Optimal parameter selection for the alternating direction method of multipliers (ADMM): quadratic problems

TL;DR

This paper derives optimal parameter choices for ADMM in quadratic problems, significantly improving convergence speed by minimizing the convergence factor, with practical benefits demonstrated through numerical examples.

Contribution

It provides a quantitative method for selecting ADMM parameters that optimize convergence in quadratic problems, filling a gap in existing literature.

Findings

Optimal parameters reduce convergence time

Numerical examples confirm improved performance

Outperforms existing parameter selection methods

Abstract

The alternating direction method of multipliers (ADMM) has emerged as a powerful technique for large-scale structured optimization. Despite many recent results on the convergence properties of ADMM, a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking. In this paper we find the optimal algorithm parameters that minimize the convergence factor of the ADMM iterates in the context of l2-regularized minimization and constrained quadratic programming. Numerical examples show that our parameter selection rules significantly outperform existing alternatives in the literature.