Generalized Peaceman–Rachford splitting method with substitution for convex programming

The generalized alternating direction method of multipliers (GADMM), which expands the dual step length to (0,2), is a benchmark for solving the two-block separable convex programming. Recently, there are many ADMM-based improved algorithms with indefinite term, that is, the second subproblem is linearized by a specialized indefinite matrix. In this paper, we propose a generalized proximal Peaceman–Rachford splitting method (abbreviated as GPRSM-S) with substitution step and indefinite term. We will find out the relationship between linearized parameter, dual step length and substitution factor. The global convergence and the worst-case convergence rate in nonergodic sense are established theoretically by variational inequality. Finally, some numerical results on LASSO and total variation based denoising problems are presented to verify the feasibility of the introduced method.