A proximal Peaceman-Rachford splitting method for solving the multi-block separable convex minimization problems

The Peaceman-Rachford splitting method (PRSM) is well studied for solving the two-block separable convex minimization problems with linear constraints recently. In this paper, we consider the separable convex minimization problem where its objective function is the sum of more than two functions without coupled variables, when applying the PRSM to this case directly, it is not necessarily convergent. To remedy this difficulty, we propose a proximal Peaceman-Rachford splitting method for solving this multi-block separable convex minimization problems, which updates the Lagrangian multiplier two times at each iteration and solves some subproblems parallelly. Under some mild conditions, we prove global convergence of the new method and analyse the worst-case convergence rate in both ergodic and nonergodic senses. In addition, we apply the new method to solve the robust principal component analysis problem and report some preliminary numerical results to indicate the feasibility and effectiveness of the proposed method.