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

TL;DR

This paper provides a comprehensive analysis of the iteration complexity of a generalized ADMM variant with relaxation, establishing ergodic and pointwise complexity bounds by framing it within a hybrid proximal extragradient framework.

Contribution

It introduces a new analysis of G-ADMM's iteration complexity, extending existing results by incorporating a relaxation parameter and connecting it to a hybrid extragradient framework.

Findings

Ergodic iteration-complexity bounds for G-ADMM with relaxation parameter in (0,2]

Pointwise iteration-complexity results for G-ADMM

G-ADMM is shown to be an instance of a hybrid proximal extragradient method

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.