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Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights
Chen, Caihua1; Li, Min1; Liu, Xin2,3; Ye, Yinyu1,4
2019
Source PublicationMATHEMATICAL PROGRAMMING
ISSN0025-5610
Volume173Issue:1-2Pages:37-77
AbstractIn 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.
KeywordNonseparable convex minimization Alternating direction method of multipliers Block coordinate descent method Iterate convergence Random permutation
DOI10.1007/s10107-017-1205-9
Language英语
Funding ProjectNational 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 Research AreaComputer Science ; Operations Research & Management Science ; Mathematics
WOS SubjectComputer Science, Software Engineering ; Operations Research & Management Science ; Mathematics, Applied
WOS IDWOS:000456970900002
PublisherSPRINGER HEIDELBERG
Citation statistics
Cited Times:1[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/32389
Collection计算数学与科学工程计算研究所
Affiliation1.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
Recommended Citation
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.
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