动态规划法研究论文

Dynamic programming (Dynamic Programming) the method was the 1950s proposed, and by Richard•Bellman (Richard Bellman) introduces the optimized principle, has laid the solid foundation for the dynamic programming. In the past more than 50 years advancement, the dynamic programming in domain in and so on operations research, cybernetics, management science development, displayed incomparably has commanded troops the function, became one of solution mathematics modelling question most commonly used optimization methods. But, takes an important optimized method, it has the problem which urgently awaits to be solved appears is very imperfect. Therefore, in utilizes this method in the process, the people have devoted to consummate the application dynamic programming unceasingly the condition as well as the dynamic programming question solution method. This article has conducted the deep research and the summary to the dynamic programming in the past 50 year development, specially paid attention has studied the algorithm which its each kind produced newly. Such as deterministic univariate algorithms and so on dynamic programming analysis law, computation method; The deterministic multi-dimensional dynamic programming's Lagrange multiplier, iterates, the strategy and the function space gradually is approximate, the multinomial approaches, algorithms and so on hypersurface search; Stochastic dynamic programming primary algorithm and so on. The article solves a mathematics modelling problem finally through the use dynamic programming method, manifests the dynamic programming in the practical application : Dynamic programming; Deterministic dynamic programming; Univariate dynamic programming solution method; Multi-dimensional dynamic programming processing method; Stochastic dynamic programming

Dynamic programming (Dynamic Programming) the method is the 20th century 50's proposed, and by Richard•Bellman (Richard Bellman) introduces the optimized principle, has laid the solid foundation for the dynamic programming. In the past more than 50 years advancement, the dynamic programming in domain in and so on operations research, cybernetics, management science development, all displayed incomparably has commanded troops the function, became one of solution mathematics modelling question most commonly used optimization methods. But, took an important optimized method, it has the problem which very urgently awaits to be solved to appear is very imperfect. Therefore, in utilizes this method in the process, the people devote continuously to consummate the application dynamic programming unceasingly the condition as well as the dynamic programming question solution article has conducted the thorough research and the summary to the dynamic programming in the past 50 years development, specially paid attention has studied the algorithm which its each kind produced newly. Such as deterministic unidimensional algorithms and so on dynamic programming analysis law, computation method; Deterministic multi-dimensional dynamic programming Lagrange while the child, iterates, the strategy and the functional space gradually approximate, the multinomial approaches, algorithms and so on hypersurface search; Stochastic dynamic programming basic algorithm and so on. The article finally solves a mathematics modelling problem through the use dynamic programming method, manifests the dynamic programming in the practical application superiority. Key word: Dynamic programming; Deterministic dynamic programming; Unidimensional dynamic programming solution method; Multi-dimensional dynamic programming processing method; Stochastic dynamic programming

Dynamic programming the method was the 1950s proposed, and by Richard•Bellmanintroduces the optimized principle, has laid the solid foundation for the dynamic programming. In the past more than 50 years advancement, the dynamic programming in domain in and so on operations research, cybernetics, management science development, displayed incomparably has commanded troops the function, became one of solution mathematics modelling question most commonly used optimization methods. But, takes an important optimized method, it has the problem which urgently awaits to be solved appears is very imperfect. Therefore, in utilizes this method in the process, the people have devoted to consummate the application dynamic programming unceasingly the condition as well as the dynamic programming question solution method. This article has conducted the deep research and the summary to the dynamic programming in the past 50 year development, specially paid attention has studied the algorithm which its each kind produced newly. Such as deterministic univariate algorithms and so on dynamic programming analysis law, computation method; The deterministic multi-dimensional dynamic programming's Lagrange multiplier, iterates, the strategy and the function space gradually is approximate, the multinomial approaches, algorithms and so on hypersurface search; Stochastic dynamic programming primary algorithm and so on. The article solves a mathematics modelling problem finally through the use dynamic programming method, manifests the dynamic programming in the practical application superiority. Keywords: Dynamic programming; Deterministic dynamic programming; Univariate dynamic programming solution method; Multi-dimensional dynamic programming processing method; Stochastic dynamic programming

个人认为动态规划可以分为三个大的阶段，以创立者Bellman为划分界限，一是前Bellman时代，二是Bellman时代，三是后Bellman时代。1.前Bellman时代20世纪40年代，许多经济学和统计学领域的学者发现逆序归纳法（backwardinduction）可以解决一些含风险和不去定性的多阶段决策问题。vonNeumann和Morgenstern在他们的博弈论研究中，用逆序归纳法找到了扩展性博弈的子博弈精炼均衡（subgameperfectequilibriaofextensiveformgames）。统计决策理论（statisticaldecisiontheory）的创立者AbrahamWald也将其理论应用到多阶段决策问题。之后，Arrow,Blackwell和Girshick研究了统计决策问题的一般形式，用现代动态规划的方法阐明并解决了这一问题。另外，Arrow,Harris和Marschak还用逆序归纳法研究过最优库存策略（optimalinventorypolicy）。时代几乎同一时期，也就是20世纪40年代末开始，RichardBellman逐渐发现了多阶段决策问题的背后结构，并指出逆序归纳法到底是如何求解一大类多阶段决策问题的。从1949年开始，Bellman在兰德公司（RANDCorporation）开始了他对动态规划的研究，当时尚没有动态规划这一名词，此称呼是后来Bellman命名的。Bellman最为核心的工作就是给出了动态规划的最优性原理。此处有重要的文献，即1953,1954年发表在.和OperationsResearch上的几篇重要论文，包括像Thetheoryofdynamicprogramming，SomeApplicationsoftheTheoryofDynamicProgramming-AReview。具有标志性意义的是1957年PrincetonUniversityPress出版的DynamicProgramming，这标志着动态规划理论的正式建立。之后，Bellman的研究一直活跃到20世界60年代，其间对于动态规划应用范围的扩展做了一系列的研究工作，也有不少重要论文，此处不再罗列。3.后Bellman时代在基本的理论体系建立之后，动态规划进入了一个全面发展的时期，研究工作也有多个方向，总结来看，一是理论体系的一般化，二是修正原有理论的某些条件，包括像最优性原理来适应某些特殊的新问题。这一阶段的研究工作相当庞杂，此处很难一一罗列，简单举几个分支。一是对于所谓维数灾的研究，二是在大规模可分非线性整数规划问题中的应用，三是模糊动态规划，四是与其他数学领域像偏微分方程的交叉研究等等。此种划分纯粹是在下出于历史先后上的考虑而作出的，从学科内容上来说，可能不太合理。但是对于初级研究者理清基本脉络我想是有帮助的。