A proximal partially parallel splitting method for separable convex programs. (2nd January 2017)

• Record Type: Journal Article
• Authors: Wang, Kai; Desai, Jitamitra; He, Hongjin
• Is Part Of: Optimization methods and software. Volume 32:Number 1(2017)
• Page Start: 39
• Page End: 68
• Publication Date: 2017-01-02
• Subjects: augmented Lagrangian method -- partially parallel splitting algorithm -- separable convex programs -- robust PCA problem -- global convergence -- convergence rate; 65K05 -- 90C30; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2016.1200044
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : In this paper, we propose a proximal partially parallel splitting method for solving convex minimization problems, where the objective function is separable into m individual operators without any coupled variables, and the structural constraint set comprises only linear functions. At each iteration of this algorithm, one selected subproblem is solved, and subsequently the remaining subproblems are solved in parallel, utilizing the new iterate information. Hence, the proposed method is a hybrid mechanism that combines the nice features of parallel decomposition methods and alternating direction methods, while simultaneously adopting the predictor–corrector strategy to ensure convergence of the algorithm. Our algorithmic framework is also amenable to admitting linearized versions of the subproblems, which frequently have closed-form solutions, thereby making the proposed method more implementable in practice. Furthermore, the worst-case O ( 1 / k ) convergence rate of the proposed method is obtained under both ergodic and nonergodic conditions. The efficiency of the proposed algorithm is also demonstrated by solving several instances of the robust PCA problem.