Studios on theory of optimization with utility in stochastic model

研究结果总结如下：1.研究了具有目标集的非折扣马氏决策过程的风险最小化问题。我们将问题表示为具有递归类的无限水平情形。证明了最优值函数是最优性方程的唯一解，并且存在平稳最优策略。给出了几种数值迭代方法和一种策略改进方法。我们还考虑了具有有界报酬集的折扣马尔可夫决策过程中阈值概率最大化或最小化的八个问题。我们将这类问题分为两个等价类，并给出了每个等价类中问题的最优值和最优策略之间的关系。我们还给出了最优策略存在的两个充分条件。最后给出了第一等价类和第二等价类的最优值之间的关系。2.利用动态规划方法求解了一类具有前向递归准则的有限时间随机优化问题。3.利用正则性和弱零可加性将Lusin定理推广到模糊度量空间。4.在拓扑空间中引入有限交族的概念，用有限交族刻画了拓扑空间中的几个概念，并举例说明了有限交族的一些应用。5.EM算法的使用者认为Wu(1983)的条件保证了GEM序列的收敛，但本文给出了一个简短的反例，它满足Wu的条件但不收敛于Mille或任何最优解。并修正了他关于EM序列收敛的证明。

The summary of research results is as follows.1.We consider risk minimizing problems in undiscounted Markov decisions processes with a target set. We formulate the problem as an infinite horizon case with a recurrent class. We show that an optimal value function is a unique solution to an optimality equation and there exists an stationary optimal policy. Also we give several value iteration methods and a policy improvement method. We also consider eight problems in which we maximize or minimize threshold probabilities in discounted Markov decision processes with bounded reward set. We show that such problems are classified to two equivalence classes and give a relationship between optimal values and optimal policies of problems in each equivalence class. We also give two sufficient conditions for the existence of an optimal policy. Finally we give a relationship of optimal values between first and second equivalence classes.2.We solves a finite horizon stochastic optimization problem with forward recursive criterion through dynamic programming. The basic idea is to apply invariant imbedding method for stochastic programming.3.We show that weakly null-additive fuzzy measures on metric spaces posses regularity Lusin's theorem is generalized to fuzzy measure space by using the regularity and weakly null-additivity4.We introduce an idea of finite intersection family into a topological space, characterize several concepts in a topological space by mean of finite intersection family and illustrate some applications of finite intersection family.5.EM-algorithm users believe that the conditions of Wu(1983) assure the convergence of GEM sequence, but this paper gives a brief counter example which satisfies Wu's conditions but not converge to MILE or any optimal solutions. It also gives a correction of his proof for the convergence of EM sequence.