学术报告-杨磊

发布人:
责任人:
点击数:
更新时间:
2022-09-21 16:19:00


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.