随机广义方程相对于概率分布的稳定性分析及应用

项目名称： 随机广义方程相对于概率分布的稳定性分析及应用

项目编号： No.11201044

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 刘永朝

作者单位： 大连海事大学

项目金额： 22万元

中文摘要： 广义方程（GE）能够刻画变分不等式与互补系统、优化问题的一阶最优性条件、Nash均衡，在工程、经济等方面有着广泛的应用，是目前较为活跃的研究课题之一。由于很多实际问题会涉及随机因素，研究含有随机变量的GE（SGE）非常具有现实意义。另一方面，实际问题中随机变量的分布大多只能通过历史数据了解或估计，难免有偏差，而概率分布的变化势必会对SGE带来相应的影响。本项目拟分析SGE相对于随机变量的概率分布的稳定性。特别地，我们将研究欧式空间和Banach空间中SGE的解集相对于概率分布的度量正则性、H?lder连续性、Lipschitz连续性等。鉴于经验概率近似在随机优化中的广泛应用，我们将其作为特殊例子进行稳定性分析。此外，本项目还将利用SGE的稳定性理论分析随机均衡约束数学规划、随机Nash均衡等优化问题的稳定点以及最优解相对于概率分布的稳定性。

中文关键词： 随机广义方程；稳定性分析；稳定点；均衡问题；鲁棒均衡约束数学规划

英文摘要： Generalized equations (GE) can be used to characterize variational inequality and complementarity systems, first order optimality conditions and Nash equilibrium problems. It has found extensive applications in a number of areas such as engineering, economics and is one of the most important areas in optimization. In practice, many decision making problems are often subject to uncertain factors. Consequently, the stochastic version of GE (SGE) is needed. However, there are inevitably some deviations when the true probability distributions of random variables are approximated through empirical data. The deviation of the distributions of random variables will affect the solution of the SGE. The aim of this project is to study the stability of the solution set of SGE in Euclidean spaces and Banach spaces with respect to the perturbation of probability distributions. Specially, we will study the metric regularity, H?lder continuity and Lipschitz continuity of the solution set of SGE with respect to variation of the probability distribution. A particular focus will be given to empirical probability approximation which is a popular approach in stochastic programming. Moreover, the established results of SGE are applied to stability analysis of stationary points and optimal solutions of optimization problems such as s

英文关键词： stochastic generalized equations；stability analysis；stationary points；equilibrium problem；Robust MPEC