Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

A strictly contractive Peaceman-Rachford splitting method for convex programming

Language

English

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.

Keywords

Contraction, Convergence rate, Convex programming, Peaceman-Rachford splitting method

Publication Date

2014

Source Publication Title

SIAM Journal on Optimization

Volume

24

Issue

3

Start Page

1011

End Page

1040

Publisher

Society for Industrial and Applied Mathematics

DOI

10.1137/13090849X

Link to Publisher's Edition

http://dx.doi.org/10.1137/13090849X

ISSN (print)

10526234

ISSN (electronic)

10957189

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