非光滑矩阵优化问题的理论与算法研究

In this project, we will study a class of optimization problems involving the matrix unknown variables and the nonsmooth objective function. Such kind of matrix optimization problems (MOPs) has recently been found to have many important applications in different fields such as electronic engineering, communication technology, financial management, applied statistics, machine Learning, data mining and control theory. However, for MOPs, there is not much work done on both theoretical part and the algorithm design. Therefore, in the theoretical part, we will focus on the perturbation analysis of the MOP. The systematical theoretical study of the MOP is not only of crucial important itself but also the foundation of the convergence and stability study of algorithms. For algorithms, by applying the obtained theoretical results, we will try to design a framework of solving the general MOP. Also, for some special MOPs, we will design the semismooth Newton based argument Lagrange method to solve the large scale problems. Overall, after this project, we will try to build up the theoretical foundation of MOPs and design the efficient algorithms for to solve the problems.

本项目旨在研究一类自变量为矩阵，目标函数为一般非光滑凸函数且带约束的优化问题。这类矩阵优化问题在电子通讯工程、金融管理和应用统计学，机器学习、数据挖掘和控制论等领域都有着广泛的应用。然而，目前对这类非光滑优化问题还没有系统的理论分析以及统一的算法研究。在理论分析方面，本项目将重点研究这类问题的扰动分析。这一研究不仅有其重要的理论意义，而且为算法的收敛性和稳定性研究提供了理论。在算法研.究方面，本项目将利用理论分析的成果，设计求解这类矩阵问题的统一算法框架。同时，本项目还将针对实际应用中的若干大规模问题，利用其数据结构特点，设计基于半光滑牛顿共轭梯度的增广拉格朗日乘子法的有效算法。总之，本项目将建立系统的非光滑矩阵优化问题的理论，提出有效的求解算法，并为求解实际应用问题提供可行的方案。

非光滑矩阵优化问题，由于其在大数据科学相关的实际领域中的重要应用，成为近来一个热点研究邻域。本项目针对非光滑矩阵优化问题，系统研究了其最优性条件以及扰动性分析，为设计算法和分析算法收敛性提供了重要理论依据。此外，针对非光滑矩阵优化的特点，我们设计了基于半光滑牛顿共轭梯度的增广拉格朗日乘子法的求解算法，并通过实际问题验证了相关理论与设计算法的有效性。