Tianyuan Mathematical Center
in Southwest China

On Convergence of Iterative Thresholding Algorithms to Global Solution for Nonconvex Sparse Optimization

报告题目：On Convergence of Iterative Thresholding Algorithms to Global Solution for Nonconvex Sparse Optimization

报告专家：Prof. Yaohua Hu（College of Mathematics and Statistics Shenzhen University）

报告时间：2023年5月11日（星期四）16:30-17:30      

报告地点：数学学院东409

报告摘要：

Sparse optimization is a popular research topic in applied mathematics and optimization, and nonconvex sparse regularization problems have been extensively studied to ameliorate the statistical bias and enjoy robust sparsity promotion capability in vast applications. However, puzzled by the nonconvex and nonsmooth structure in nonconvex regularization problems, the convergence theory of their optimization algorithms is still far from completion: only the convergence to a stationary point was established in the literature, while there is still no theoretical evidence to guarantee the convergence to a global minimum or a true sparse solution.

This talk aims to find an approximate global solution or true sparse solution of an under-determined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage thresholding operator and apply it, together with the restricted isometry property, to show that the proposed algorithms converge to an approximate global solution or true sparse solution within a tolerance relevant to the noise level and the limited shrinkage magnitude. Applying the obtained results to nonconvex regularization problems with SCAD, MCP and Lp penalty and utilizing the recovery bound theory, we establish the convergence of their proximal gradient algorithms to an approximate global solution of nonconvex regularization problems.

专家简介：

胡耀华，先后于浙江大学获得学士与硕士学位，香港理工大学获得博士学位，现任深圳大学数学与统计学院教授，博士生导师，香港理工大学兼职博导，兼任中国运筹学会数学规划分会青年理事，中国运筹学会科普工作委员会委员。主要从事连续优化理论、方法与应用研究，在非凸稀疏优化方法与交叉学科应用方面取得了一系列研究成果，代表性成果发表在SIAM Journal on Optimization, Inverse Problems, Journal of Machine Learning Research等知名期刊，申请3项国家发明专利，参与开发多个生物信息学工具包与网页服务器，先后主持国家自然科学基金优秀青年科学基金等10余项国家与省市级科研项目。

邀请人：李雪松