Convergence of Bregman alternating direction method with multipliers for nonconvex composite problems

TL;DR

This paper analyzes the convergence of a Bregman-modified ADMM algorithm for nonconvex composite problems, demonstrating that it converges to a stationary point under certain conditions, thus extending ADMM's applicability.

Contribution

It introduces and proves convergence of BADMM, a Bregman modification of ADMM, for nonconvex problems, which was previously less understood.

Findings

BADMM converges to stationary points in nonconvex settings

The method generalizes conventional ADMM with improved performance

Convergence is established under specific assumptions

Abstract

The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various convex or nonconvex composite problems that arise in the fields of image & signal processing and machine learning. In convex settings, numerous convergence results have been established for ADMM as well as its varieties. However, due to the absence of convexity, the convergence analysis of nonconvex ADMM is generally very difficult. In this paper we study the Bregman modification of ADMM (BADMM), which includes the conventional ADMM as a special case and often leads to an improvement of the performance of the algorithm. Under certain assumptions, we prove that the iterative sequence generated by BADMM converges to a stationary point of the associated augmented Lagrangian function. The obtained results underline the feasibility of ADMM in applications under nonconvex settings.