Modified proximal symmetric ADMMs for multi-block separable convex optimization with linear constraints

Abstract

We consider the linearly constrained separable convex optimization problem whose objective function is separable with respect to m blocks of variables. A bunch of methods have been proposed and extensively studied in the past decade. Specifically, a modified strictly contractive Peaceman–Rachford splitting method (SC-PRCM) [S. H. Jiang and M. Li, A modified strictly contractive Peaceman–Rachford splitting method for multi-block separable convex programming, J. Ind. Manag. Optim. 14(1) (2018) 397-412] has been well studied in the literature for the special case of m = 3. Based on the modified SC-PRCM, we present modified proximal symmetric ADMMs (MPSADMMs) to solve the multi-block problem. In MPSADMMs, all subproblems but the first one are attached with a simple proximal term, and the multipliers are updated twice. At the end of each iteration, the output is corrected via a simple correction step. Without stringent assumptions, we establish the global convergence result and the O(1/t) convergence rate in the ergodic sense for the new algorithms. Preliminary numerical results show that our proposed algorithms are effective for solving the linearly constrained quadratic programming and the robust principal component analysis problems.