Convergence of multi-block Bregman ADMM for nonconvex composite problems

TL;DR

This paper proves the convergence of a Bregman 3-block ADMM and its extension to N-blocks for nonconvex problems, supported by simulations and real-world applications, advancing nonconvex optimization methods.

Contribution

It introduces a Bregman modification of multi-block ADMM and establishes its convergence for a broad class of nonconvex functions, extending existing theory.

Findings

Convergence of 3-block Bregman ADMM for nonconvex functions.

Extension of convergence results to N-block ADMM ($N \\geq 3$).

Validation through simulations and real-world application.

Abstract

The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various composite problems. The convergence of the conventional ADMM (i.e., 2-block) for convex objective functions has been justified for a long time, and its convergence for nonconvex objective functions has, however, been established very recently. The multi-block ADMM, a natural extension of ADMM, is a widely used scheme and has also been found very useful in solving various nonconvex optimization problems. It is thus expected to establish convergence theory of the multi-block ADMM under nonconvex frameworks. In this paper we present a Bregman modification of 3-block ADMM and establish its convergence for a large family of nonconvex functions. We further extend the convergence results to the $N$-block case ($N \geq 3$), which underlines the feasibility of multi-block ADMM applications in nonconvex settings. Finally, we present a simulation study and a real-world application to support the correctness of the obtained theoretical assertions.