Studies on Flexible Structure of Dynamic Programming

在这个项目中，我们的目标是建立结构更灵活的动态规划（DP）数学理论，适用于软或粗糙的数学模型。为此，我们对柔性进行了理论研究，并利用区间和模糊集对各种柔性DP模型进行了分析研究。主要研究成果如下：(1)柔性的理论研究我们成功地将一维模糊最大阶的概念推广到多维模糊集。此外，我们还发展了多维模糊集序列的极限定理和势理论。此外，我们还推导了平均准则下约束马尔可夫决策过程（简称mdp）的鞍点定理。(2)不确定mdp应用区间分析和模糊理论，提出了一种不确定mdp的近似模型，称为控制马尔可夫集链或模糊处理模型，并证明了DP方法的有效性。(3)模糊停止模型引入模糊停止时间，用最优DP方程对动态模糊系统进行分析，扩大了DP方法的应用范围。(4)模糊聚类模型应用DP方法解决了任意两个元素之间的度量由模糊度量给出的最优聚类问题。(5)序列马尔可夫对策模型考虑具有区间值收益的区间矩阵对策，引入鞍点的概念，并将其表征为相应的非零和参数对策的平衡点。将这些结果推广到模糊序列对策。

In this project, our objective is to establish the mathematical theory on Dynamic Programming (DP in short) with more flexible structure, applicable to soft or rough mathematical models. To this end, we have done theoretical studies on flexibility and using intervals and fuzzy sets tried to develop analytical studies on various flexible DP models. The main research results are as follows:(1) Theoretical studies on flexibilityWe have succeeded in extending the concept of one-dimensional fuzzy max order to multi-dimensional fuzzy sets. Moreover, we develop limit theorems and potential theory for a sequence of multi dimensional fuzzy sets. Also, we derived a saddle point theorem for constrained Markov decision processes (MDPs in short) under average criteria.(2) Uncertain MDPsApplying interval analysis and fuzzy theory, we proposed some approximation model for uncertain MDPs, called controlled Markov set-chains or fuzzy treatment models, in which the validity of DP methods was proved.(3) Fuzzy stopping modelsIntroducing fuzzy stopping time, dynamic fuzzy systems have been analyzed by optimal DP equations, which enlarge the application of DP methods.(4) Fuzzy clustering modelsApplying DP methods we solved the optimal clustering problem in which the metric between any two elements is given by a fuzzy metric.(5) Sequential Markov game modelsWe considered interval matrix games with interval valued payoffs, for which the concept of saddle points is introduced and characterized as equilibrium points of corresponding non-zero sum parametric games. These results are extended to fuzzy sequential games.