A multi-step inertial generalized Peaceman-Rachford splitting method and its applications

As an effective acceleration technique, multi-step inertia has attracted increasing attention in the development of first-order methods. In this paper, we propose a multi-step inertial generalized Peaceman-Rachford splitting method (abbreviated as MIGPRSM) for solving a family of separable convex programming problems subject to linear constraints. The involved subproblems are linearized by tailored proximal terms, which could be solved possibly easier than that without employing proximal terms. The global convergence and sublinear convergence rate of MIGPRSM are analysed by variational characterization for both the saddle point of the problem and the iterative sequence. Numerical experiments on LASSO and low patch rank image decomposition problems are performed to verify the efficiency of our proposed method.