title: A STRICTLY CONTRACTIVE PEACEMAN-RACHFORD SPLITTING METHOD FOR CONVEX PROGRAMMING.
abstract: In this paper, we focus on the application of 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 firsttiterations of PRSM still enable us to find an approximate solution with an accuracy ofO(1/t). A worst-caseO(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-caseO(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.
authors.affiliation: International Centre of Management Science and Engineering, and Department of Mathematics, Nanjing University, Nanjing, 200093, China. This author was supported by NSFC grant 91130007 and MOEC fund 20110091110004.
authors.forename: He
authors.initials: H
authors.lastname: Bingsheng
authors.affiliation: Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544. This author was supported by NSF grant III-1116730.
authors.forename: Han
authors.initials: H
authors.lastname: Liu
authors.affiliation: Department of Electrical Engineering, Princeton University, Princeton, NJ 08544.
authors.forename: Zhaoran
authors.initials: Z
authors.lastname: Wang
authors.affiliation: Department of Mathematics, Hong Kong Baptist University, Hong Kong. This author was partially supported by the General Research Fund from Hong Kong Research Grants Council: 12302514.
authors.forename: Xiaoming
authors.initials: X
authors.lastname: Yuan
keywords: Peaceman–Rachford splitting method; contraction; convergence rate; convex programming