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Article: A strictly contractive Peace man Rachford splitting method for convex programming

TitleA strictly contractive Peace man Rachford splitting method for convex programming
Authors
KeywordsContraction
Peaceman-Rachford splitting method
Convex programming
Convergence rate
Issue Date2014
PublisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/siopt.php
Citation
SIAM Journal on Optimization, 2014, v. 24, n. 3, p. 1011-1040 How to Cite?
Abstract© 2014 Society for Industrial and Applied Mathematics. In this paper, we focus on the application o f the Peaceman-Rachford splitting method (PRSM) to a convex minimization model with linear constraints and a separable objective function. Compared to the Douglas-Rachford splitting method (DRSM), another splitting method from which the alternating direction method of multipliers originates, PRSM requires more restrictive assumptions to ensure its convergence, while it is always faster whenever it is convergent. We first illustrate that the reason for this difference is that the iterative sequence generated by DRSM is strictly contractive, while that generated by PRSM is only contractive with respect to the solution set of the model. With only the convexity assumption on the objective function of the model under consideration, the convergence of PRSM is not guaranteed. But for this case, we show that the first t iterations of PRSM still enable us to find an approximate solution with an accuracy of O(1/t). A worst-case O(1/t) convergence rate of PRSM in the ergodic sense is thus established under mild assumptions. After that, we suggest attaching an underdetermined relaxation factor with PRSM to guarantee the strict contraction of its iterative sequence and thus propose a strictly contractive PRSM. A worst-case O(1/t) convergence rate of this strictly contractive PRSM in a nonergodic sense is established. We show the numerical efficiency of the strictly contractive PRSM by some applications in statistical learning and image processing.
Persistent Identifierhttp://hdl.handle.net/10722/250878
ISSN
2020 Impact Factor: 2.85
2020 SCImago Journal Rankings: 2.066
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorHe, Bingsheng-
dc.contributor.authorLiu, Han-
dc.contributor.authorWang, Zhaoran-
dc.contributor.authorYuan, Xiaoming-
dc.date.accessioned2018-02-01T01:53:58Z-
dc.date.available2018-02-01T01:53:58Z-
dc.date.issued2014-
dc.identifier.citationSIAM Journal on Optimization, 2014, v. 24, n. 3, p. 1011-1040-
dc.identifier.issn1052-6234-
dc.identifier.urihttp://hdl.handle.net/10722/250878-
dc.description.abstract© 2014 Society for Industrial and Applied Mathematics. In this paper, we focus on the application o f the Peaceman-Rachford splitting method (PRSM) to a convex minimization model with linear constraints and a separable objective function. Compared to the Douglas-Rachford splitting method (DRSM), another splitting method from which the alternating direction method of multipliers originates, PRSM requires more restrictive assumptions to ensure its convergence, while it is always faster whenever it is convergent. We first illustrate that the reason for this difference is that the iterative sequence generated by DRSM is strictly contractive, while that generated by PRSM is only contractive with respect to the solution set of the model. With only the convexity assumption on the objective function of the model under consideration, the convergence of PRSM is not guaranteed. But for this case, we show that the first t iterations of PRSM still enable us to find an approximate solution with an accuracy of O(1/t). A worst-case O(1/t) convergence rate of PRSM in the ergodic sense is thus established under mild assumptions. After that, we suggest attaching an underdetermined relaxation factor with PRSM to guarantee the strict contraction of its iterative sequence and thus propose a strictly contractive PRSM. A worst-case O(1/t) convergence rate of this strictly contractive PRSM in a nonergodic sense is established. We show the numerical efficiency of the strictly contractive PRSM by some applications in statistical learning and image processing.-
dc.languageeng-
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/siopt.php-
dc.relation.ispartofSIAM Journal on Optimization-
dc.subjectContraction-
dc.subjectPeaceman-Rachford splitting method-
dc.subjectConvex programming-
dc.subjectConvergence rate-
dc.titleA strictly contractive Peace man Rachford splitting method for convex programming-
dc.typeArticle-
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1137/13090849X-
dc.identifier.scopuseid_2-s2.0-84910619550-
dc.identifier.volume24-
dc.identifier.issue3-
dc.identifier.spage1011-
dc.identifier.epage1040-
dc.identifier.isiWOS:000343229000004-
dc.identifier.issnl1052-6234-

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