An indefinite proximal Peaceman–Rachford splitting method with substitution procedure for convex programming

The strictly contractive Peaceman–Rachford splitting method (SCPRSM) has received a tremendous amount of attention for solving linearly constrained separable convex optimization problems. In this paper, we propose an indefinite proximal SCPRSM with substitution procedure (abbreviated as PPRSM-S) to improve numerical results. The prediction step takes a proximal SCPRSM cycle to update the variable blocks, then the correction step corrects the output slightly by computing a combination of the prediction step and the previous iteration. We derive the global convergence of the proposed method and analyze the convergence rate results under much mild conditions. Some experimental results on LASSO and total variation-based denoising problems demonstrate the efficiency of the substitution step and the indefinite proximal term.