An inexact ADMM for separable nonconvex and nonsmooth optimization

作     者：Bai, Jianchao Zhang, Miao Zhang, Hongchao 

作者机构：Northwestern Polytech Univ Shenzhen Res & Dev Inst Shenzhen 518057 Peoples R China Northwestern Polytech Univ Sch Math & Stat Xian 710072 Peoples R China Louisiana State Univ Dept Math Baton Rouge LA 70803 USA 

出 版 物：《COMPUTATIONAL OPTIMIZATION AND APPLICATIONS》 (Comput Optim Appl)

年 卷 期：2025年第90卷第2期

页      面：445-479页

核心收录：

学科分类：1201[管理学-管理科学与工程(可授管理学、工学学位)] 07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：National Natural Science Foundation of China Shaanxi Fundamental Science Research Project for Mathematics and Physics [23JSQ031] Guangdong Basic and Applied Basic Research Foundation [2023A1515012405] USA National Science Foundation [DMS-2110722, DMS-2309549] 

主　　题：Nonconvex optimization Nonsmooth optimization Separable structure Lipschitz continuous Inexact ADMM Accelerated gradient method Global convergence Linear convergence rate 

摘      要：An inexact alternating direction method of multiplies (I-ADMM) with an expansion linesearch step was developed for solving a family of separable minimization problems subject to linear constraints, where the objective function is the sum of a smooth but possibly nonconvex function and a possibly nonsmooth nonconvex function. Global convergence and linear convergence rate of the I-ADMM were established under proper conditions while inexact relative error criterion was used for solving the subproblems. In addition, a unified proximal gradient (UPG) method with momentum acceleration was proposed for solving the smooth but possibly nonconvex subproblem. This UPG method guarantees global convergence and will automatically reduce to an optimal accelerated gradient method when the smooth function in the objective is convex. Our numerical experiments on solving nonconvex quadratic programming problems and sparse optimization problems from statistical learning show that the proposed I-ADMM is very effective compared with other state-of-the-art algorithms in the literature.