A note on the ergodic convergence of symmetric alternating proximal gradient method

We consider the alternating proximal gradient method (APGM) proposed to solve a convex minimization model with linear constraints and separable objective function which is the sum of two functions without coupled variables. Inspired by Peaceman-Rachford splitting method (PRSM), a nature idea is to extend APGM to the symmetric alternating proximal gradient method (SAPGM), which can be viewed as an extension of APGM by performing an additional intermediate multiplier updating step at each iteration. This note shows that SAPGM possesses an ergodic convergence rate of O(t−1) established under the analytic framework of contraction type methods. The efficiency of the proposed method is verified by solving the stable principal component pursuit problem.