An inertial proximal Peaceman-Rachford splitting method

Abstract

The strictly contractive Peaceman-Rachford splitting method (SC-PRSM) is a very efficient first-order approach for linearly constrained separable convex optimization problems, and its convergence rate is faster than that of ADMM. Recently, a semi-proximal Peaceman-Rachford splitting method was proposed to modify SC-PRSM by introducing semi-proximal terms to the subproblems. In this paper, motivated by the idea of the inertial proximal ADMM, we improve the semi-proximal Peaceman-Rachford splitting method by employing an inertial technique to accelerate its convergence, i.e., at each iteration the semi-proximal Peaceman-Rachford splitting method is applied to a point extrapolated at the current iterate in the direction of last movement. The proposed algorithmic framework is very general so that SC-PRSM and semi-proximal Peaceman-Rachford splitting methods are covered as special cases. Based on the asymptotic feasibility of the iterative sequence and the convergence of the function values, we establish the convergence of the whole sequence generated by the proposed algorithm under very mild assumptions. We demonstrate the efficiency of the inertial extrapolation step via experimental results on least squares semi-definite programming problems and total variation based medical image reconstruction problems.