Iteration-complexity analysis of a generalized alternating direction method of multipliers

Abstract: This paper analyzes the iteration-complexity of a generalized alternating direction method of multipliers (G-ADMM) for solving linearly constrained convex problems. This ADMM variant, which was first proposed by Bertsekas and Eckstein, introduces a relaxation parameter $\alpha \in (0,2)$ into the second ADMM subproblem. Our approach is to show that the G-ADMM is an instance of a hybrid proximal extragradient framework with some special properties, and, as a by product, we obtain ergodic iteration-complexity for the G-ADMM with $\alpha\in (0,2]$, improving and complementing related results in the literature. Additionally, we also present pointwise iteration-complexity for the G-ADMM.