半无限极大极小问题的非精确束方法及应用

The semi-infinite minimax problem falls into the class of nonsmooth optimization problems, which is widely used in, for instance, engineering design, control system, and investment portfolio allocation. This project will focus on an inexact bundle method for semi-infinite minimax problems and its applications. Firstly, via nonsmooth analysis, this project will study the Clarke subdifferential and approximate Goldstein subdifferential for the supremun function, which is the pointwise supremum of a infinite collection of nonconvex functions, and consider the ε—enlarged subdifferential to develop inner approximations of the approximate subdifferential. These results, in turn, will imply a nonpolyhedral approximation of the supremun function. We will also investigate the global error bound of this nonpolyhedral approximation, the monotonicity of the regularization of this nonpolyhedral approximation with suitalbe metric. Then we will present an inexact bundle method for solving semi-infinite minimax problems, prove the global convergence of the bundle method, and give its convergence rate and error bounds. On the other hand, using a well-known variational description of the largest eigenvalue and the improvement function given by Kiwiel, we will cast nonconvex semidefinite programming problems with inclusion constraints for the cone of positive semidefinite matrices as semi-infinite minimax problems, and then solve them by the inexact bundle method. Numerical experiments for nonconvex semidefinite programming problems will be performed to demonstrate the effectiveness of this method.

半无限极大极小问题是一类非光滑优化问题， 广泛见于工程设计、 控制系统、 投资组合配置等领域。本项目主要研究半无限极大极小问题的非精确束方法及其应用: (1) 利用非光滑分析，研究非凸函数族的上确界函数的Clarke 次微分和 Goldstein近似次微分， 构造次微分的ε—扩大集以逼近近似次微分， 从而构造上确界函数的非多面逼近函数， 并研究其全局误差界； 选择合适的度量将该逼近函数正规化， 研究其在点列上的单调性等。 在此基础上, 建立起相应的基于非精确的函数值及次微分的束方法来求解半无限极大极小问题并分析算法的全局收敛性和收敛速率， 并进行误差分析； (2) 利用最大特征值函数的变分表示形式和Kiwiel 提出的改进函数， 将一类具有半定矩阵锥约束的非凸半定规划问题转化为等价的半无限极大极小问题， 并用所建立的束方法求解， 同时通过数值实验验证所作的理论分析。