一类具有光滑结构的非光滑随机优化的分解方法

Nonsmooth optimization problems with smooth substructures arise in many field of Science and Technology, such as minimax stochastic optimization in robust control, optimization problem of maximum eigenvalue function of stochastic matrix and stochastic semi-infinite programming in the field of civil engineering. The project focuses on the sample average approximation decomposition method for the three nonsmooth stochastic optimizations which have smooth structures. The main research contents are as follows. Firstly, the convergence and the convergence rate with probability 1,and the exponential convergence rate of probability for the sample average approximation decomposition method as the SAA subproblem is solved exactly, are presented. Secondly, the criterion is established for the SAA subproblem being solved inexactly, and then the convergence and the convergence rate with probability 1, and the exponential convergence rate of probability for the sample average approximation decomposition method under the inexact criterion, are proved. Lastly, the effective procedure for the concrete algorithms are programmed by Matlab language. The anticipated results will play an active promoting role for the development of stochastic optimization.

很多科学技术领域中都存在着具有光滑结构的非光滑随机优化问题，例如鲁棒控制中的极大极小随机优化，随机矩阵最大特征值函数优化和土木工程领域中的随机半无限规划等。本项目将研究上述三种具有光滑结构的非光滑随机优化的样本均值分解方法。主要研究内容包括：三种非光滑随机优化的光滑结构与光滑性质；SAA子问题精确求解时样本均值分解方法的概率1意义下的收敛性与收敛速度，以及概率的指数收敛率；建立SAA子问题非精确求解的准则，证明在此非精确准则下的样本均值分解方法的概率1意义下的收敛性与收敛速度，以及概率的指数收敛率；以MATLAB语言为工具，编制三种非光滑随机优化的具体有效的算法程序。期望项目的研究可对随机优化的发展起积极促进作用。

极大极小随机优化，随机半无限规划和随机矩阵最大特征值函数优化都是具有光滑结构的非光滑随机优化问题。这三类问题具有重大理论和实用价值，如在鲁棒控制和土木工程中，许多有重大价值的理论问题和实际问题的模型都是非光滑随机优化问题，并且这三类问题具有与空间分解相关的PDG结构。因此利用空间分解理论系统的研究这三类随机优化问题的理论、算法和应用意义重大。本项目取得的主要成果概括如下：.1、系统地研究了极大极小随机凸优化问题的空间分解方法；.2、讨论了随机二阶锥规划问题的快速空间分解算法；.3、系统地研究了一类随机约束非光滑凸优化的空间分解方法；.4、讨论了一类非凸特征值优化问题的快速空间分解方法；.5、对任意特征值优化问题的束方法进行了研究；.6、讨论了一类非光滑凸优化问题的交替线性化算法。