71518,360],{"className":71519},[462],[238,71521],{"className":71522,"style":660},[429],[238,71524,987],{"className":71525},[664],[238,71527],{"className":71528,"style":660},[429],[238,71530,71532,71535,71538,71541,71544,71576],{"className":71531},[273],[238,71533],{"className":71534,"style":373},[277],[238,71536,730],{"className":71537,"style":764},[282,283],[238,71539,4429],{"className":71540,"style":660},[282,283],[238,71542,335],{"className":71543},[381],[238,71545,71547,71550],{"className":71546},[282],[238,71548,341],{"className":71549},[282,283],[238,71551,71553],{"className":71552},[391],[238,71554,71556],{"className":71555},[395],[238,71557,71559],{"className":71558},[399],[238,71560,71562],{"className":71561,"style":404},[403],[238,71563,71564,71567],{"style":407},[238,71565],{"className":71566,"style":412},[411],[238,71568,71570],{"className":71569},[416,417,418,419],[238,71571,71573],{"className":71572},[282,419],[238,71574,346],{"className":71575},[282,419],[238,71577,360],{"className":71578},[462]," 可以成立。但在随机转移下，必须对下一状态取期望。遗漏期望会把随机问题误写成确定性问题。",[11,71581,71582,71585,71586,71667,71668,71749],{},[1688,71583,71584],{},"把最大值放进期望内部。"," 最优方程是 ",[238,71587,71589,71612],{"className":71588,"translate":242},[241],[238,71590,71592],{"className":71591},[246],[248,71593,71594],{"xmlns":250},[252,71595,71596,71610],{},[255,71597,71598,71608],{},[812,71599,71600,71606],{},[255,71601,71602,71604],{},[258,71603,1886],{},[332,71605,1889],{},[258,71607,357],{},[258,71609,4452],{"mathvariant":620},[262,71611,45576],{"encoding":264},[238,71613,71615],{"className":71614,"ariaHidden":269},[268],[238,71616,71618,71621,71661,71664],{"className":71617},[273],[238,71619],{"className":71620,"style":45586},[277],[238,71622,71624,71627],{"className":71623},[1943],[238,71625,1886],{"className":71626},[1943],[238,71628,71630],{"className":71629},[391],[238,71631,71633,71653],{"className":71632},[395,848],[238,71634,71636,71650],{"className":71635},[399],[238,71637,71639],{"className":71638,"style":1927},[403],[238,71640,71641,71644],{"style":44220},[238,71642],{"className":71643,"style":412},[411],[238,71645,71647],{"className":71646},[416,417,418,419],[238,71648,357],{"className":71649},[282,283,419],[238,71651,872],{"className":71652},[871],[238,71654,71656],{"className":71655},[399],[238,71657,71659],{"className":71658,"style":879},[403],[238,71660],{},[238,71662],{"className":71663,"style":455},[429],[238,71665,4452],{"className":71666},[282,714],"，不是 ",[238,71669,71671,71694],{"className":71670,"translate":242},[241],[238,71672,71674],{"className":71673},[246],[248,71675,71676],{"xmlns":250},[252,71677,71678,71692],{},[255,71679,71680,71682],{},[258,71681,4452],{"mathvariant":620},[812,71683,71684,71690],{},[255,71685,71686,71688],{},[258,71687,1886],{},[332,71689,1889],{},[258,71691,357],{},[262,71693,45660],{"encoding":264},[238,71695,71697],{"className":71696,"ariaHidden":269},[268],[238,71698,71700,71703,71706,71709],{"className":71699},[273],[238,71701],{"className":71702,"style":45586},[277],[238,71704,4452],{"className":71705},[282,714],[238,71707],{"className":71708,"style":455},[429],[238,71710,71712,71715],{"className":71711},[1943],[238,71713,1886],{"className":71714},[1943],[238,71716,71718],{"className":71717},[391],[238,71719,71721,71741],{"className":71720},[395,848],[238,71722,71724,71738],{"className":71723},[399],[238,71725,71727],{"className":71726,"style":1927},[403],[238,71728,71729,71732],{"style":44220},[238,71730],{"className":71731,"style":412},[411],[238,71733,71735],{"className":71734},[416,417,418,419],[238,71736,357],{"className":71737},[282,283,419],[238,71739,872],{"className":71740},[871],[238,71742,71744],{"className":71743},[399],[238,71745,71747],{"className":71746,"style":879},[403],[238,71748],{},"。4.3 节的例子说明，二者一般相差很大。动作必须在观察到下一状态之前选定，这个时序约束正是最大值位于期望之外的原因。",[11,71751,71752,2282,71811,71885,71886,71972,71973,72125,72126,72235,72236,72280],{},[1688,71753,71754,71755,1838,71783,1806],{},"混淆 ",[238,71756,71758,71771],{"className":71757,"translate":242},[241],[238,71759,71761],{"className":71760},[246],[248,71762,71763],{"xmlns":250},[252,71764,71765,71769],{},[255,71766,71767],{},[258,71768,4429],{},[262,71770,4429],{"encoding":264},[238,71772,71774],{"className":71773,"ariaHidden":269},[268],[238,71775,71777,71780],{"className":71776},[273],[238,71778],{"className":71779,"style":278},[277],[238,71781,4429],{"className":71782,"style":660},[282,283],[238,71784,71786,71799],{"className":71785,"translate":242},[241],[238,71787,71789],{"className":71788},[246],[248,71790,71791],{"xmlns":250},[252,71792,71793,71797],{},[255,71794,71795],{},[258,71796,4494],{},[262,71798,4494],{"encoding":264},[238,71800,71802],{"className":71801,"ariaHidden":269},[268],[238,71803,71805,71808],{"className":71804},[273],[238,71806],{"className":71807,"style":6651},[277],[238,71809,4494],{"className":71810},[282,283],[238,71812,71814,71838],{"className":71813,"translate":242},[241],[238,71815,71817],{"className":71816},[246],[248,71818,71819],{"xmlns":250},[252,71820,71821,71835],{},[255,71822,71823,71829,71831,71833],{},[337,71824,71825,71827],{},[258,71826,4429],{},[258,71828,2297],{},[332,71830,335],{"stretchy":334},[258,71832,341],{},[332,71834,360],{"stretchy":334},[262,71836,71837],{"encoding":264},"V^\\pi(s)",[238,71839,71841],{"className":71840,"ariaHidden":269},[268],[238,71842,71844,71847,71876,71879,71882],{"className":71843},[273],[238,71845],{"className":71846,"style":444},[277],[238,71848,71850,71853],{"className":71849},[282],[238,71851,4429],{"className":71852,"style":660},[282,283],[238,71854,71856],{"className":71855},[391],[238,71857,71859],{"className":71858},[395],[238,71860,71862],{"className":71861},[399],[238,71863,71865],{"className":71864,"style":6609},[403],[238,71866,71867,71870],{"style":407},[238,71868],{"className":71869,"style":412},[411],[238,71871,71873],{"className":71872},[416,417,418,419],[238,71874,2297],{"className":71875,"style":2313},[282,283,419],[238,71877,335],{"className":71878},[381],[238,71880,341],{"className":71881},[282,283],[238,71883,360],{"className":71884},[462]," 已经对策略的动作分布取了平均，因此它不再区分同一状态下的不同动作。要选择动作，应使用 ",[238,71887,71889,71916],{"className":71888,"translate":242},[241],[238,71890,71892],{"className":71891},[246],[248,71893,71894],{"xmlns":250},[252,71895,71896,71914],{},[255,71897,71898,71904,71906,71908,71910,71912],{},[337,71899,71900,71902],{},[258,71901,4494],{},[332,71903,22083],{},[332,71905,335],{"stretchy":334},[258,71907,341],{},[332,71909,354],{"separator":269},[258,71911,357],{},[332,71913,360],{"stretchy":334},[262,71915,45368],{"encoding":264},[238,71917,71919],{"className":71918,"ariaHidden":269},[268],[238,71920,71922,71925,71954,71957,71960,71963,71966,71969],{"className":71921},[273],[238,71923],{"className":71924,"style":444},[277],[238,71926,71928,71931],{"className":71927},[282],[238,71929,4494],{"className":71930},[282,283],[238,71932,71934],{"className":71933},[391],[238,71935,71937],{"className":71936},[395],[238,71938,71940],{"className":71939},[399],[238,71941,71943],{"className":71942,"style":22096},[403],[238,71944,71945,71948],{"style":407},[238,71946],{"className":71947,"style":412},[411],[238,71949,71951],{"className":71950},[416,417,418,419],[238,71952,22083],{"className":71953},[664,419],[238,71955,335],{"className":71956},[381],[238,71958,341],{"className":71959},[282,283],[238,71961,354],{"className":71962},[451],[238,71964],{"className":71965,"style":455},[429],[238,71967,357],{"className":71968},[282,283],[238,71970,360],{"className":71971},[462],"，最优动作是 ",[238,71974,71976,72018],{"className":71975,"translate":242},[241],[238,71977,71979],{"className":71978},[246],[248,71980,71981],{"xmlns":250},[252,71982,71983,72015],{},[255,71984,71985,71987,71989,71999,72005,72007,72009,72011,72013],{},[258,71986,39246],{},[332,71988,1889],{},[812,71990,71991,71997],{},[255,71992,71993,71995],{},[258,71994,1886],{},[332,71996,1889],{},[258,71998,357],{},[337,72000,72001,72003],{},[258,72002,4494],{},[332,72004,22083],{},[332,72006,335],{"stretchy":334},[258,72008,341],{},[332,72010,354],{"separator":269},[258,72012,357],{},[332,72014,360],{"stretchy":334},[262,72016,72017],{"encoding":264},"\\arg\\max_a Q^*(s,a)",[238,72019,72021],{"className":72020,"ariaHidden":269},[268],[238,72022,72024,72027,72032,72035,72075,72078,72107,72110,72113,72116,72119,72122],{"className":72023},[273],[238,72025],{"className":72026,"style":444},[277],[238,72028,39405,72030],{"className":72029},[1943],[238,72031,39408],{"style":35277},[238,72033],{"className":72034,"style":455},[429],[238,72036,72038,72041],{"className":72037},[1943],[238,72039,1886],{"className":72040},[1943],[238,72042,72044],{"className":72043},[391],[238,72045,72047,72067],{"className":72046},[395,848],[238,72048,72050,72064],{"className":72049},[399],[238,72051,72053],{"className":72052,"style":1927},[403],[238,72054,72055,72058],{"style":44220},[238,72056],{"className":72057,"style":412},[411],[238,72059,72061],{"className":72060},[416,417,418,419],[238,72062,357],{"className":72063},[282,283,419],[238,72065,872],{"className":72066},[871],[238,72068,72070],{"className":72069},[399],[238,72071,72073],{"className":72072,"style":879},[403],[238,72074],{},[238,72076],{"className":72077,"style":455},[429],[238,72079,72081,72084],{"className":72080},[282],[238,72082,4494],{"className":72083},[282,283],[238,72085,72087],{"className":72086},[391],[238,72088,72090],{"className":72089},[395],[238,72091,72093],{"className":72092},[399],[238,72094,72096],{"className":72095,"style":22096},[403],[238,72097,72098,72101],{"style":407},[238,72099],{"className":72100,"style":412},[411],[238,72102,72104],{"className":72103},[416,417,418,419],[238,72105,22083],{"className":72106},[664,419],[238,72108,335],{"className":72109},[381],[238,72111,341],{"className":72112},[282,283],[238,72114,354],{"className":72115},[451],[238,72117],{"className":72118,"style":455},[429],[238,72120,357],{"className":72121},[282,283],[238,72123,360],{"className":72124},[462],"。把 ",[238,72127,72129,72163],{"className":72128,"translate":242},[241],[238,72130,72132],{"className":72131},[246],[248,72133,72134],{"xmlns":250},[252,72135,72136,72160],{},[255,72137,72138,72140,72142,72152,72154,72156,72158],{},[258,72139,39246],{},[332,72141,1889],{},[812,72143,72144,72150],{},[255,72145,72146,72148],{},[258,72147,1886],{},[332,72149,1889],{},[258,72151,357],{},[258,72153,4429],{},[332,72155,335],{"stretchy":334},[258,72157,341],{},[332,72159,360],{"stretchy":334},[262,72161,72162],{"encoding":264},"\\arg\\max_a V(s)",[238,72164,72166],{"className":72165,"ariaHidden":269},[268],[238,72167,72169,72172,72177,72180,72220,72223,72226,72229,72232],{"className":72168},[273],[238,72170],{"className":72171,"style":444},[277],[238,72173,39405,72175],{"className":72174},[1943],[238,72176,39408],{"style":35277},[238,72178],{"className":72179,"style":455},[429],[238,72181,72183,72186],{"className":72182},[1943],[238,72184,1886],{"className":72185},[1943],[238,72187,72189],{"className":72188},[391],[238,72190,72192,72212],{"className":72191},[395,848],[238,72193,72195,72209],{"className":72194},[399],[238,72196,72198],{"className":72197,"style":1927},[403],[238,72199,72200,72203],{"style":44220},[238,72201],{"className":72202,"style":412},[411],[238,72204,72206],{"className":72205},[416,417,418,419],[238,72207,357],{"className":72208},[282,283,419],[238,72210,872],{"className":72211},[871],[238,72213,72215],{"className":72214},[399],[238,72216,72218],{"className":72217,"style":879},[403],[238,72219],{},[238,72221],{"className":72222,"style":455},[429],[238,72224,4429],{"className":72225,"style":660},[282,283],[238,72227,335],{"className":72228},[381],[238,72230,341],{"className":72231},[282,283],[238,72233,360],{"className":72234},[462]," 当作最优动作是没有意义的，因为 ",[238,72237,72239,72259],{"className":72238,"translate":242},[241],[238,72240,72242],{"className":72241},[246],[248,72243,72244],{"xmlns":250},[252,72245,72246,72256],{},[255,72247,72248,72250,72252,72254],{},[258,72249,4429],{},[332,72251,335],{"stretchy":334},[258,72253,341],{},[332,72255,360],{"stretchy":334},[262,72257,72258],{"encoding":264},"V(s)",[238,72260,72262],{"className":72261,"ariaHidden":269},[268],[238,72263,72265,72268,72271,72274,72277],{"className":72264},[273],[238,72266],{"className":72267,"style":444},[277],[238,72269,4429],{"className":72270,"style":660},[282,283],[238,72272,335],{"className":72273},[381],[238,72275,341],{"className":72276},[282,283],[238,72278,360],{"className":72279},[462]," 中已经没有动作变量。",[11,72282,72283,72286,72287,72337],{},[1688,72284,72285],{},"混淆必要性、充分性与唯一性。"," 在折扣有界 MDP 中，贝尔曼方程既必要又充分，而且解唯一。但在 ",[238,72288,72290,72307],{"className":72289,"translate":242},[241],[238,72291,72293],{"className":72292},[246],[248,72294,72295],{"xmlns":250},[252,72296,72297,72305],{},[255,72298,72299,72301,72303],{},[258,72300,730],{},[332,72302,992],{},[738,72304,745],{},[262,72306,29189],{"encoding":264},[238,72308,72310,72328],{"className":72309,"ariaHidden":269},[268],[238,72311,72313,72316,72319,72322,72325],{"className":72312},[273],[238,72314],{"className":72315,"style":2853},[277],[238,72317,730],{"className":72318,"style":764},[282,283],[238,72320],{"className":72321,"style":430},[429],[238,72323,992],{"className":72324},[434],[238,72326],{"className":72327,"style":430},[429],[238,72329,72331,72334],{"className":72330},[273],[238,72332],{"className":72333,"style":2672},[277],[238,72335,745],{"className":72336},[282],"、奖励无界或状态空间不满足可测选择条件时，这些结论可能分别失效。讨论贝尔曼方程时，应明确所依赖的假设。",[11,72339,72340,72343,72344,72372],{},[1688,72341,72342],{},"认为最优策略唯一。"," 最优值函数是唯一的，但最优策略不一定唯一。例如只有一个状态、两个动作，两个动作都使状态保持不变且都获得奖励 ",[238,72345,72347,72360],{"className":72346,"translate":242},[241],[238,72348,72350],{"className":72349},[246],[248,72351,72352],{"xmlns":250},[252,72353,72354,72358],{},[255,72355,72356],{},[738,72357,745],{},[262,72359,745],{"encoding":264},[238,72361,72363],{"className":72362,"ariaHidden":269},[268],[238,72364,72366,72369],{"className":72365},[273],[238,72367],{"className":72368,"style":2672},[277],[238,72370,745],{"className":72371},[282],"。此时",[238,72374,72376],{"className":72375,"translate":242},[945],[238,72377,72379,72413],{"className":72378,"translate":242},[241],[238,72380,72382],{"className":72381},[246],[248,72383,72384],{"xmlns":250,"display":954},[252,72385,72386,72410],{},[255,72387,72388,72394,72396,72408],{},[337,72389,72390,72392],{},[258,72391,4429],{},[332,72393,22083],{},[332,72395,992],{},[5561,72397,72398,72400],{},[738,72399,745],{},[255,72401,72402,72404,72406],{},[738,72403,745],{},[332,72405,5577],{},[258,72407,730],{},[332,72409,354],{"separator":269},[262,72411,72412],{"encoding":264},"V^*=\\frac{1}{1-\\gamma},",[238,72414,72416,72460],{"className":72415,"ariaHidden":269},[268],[238,72417,72419,72422,72451,72454,72457],{"className":72418},[273],[238,72420],{"className":72421,"style":22441},[277],[238,72423,72425,72428],{"className":72424},[282],[238,72426,4429],{"className":72427,"style":660},[282,283],[238,72429,72431],{"className":72430},[391],[238,72432,72434],{"className":72433},[395],[238,72435,72437],{"className":72436},[399],[238,72438,72440],{"className":72439,"style":22441},[403],[238,72441,72442,72445],{"style":1226},[238,72443],{"className":72444,"style":412},[411],[238,72446,72448],{"className":72447},[416,417,418,419],[238,72449,22083],{"className":72450},[664,419],[238,72452],{"className":72453,"style":430},[429],[238,72455,992],{"className":72456},[434],[238,72458],{"className":72459,"style":430},[429],[238,72461,72463,72467,72541],{"className":72462},[273],[238,72464],{"className":72465,"style":72466},[277],"height:2.2019em;vertical-align:-0.8804em;",[238,72468,72470,72473,72538],{"className":72469},[282],[238,72471],{"className":72472},[381,5662],[238,72474,72476],{"className":72475},[5561],[238,72477,72479,72530],{"className":72478},[395,848],[238,72480,72482,72527],{"className":72481},[399],[238,72483,72485,72508,72516],{"className":72484,"style":18439},[403],[238,72486,72487,72490],{"style":5678},[238,72488],{"className":72489,"style":1134},[411],[238,72491,72493,72496,72499,72502,72505],{"className":72492},[282],[238,72494,745],{"className":72495},[282],[238,72497],{"className":72498,"style":660},[429],[238,72500,5577],{"className":72501},[664],[238,72503],{"className":72504,"style":660},[429],[238,72506,730],{"className":72507,"style":764},[282,283],[238,72509,72510,72513],{"style":5702},[238,72511],{"className":72512,"style":1134},[411],[238,72514],{"className":72515,"style":5710},[5709],[238,72517,72518,72521],{"style":5713},[238,72519],{"className":72520,"style":1134},[411],[238,72522,72524],{"className":72523},[282],[238,72525,745],{"className":72526},[282],[238,72528,872],{"className":72529},[871],[238,72531,72533],{"className":72532},[399],[238,72534,72536],{"className":72535,"style":5781},[403],[238,72537],{},[238,72539],{"className":72540},[462,5662],[238,72542,354],{"className":72543},[451],[11,72545,72546],{},"两个动作在每个状态都达到最大值，因此都是最优的。唯一性属于值函数，而不属于策略。",[220,72548,72549],{"id":72549},"总结",[11,72551,72552,72553,72768,72769,72825],{},"贝尔曼方程的核心可以概括为三步。第一步，利用折扣回报的递归分解 ",[238,72554,72556,72597],{"className":72555,"translate":242},[241],[238,72557,72559],{"className":72558},[246],[248,72560,72561],{"xmlns":250},[252,72562,72563,72595],{},[255,72564,72565,72571,72573,72579,72581,72583],{},[812,72566,72567,72569],{},[258,72568,3380],{},[258,72570,820],{},[332,72572,992],{},[812,72574,72575,72577],{},[258,72576,600],{},[258,72578,820],{},[332,72580,987],{},[258,72582,730],{},[812,72584,72585,72587],{},[258,72586,3380],{},[255,72588,72589,72591,72593],{},[258,72590,820],{},[332,72592,987],{},[738,72594,745],{},[262,72596,7662],{"encoding":264},[238,72598,72600,72655,72710],{"className":72599,"ariaHidden":269},[268],[238,72601,72603,72606,72646,72649,72652],{"className":72602},[273],[238,72604],{"className":72605,"style":833},[277],[238,72607,72609,72612],{"className":72608},[282],[238,72610,3380],{"className":72611},[282,283],[238,72613,72615],{"className":72614},[391],[238,72616,72618,72638],{"className":72617},[395,848],[238,72619,72621,72635],{"className":72620},[399],[238,72622,72624],{"className":72623,"style":855},[403],[238,72625,72626,72629],{"style":1165},[238,72627],{"className":72628,"style":412},[411],[238,72630,72632],{"className":72631},[416,417,418,419],[238,72633,820],{"className":72634},[282,283,419],[238,72636,872],{"className":72637},[871],[238,72639,72641],{"className":72640},[399],[238,72642,72644],{"className":72643,"style":879},[403],[238,72645],{},[238,72647],{"className":72648,"style":430},[429],[238,72650,992],{"className":72651},[434],[238,72653],{"className":72654,"style":430},[429],[238,72656,72658,72661,72701,72704,72707],{"className":72657},[273],[238,72659],{"className":72660,"style":7727},[277],[238,72662,72664,72667],{"className":72663},[282],[238,72665,600],{"className":72666,"style":637},[282,283],[238,72668,72670],{"className":72669},[391],[238,72671,72673,72693],{"className":72672},[395,848],[238,72674,72676,72690],{"className":72675},[399],[238,72677,72679],{"className":72678,"style":855},[403],[238,72680,72681,72684],{"style":3136},[238,72682],{"className":72683,"style":412},[411],[238,72685,72687],{"className":72686},[416,417,418,419],[238,72688,820],{"className":72689},[282,283,419],[238,72691,872],{"className":72692},[871],[238,72694,72696],{"className":72695},[399],[238,72697,72699],{"className":72698,"style":879},[403],[238,72700],{},[238,72702],{"className":72703,"style":660},[429],[238,72705,987],{"className":72706},[664],[238,72708],{"className":72709,"style":660},[429],[238,72711,72713,72716,72719],{"className":72712},[273],[238,72714],{"className":72715,"style":7783},[277],[238,72717,730],{"className":72718,"style":764},[282,283],[238,72720,72722,72725],{"className":72721},[282],[238,72723,3380],{"className":72724},[282,283],[238,72726,72728],{"className":72727},[391],[238,72729,72731,72760],{"className":72730},[395,848],[238,72732,72734,72757],{"className":72733},[399],[238,72735,72737],{"className":72736,"style":1162},[403],[238,72738,72739,72742],{"style":1165},[238,72740],{"className":72741,"style":412},[411],[238,72743,72745],{"className":72744},[416,417,418,419],[238,72746,72748,72751,72754],{"className":72747},[282,419],[238,72749,820],{"className":72750},[282,283,419],[238,72752,987],{"className":72753},[664,419],[238,72755,745],{"className":72756},[282,419],[238,72758,872],{"className":72759},[871],[238,72761,72763],{"className":72762},[399],[238,72764,72766],{"className":72765,"style":1193},[403],[238,72767],{}," 与马尔可夫性，把值函数写成当前奖励与后继值函数的期望。第二步，把该递归式视为有界函数空间上的算子方程 ",[238,72770,72772,72792],{"className":72771,"translate":242},[241],[238,72773,72775],{"className":72774},[246],[248,72776,72777],{"xmlns":250},[252,72778,72779,72789],{},[255,72780,72781,72783,72785,72787],{},[258,72782,4429],{},[332,72784,992],{},[258,72786,11971],{"mathvariant":816},[258,72788,4429],{},[262,72790,72791],{"encoding":264},"V=\\mathcal{T}V",[238,72793,72795,72813],{"className":72794,"ariaHidden":269},[268],[238,72796,72798,72801,72804,72807,72810],{"className":72797},[273],[238,72799],{"className":72800,"style":278},[277],[238,72802,4429],{"className":72803,"style":660},[282,283],[238,72805],{"className":72806,"style":430},[429],[238,72808,992],{"className":72809},[434],[238,72811],{"className":72812,"style":430},[429],[238,72814,72816,72819,72822],{"className":72815},[273],[238,72817],{"className":72818,"style":278},[277],[238,72820,11971],{"className":72821,"style":11992},[282,840],[238,72823,4429],{"className":72824,"style":660},[282,283],"。第三步，利用算子的单调性、常数平移性与压缩性，证明不动点存在唯一，并由 Banach 不动点定理得到几何收敛。",[11,72827,72828,72829,72914],{},"贝尔曼期望方程刻画固定策略的值函数，其有限状态形式是一个线性方程组，系数矩阵为 ",[238,72830,72832,72855],{"className":72831,"translate":242},[241],[238,72833,72835],{"className":72834},[246],[248,72836,72837],{"xmlns":250},[252,72838,72839,72853],{},[255,72840,72841,72843,72845,72847],{},[258,72842,14572],{},[332,72844,5577],{},[258,72846,730],{},[337,72848,72849,72851],{},[258,72850,330],{},[258,72852,2297],{},[262,72854,14853],{"encoding":264},[238,72856,72858,72876],{"className":72857,"ariaHidden":269},[268],[238,72859,72861,72864,72867,72870,72873],{"className":72860},[273],[238,72862],{"className":72863,"style":653},[277],[238,72865,14572],{"className":72866,"style":14619},[282,283],[238,72868],{"className":72869,"style":660},[429],[238,72871,5577],{"className":72872},[664],[238,72874],{"className":72875,"style":660},[429],[238,72877,72879,72882,72885],{"className":72878},[273],[238,72880],{"className":72881,"style":6651},[277],[238,72883,730],{"className":72884,"style":764},[282,283],[238,72886,72888,72891],{"className":72887},[282],[238,72889,330],{"className":72890,"style":377},[282,283],[238,72892,72894],{"className":72893},[391],[238,72895,72897],{"className":72896},[395],[238,72898,72900],{"className":72899},[399],[238,72901,72903],{"className":72902,"style":6609},[403],[238,72904,72905,72908],{"style":407},[238,72906],{"className":72907,"style":412},[411],[238,72909,72911],{"className":72910},[416,417,418,419],[238,72912,2297],{"className":72913,"style":2313},[282,283,419],"。贝尔曼最优方程刻画最优值函数，其算子在动作上取最大值。两者的共同点是都把无限期问题转化为一个不动点问题。区别在于前者对策略取期望，因而在有限状态下是线性方程。后者对动作取最大值，因而含有非线性项。",[11,72916,72917],{},"最优子结构是这一整套理论的解释性核心。它说明，在可加折扣效用、马尔可夫转移与有界回报的条件下，整体最优决策可以沿着实际到达的后继状态分解为一系列子问题的最优决策。它不是一个无条件的哲学原则，而是一个由回报结构、状态表示与折扣条件共同保证的数学定理。",{"title":19,"searchDepth":190,"depth":190,"links":72919},[72920,72926,72933,72940,72947,72953,72958,72963],{"id":222,"depth":190,"text":223,"children":72921},[72922,72924,72925],{"id":227,"depth":72923,"text":228},3,{"id":2785,"depth":72923,"text":2786},{"id":6687,"depth":72923,"text":6688},{"id":10237,"depth":190,"text":10238,"children":72927},[72928,72929,72930,72931,72932],{"id":10241,"depth":72923,"text":10242},{"id":11947,"depth":72923,"text":11948},{"id":13127,"depth":72923,"text":13128},{"id":17529,"depth":72923,"text":17530},{"id":19942,"depth":72923,"text":19943},{"id":21548,"depth":190,"text":21549,"children":72934},[72935,72936,72937,72938,72939],{"id":21552,"depth":72923,"text":21553},{"id":23876,"depth":72923,"text":23877},{"id":26830,"depth":72923,"text":26831},{"id":29595,"depth":72923,"text":29596},{"id":34281,"depth":72923,"text":34282},{"id":36734,"depth":190,"text":36735,"children":72941},[72942,72943,72944,72945,72946],{"id":36738,"depth":72923,"text":36739},{"id":37548,"depth":72923,"text":37549},{"id":45545,"depth":72923,"text":45546},{"id":47367,"depth":72923,"text":47368},{"id":48616,"depth":72923,"text":48617},{"id":51227,"depth":190,"text":51228,"children":72948},[72949,72950,72951,72952],{"id":51231,"depth":72923,"text":51232},{"id":53565,"depth":72923,"text":53566},{"id":58782,"depth":72923,"text":58783},{"id":61800,"depth":72923,"text":61801},{"id":62886,"depth":190,"text":62887,"children":72954},[72955,72956,72957],{"id":62890,"depth":72923,"text":62891},{"id":68208,"depth":72923,"text":68209},{"id":68496,"depth":72923,"text":68497},{"id":69104,"depth":190,"text":69105,"children":72959},[72960,72961,72962],{"id":69108,"depth":72923,"text":69109},{"id":70097,"depth":72923,"text":70098},{"id":71402,"depth":72923,"text":71403},{"id":72549,"depth":190,"text":72549},{},"\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties",{"title":207,"description":212},"blog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties","贝尔曼方程（Bellman equation）是动态规划的核心，本质是把一个多阶段的决策问题拆成当前决策 + 剩余子问题的递归关系。它由理查德·贝尔曼在 20 世纪 50 年代提出，是强化学习、最优控制、经济学等领域的基础工具。",[72970,72971],"machine-learning","mathematics","2026-11-09T23:49:14+08:00","hpJs47-d_ZZWWr8USnYwX8xzwtvDogQjVGqetsjCV2g",{"left":72975,"top":72975,"width":72976,"height":72976,"rotate":72975,"vFlip":192,"hFlip":192,"body":72977},0,24,"\u003Cg fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\">\u003Cpath d=\"m4.748 7.645l-.331 3.644c-.05.54-.074.811-.03 1.07a2 2 0 0 0 .238.655c.131.228.325.422.711.808l5.176 5.176c.787.787 1.18 1.18 1.636 1.328c.402.131.835.131 1.237 0c.456-.148.853-.544 1.645-1.336l3.96-3.96c.792-.792 1.187-1.188 1.336-1.644a2 2 0 0 0-.001-1.236c-.148-.457-.543-.853-1.335-1.645l-5.163-5.163c-.39-.39-.584-.584-.813-.716a2 2 0 0 0-.656-.238c-.26-.045-.535-.02-1.084.03l-3.63.33c-.944.086-1.417.129-1.787.335a2 2 0 0 0-.775.775c-.205.368-.248.838-.333 1.773z\"\u002F>\u003Cpath d=\"M9.713 9.713a1 1 0 1 0-1.415-1.414a1 1 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