ADMM for Multiaffine Constrained Optimization

TL;DR

This paper extends ADMM to solve multiaffine constrained problems with convergence guarantees under weak assumptions, applicable to nonconvex, nonsmooth, and multi-block scenarios, with practical examples.

Contribution

It provides the first convergence analysis of ADMM for multiaffine constraints with weak assumptions, including nonconvex and nonsmooth problems, and demonstrates broad applicability.

Findings

ADMM converges to stationary points under certain conditions.

Applicable to nonconvex, nonsmooth, multi-block problems.

Subproblems have closed-form solutions in examples.

Abstract

We expand the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka-\L{}ojasiewicz (K-\L{}) property holds, this is strengthened to convergence to a single constrained stationary point. Our analysis applies under assumptions that we have endeavored to make as weak as possible. It applies to problems that involve nonconvex and/or nonsmooth objective terms, in addition to the multiaffine constraints that can involve multiple (three or more) blocks of variables. To illustrate the applicability of our results, we describe examples including nonnegative matrix factorization, sparse learning, risk parity portfolio selection, nonconvex formulations of convex problems, and neural network training. In each case, our ADMM approach encounters only subproblems that have closed-form solutions.