On the Convergence Rate of Inexact Majorized sGS ADMM with Indefinite Proximal Terms for Convex Composite Programming

In this paper, we propose an inexact majorized symmetric Gauss–Seidel (sGS) alternating direction method of multipliers (ADMM) with indefinite proximal terms for multi-block convex composite programming. This method is a specific form of the inexact majorized ADMM which is further proposed to solve a general two-block separable optimization problem. The new methods adopt certain relative error criteria to solve the involving subproblems approximately, and the step-sizes allow to choose in the scope [Formula: see text]. Under more general conditions, we establish the global convergence and Q-linear convergence rate of the proposed methods.