An inexact version of the symmetric proximal ADMM for solving separable convex optimization

TL;DR

This paper introduces and analyzes the first inexact symmetric proximal ADMM for linearly constrained convex optimization, achieving optimal convergence rates and demonstrating practical advantages through numerical experiments.

Contribution

It presents the first inexact version of the symmetric proximal ADMM with proven convergence rates, extending the applicability of ADMM variants.

Findings

Achieves global $ ext{O}(1/\sqrt{k})$ pointwise convergence rate.

Achieves $ ext{O}(1/k)$ ergodic convergence rate.

Numerical experiments show practical benefits of the inexact method.

Abstract

In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number $k$, we establish global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case. Since the symmetric proximal ADMM can be seen as a class of ADMM variants, the new algorithm as well as its convergence rates generalize, in particular, many others in the literature. Numerical experiments illustrating the practical advantages of the method are reported. To the best of our knowledge, this work is the first one to study an inexact version of the symmetric proximal ADMM.