Application of the Strictly Contractive Peaceman-Rachford Splitting Method to Multi-block Separable Convex Programming

Recently, a strictly contractive Peaceman-Rachford splitting method (SCPRSM) was proposed to solve a convex minimization model with linear constraints and a separable objective function which is the sum of two functionals without coupled variables. We show by an example that the SC-PRSM cannot be directly extended to the case where the objective function is the sum of three or more functionals. To solve such a multi-block model, if we treat its variables and functions as two groups and directly apply the SC-PRSM, then at least one of SC-PRSM subproblems involves more than one function and variable which might not be easy to solve. One way to improve the solvability for this direct application of the SC-PRSM is to further decompose such a subproblem so as to generate easier decomposed subproblems which could potentially be simple enough to have closed-form solutions for some specific applications. The curse accompanying this improvement in solvability is that the SC-PRSM with further decomposed subproblems is not necessarily convergent, either. We will show its divergence by the same example. Our main goal in this chapter is to show that the convergence can be guaranteed if the further deBingsheng He International Centre of Management Science and Engineering, School of Management and Engineering, and Department of Mathematics, Nanjing University, Nanjing, 200093, China, e-mail: [email protected]. This author was supported by the NSFC grant 91130007 and the MOEC fund 20110091110004., Han Liu Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA, e-mail: [email protected]. This author was supported by NSF Grant III–1116730. Juwei Lu Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA, e-mail: [email protected] Xiaoming Yuan Department of Mathematics, Hong Kong Baptist University, Hong Kong, e-mail: xmyuan@ hkbu.edu.hk. This author was supported by the General Research Fund from Research Grants Council of Hong Kong: 12302514.