半定参数广义方程与半定锥均衡约束数学规划问题

Mathematical programs with semidefinite cone equilibrium constraints(SDCMPECs) are optimization problems whose constraints include generalized equations that are defined by semidefinite cone(SDC). Such problems are widely used in economics and engineering, and have mathematical programs with SDC complementarity constraints which arise in robust optimizations as classic examples. As the study of SDCMPECs has just started, it is of great importance to the study of theory and algorithm for SDCMPECs. The building of the optimality theory of SDCMPECs needs the help of perturbation analysis of semidefinite parameterized generalized equations which can represent optimality conditions of a lot of constraint optimization problems defined by SDC. Based on variational analysis, perturbation analysis and nonsmooth matrix analysis, this project is devoted to the study of perturbation analysis of semidefinite parameterized generalized equations, and that of optimality theory and numerical methods for SDCMPECs based on the characterizations for coderivatives of the solution set mappings of semidefinite parameterized generalized equations. The main content includes variational geometry on semidefinite complementarity set, Aubin property and strong regularity of semidefinite parameterized variational inequality, first and second order optimality conditions of mathematical programs with SDC complementarity constraints and mathematical programs with semidefinite variational inequality constraints, and numerical methods for solving these two SDCMPECs such as smoothing method, penalty method and smoothing Newton method. It is our expectation that the project will make a contribution to the study of theory and algorithm for conic equilibrium optimization.

半定锥均衡约束数学规划(半定锥MPEC)是约束含有半定锥定义的广义方程的优化问题，它在经济与工程领域有广泛应用，半定锥互补约束数学规划就是此类问题的典型例子，常出现于鲁棒优化中。目前半定锥MPEC的研究才刚开始，因此深入研究其理论与算法意义重大。半定锥MPEC的最优性理论的建立需要借助半定参数广义方程的扰动分析，该广义方程还可用来表示许多由半定锥定义的约束优化问题的最优性条件。本项目以变分分析、扰动分析和非光滑矩阵分析为理论基础，建立半定参数广义方程扰动理论。同时基于该广义方程解映射的伴同导数的刻画，研究半定锥MPEC最优性理论与数值方法。内容包括半定互补集合的变分几何，半定参数变分不等式的Aubin性质与强正则性，半定锥互补约束优化问题与半定变分不等式为约束的数学规划的一二阶最优性条件，以及求解这两个半定锥MPEC的光滑化方法、罚方法和光滑化牛顿法，为锥均衡优化的理论与算法研究做出贡献。

半定参数广义方程与半定锥互补约束数学规划问题是两类重要的矩阵优化问题。这两类问题有重要的理论与实用价值。本项目以变分分析、扰动分析和非光滑矩阵分析为理论基础，建立了半定参数广义方程扰动理论，同时研究了半定锥互补约束数学规划问题的最优性理论与数值方法。项目取得的主要成果可概述如下：.1. 给出了半定锥互补集合的切锥与法锥公式；.2. 讨论了一类半定参数广义方程的Aubin性质和强正则性；.3. 建立了半定锥互补约束数学规划问题的最优性理论；.4. 证明了求解半定锥互补约束数学规划问题的非精确牛顿法与罚方法的收敛性；.5. 对一类线性半定规划逆问题的数值方法进行了研究。