金融科技论坛第四十六讲 On Convergence of Iterative Thresholding Algorithms to Global Solution for Nonconvex Sparse Optimization（胡耀华）

“金融科技论坛”第四十六讲

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

报告人：胡耀华

时间：2026年4月24日（星期五）下午15:00-17:00
报告地点：创新港涵英楼8楼8001

报告人简介：

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。现任深圳大学数学科学学院特聘教授，副院长，博士生导师，香港理工大学兼职博导。主要从事连续优化理论、方法与应用研究，代表性成果发表在SIAM Journal on Optimization, Mathematical Programming, Mathematics of Operations Research, Inverse Problems, Journal of Machine Learning Research, Genome Biology, Bioinformatics等期刊，授权多项国家发明专利，开发多个生物信息学工具包与数据库。

讲座摘要：

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.