Alternating direction method of multipliers with variable step sizes

TL;DR

This paper introduces a variable step size approach for ADMM, enhancing its convergence and efficiency in solving convex minimization problems through an adaptive adjustment rule and improved stopping criteria.

Contribution

It proposes a novel adaptive step size rule for ADMM based on residual monotonicity, leading to better convergence and performance.

Findings

Significant improvements over existing ADMM variants

Effective step size adjustment based on residuals

Enhanced convergence properties

Abstract

The alternating direction method of multipliers (ADMM) is a flexible method to solve a large class of convex minimization problems. Particular features are its unconditional convergence with respect to the involved step size and its direct applicability. This article deals with the ADMM with variable step sizes and devises an adjustment rule for the step size relying on the monotonicity of the residual and discusses proper stopping criteria. The numerical experiments show significant improvements over established variants of the ADMM.