A partially proximal S-ADMM for separable convex optimization with linear constraints

A classical approach to solving two-block separable convex optimization could be the symmetric alternating direction method of multipliers (S-ADMM). However, its convergence may not be guaranteed for a general multi-block case without additional assumptions. Bai et al. proposed a variant of S-ADMM entitled the generalized symmetric ADMM (GS-ADMM), in which the variables are regrouped into two groups firstly. The two groups of variables are updated in a Gauss-Seidel scheme, while the variables within each group are updated in a Jacobi scheme and the Lagrangian multipliers are updated two times. In order to derive its convergence property, the authors add a special proximal term to each subproblem. In this paper, inspired by the partial PPA block-wise ADMM (PPBADMM) [32] proposed by Shen et al., we propose a partially proximal S-ADMM (PPSADMM). In PPSADMM, the special proximal term is only added to the subproblems in the first group as PPBADMM. We perform an extension step on all variables with a fixed step size at the end of each iteration. Without stringent assumptions, we establish the global convergence result and the O(1/t) convergence rate in the ergodic sense for PPSADMM. Its numerical performance is justified on two types of problems.