A Bregman-Style Improved ADMM and its Linearized Version in the Nonconvex Setting:Convergence and Rate Analyses  

作　　者：Peng-Jie Liu Jin-Bao Jian Hu Shao Xiao-Quan Wang Jia-Wei Xu Xiao-Yu Wu 

机构地区：[1]School of Mathematics,Jiangsu Center for Applied Mathematics,China University of Mining and Technology,Xuzhou 221116,Jiangsu,China [2]School of Mathematics and Physics,Center for Applied Mathematics of Guangxi,Guangxi Minzu University,Nanning 530006,Guangxi,China [3]Department of Civil and Environmental Engineering,The Hong Kong Polytechnic University,Hong Kong,China [4]School of Mathematics and Computational Science,Xiangtan University,Xiangtan 411105,Hunan,China

出　　处：《Journal of the Operations Research Society of China》2024年第2期298-340,共43页中国运筹学会会刊（英文）

基　　金：the National Natural Science Foundation of China(Nos.12171106 and 72071202);the Natural Science Foundation of Guangxi Province(No.2020GXNSFDA238017);Key Laboratory of Mathematics and Engineering Applications,Ministry of Education.

摘　　要：This work explores a family of two-block nonconvex optimization problems subject to linear constraints.We first introduce a simple but universal Bregman-style improved alternating direction method of multipliers(ADMM)based on the iteration framework of ADMM and the Bregman distance.Then,we utilize the smooth performance of one of the components to develop a linearized version of it.Compared to the traditional ADMM,both proposed methods integrate a convex combination strategy into the multiplier update step.For each proposed method,we demonstrate the convergence of the entire iteration sequence to a unique critical point of the augmented Lagrangian function utilizing the powerful Kurdyka–Łojasiewicz property,and we also derive convergence rates for both the sequence of merit function values and the iteration sequence.Finally,some numerical results show that the proposed methods are effective and encouraging for the Lasso model.

关 键 词：Nonconvex optimization Alternating direction method of multipliers Kurdyka-Lojasiewicz property Convergence rate 

分 类 号：O17[理学—数学]