学术报告

题      目： Exploring Sparse Solutions of a Class of Constrained Optimization Problems (Authors: Lei Yang, Xiaojun Chen, Shuhuang Xiang)

报  告  人：杨磊   副教授  (邀请人：陈艳男 )

                                  中山大学

时      间：2022-9-21 16:00-17:00

地      点：西楼二楼会议室 

报告人简介:

       杨磊，副教授、硕士生导师，2022年8月入选中山大学“百人计划”，加入计算机学院科学计算研究所。杨磊博士本硕毕业于天津大学，2017年底在香港理工大学获得博士学位，2018年起至2022年7月先后在新加坡国立大学(2018.01-2021.05)和香港理工大学（2021.09-2022.07）从事博士后研究工作。杨磊博士主要从事最优化理论、算法和应用研究，特别专注于重要应用领域（如机器与统计学习、数据科学、图像与信号处理等）中出现的大规模结构优化问题，致力于设计和分析高效稳健的求解算法以及相关求解器的开发。目前已在SIOPT, MOR, JMLR, SIIMS, TSP等国际重要期刊上发表多篇论文，其中1篇论文入选ESI高被引论文。其博士学位论文《First-order Splitting Algorithms for Nonconvex Matrix Optimization Problems》于2019年荣获香港数学会颁发的“最佳博士学位论文奖”。

摘      要：

       In this work, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing $\|x\|_p^p$ subject to $\|Ax-b\|_q\leq\sigma$ for given $A \in R^{m \times n}$, $b\in R^m$, $\sigma \geq0$, $0\leq p\leq 1$ and $q \geq 1$. We then study various properties of the optimal solutions of this problem. Specifically, without any condition on the matrix $A$, we provide upper bounds in cardinality and infinity norm for the optimal solutions, and show that all optimal solutions must be on the boundary of the feasible set when $0<p\leq1$. Moreover, for $q \in \{1,\infty\}$, we show that the problem with $0<p<1$ has a finite number of optimal solutions and prove that there exists $0<p^*<1$ such that the solution set of the problem with any $0<p<p^*$ is contained in the solution set of the problem with $p=0$ and there further exists $0<\bar{p}<p^*$ such that the solution set of the problem with any $0<p\leq\bar{p}$ remains unchanged. An estimation of such $p^*$ is also provided. In addition, to solve the constrained nonconvex non-Lipschitz $L_p-L_1$ problem ($0<p<1$ and $q=1$), we propose a smoothing penalty method and show that, under some mild conditions, any cluster point of the sequence generated is a stationary point of our problem. Some numerical examples are given to implicitly illustrate the theoretical results and show the efficiency of the proposed algorithm for the constrained $L_p-L_1$ problem under different noises.