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

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.

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, Inverse Problems, Journal of Machine Learning Research, 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，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。