A Partially Inexact Alternating Direction Method of Multipliers and its Iteration-Complexity Analysis

TL;DR

This paper introduces a novel inexact ADMM variant with a proximal term and stepsize adjustment, providing the first complexity analysis for such methods and demonstrating practical benefits through preliminary experiments.

Contribution

It proposes a partially inexact ADMM with a relative error criterion, proximal regularization, and stepsize tuning, along with establishing its iteration-complexity bounds.

Findings

First complexity bounds for inexact ADMM with relative error criteria.

Proximal term simplifies the second subproblem.

Numerical experiments show improved performance.

Abstract

This paper proposes a partially inexact alternating direction method of multipliers for computing approximate solution of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic interation-complexity bounds for the proposed method are established. To the best of our knowledge, this is the first time that complexity results for an inexact ADMM with relative error criteria has been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method.