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

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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.