An Inertial Proximal Peaceman-rachford Splitting Method For Solving Structured Convex Optimization

Linearly constrained separable convex optimization problem is the op-timization problem with linear constraints and convex and separable ob-jective function.This kind of problems arise from various regions such as image processing,compressive sensing and machine learning.Solving this problem rapidly has received tremendous attention in optimization field.The strictly contractive Peaceman-Rachford splitting method(PRSM)is a very efficient first-order approach for linearly constrained separable con-vex optimization problems,and its convergence rate is faster than that of ADMM.A semi-proximal PRSM was proposed to modify strictly contrac-tive PRSM.In this paper,motivated by the idea of accelerating convergence of inertial type algorithm,we improve the semi-proximal PRSM,proposing an inertial proximal PRSM.by employing an inertial technique,that is,at each iteration the semi-proximal PRSM is applied to a point extrapolat-ed at the current iterate in the direction of last movement,the proposed algorithm accelerate the convergence of semi-proximal PRSM.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.Fur-thermore,by choosing weighting matrices specially,we further propose an inertial linearized PRSM.We also demonstrate the efficiency of the inertial extrapolation step via numerical experiment.