Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights

doi:10.1007/s10107-017-1205-9

	Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights
	Chen, Caihua 1; Li, Min 1; Liu, Xin2,3 ; Ye, Yinyu 1,4
	2019
发表期刊	MATHEMATICAL PROGRAMMING
ISSN	0025-5610
卷号	173 期号:1-2 页码:37-77
摘要	In this paper, we establish the convergence of the proximal alternating direction method of multipliers (ADMM) and block coordinate descent (BCD) method for nonseparable minimization models with quadratic coupling terms. The novel convergence results presented in this paper answer several open questions that have been the subject of considerable discussion. We firstly extend the 2-block proximal ADMM to linearly constrained convex optimization with a coupled quadratic objective function, an area where theoretical understanding is currently lacking, and prove that the sequence generated by the proximal ADMM converges in point-wise manner to a primal-dual solution pair. Moreover, we apply randomly permuted ADMM (RPADMM) to nonseparable multi-block convex optimization, and prove its expected convergence for a class of nonseparable quadratic programming problems. When the linear constraint vanishes, the 2-block proximal ADMM and RPADMM reduce to the 2-block cyclic proximal BCD method and randomly permuted BCD (RPBCD). Our study provides the first iterate convergence result for 2-block cyclic proximal BCD without assuming the boundedness of the iterates. We also theoretically establish the expected iterate convergence result concerning multi-block RPBCD for convex quadratic optimization. In addition, we demonstrate that RPBCD may have a worse convergence rate than cyclic proximal BCD for 2-block convex quadratic minimization problems. Although the results on RPADMM and RPBCD are restricted to quadratic minimization models, they provide some interesting insights: (1) random permutation makes ADMM and BCD more robust for multi-block convex minimization problems; (2) cyclic BCD may outperform RPBCD for nice problems, and RPBCD should be applied with caution when solving general convex optimization problems especially with a few blocks.
关键词	Nonseparable convex minimization Alternating direction method of multipliers Block coordinate descent method Iterate convergence Random permutation
DOI	10.1007/s10107-017-1205-9
语种	英语
资助项目	National Natural Science Foundation of China[11401300] ; National Natural Science Foundation of China[71732003] ; National Natural Science Foundation of China[71673130] ; National Natural Science Foundation of China[11771078] ; National Natural Science Foundation of China[71390335] ; National Natural Science Foundation of China[71661147004] ; National Natural Science Foundation of China[11622112] ; National Natural Science Foundation of China[11471325] ; National Natural Science Foundation of China[91530204] ; National Natural Science Foundation of China[11331012] ; National Natural Science Foundation of China[11461161005] ; National Natural Science Foundation of China[11688101] ; National Center for Mathematics and Interdisciplinary Sciences, CAS ; Youth Innovation Promotion Association, CAS ; Key Research Program of Frontier Sciences, CAS ; AFOSR Grant[FA9550-12-1-0396]
WOS研究方向	Computer Science ; Operations Research & Management Science ; Mathematics
WOS类目	Computer Science, Software Engineering ; Operations Research & Management Science ; Mathematics, Applied
WOS记录号	WOS:000456970900002
出版者	SPRINGER HEIDELBERG
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/32389
专题	计算数学与科学工程计算研究所
通讯作者	Liu, Xin
作者单位	1.Nanjing Univ, Sch Management & Engn, Int Ctr Management Sci & Engn, Nanjing, Jiangsu, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, State Key Lab Sci & Engn Comp, Beijing, Peoples R China 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing, Peoples R China 4.Stanford Univ, Sch Engn, Dept Management Sci & Engn, Stanford, CA 94305 USA
推荐引用方式 GB/T 7714	Chen, Caihua,Li, Min,Liu, Xin,et al. Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights[J]. MATHEMATICAL PROGRAMMING,2019,173(1-2):37-77.
APA	Chen, Caihua,Li, Min,Liu, Xin,&Ye, Yinyu.(2019).Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights.MATHEMATICAL PROGRAMMING,173(1-2),37-77.
MLA	Chen, Caihua,et al."Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights".MATHEMATICAL PROGRAMMING 173.1-2(2019):37-77.