矩阵优化扰动性理论及其与算法应用

矩阵优化是一类变量含有矩阵的优化问题。矩阵优化是近二十年来从包含统计、数据挖掘、机器学习、图像和信号处理、金融工程、信息学等在内的实际应用的不断发展起来的数学优化研究领域热点问题之一。本项目拟围绕矩阵优化扰动分析理论及其在大规模矩阵优化问题算法设计方面展开深入研究。首先，我们拟建立矩阵优化相关非多面体变分分析理论。在此基础上系统展开针对解集映射相关类Lipschitz性的研究，然后针对若干典型大规模矩阵优化问题研究设计高效算法。通过矩阵优化理论、算法与应用研究的有机结合，研究力争在优化研究前沿领域取得突破。

Matrix optimization problems (MOPs) are a class of optimization problems whose decision variables are matrices. MOPs have many successful applications in recent advances, such as statistics, data mining, machine learning, imagine and signal processing, financial and information science, and become one of the emerging fields of optimization. This project will focus on the study of the perturbation analysis of MOPs and related applications is designing efficient algorithms for large-scale problems. We will first build up necessary foundations on variational analysis of non-polyhedral sets/functions related to MOPs. After that, we will conduct a thorough study on some important Lipschitz properties of solving mappings of MOPs and will study and introduce some efficient numerical algorithms for solving several important large-scale MOPs arising from applications. By bridging the perturbation theory and efficient algorithms designing, we try to obtain some breakthrough results in MOPs, one of important optimization research fields.