ADMM for multiaffine constrained optimization. (3rd March 2020)

• Record Type: Journal Article
• Authors: Gao, Wenbo; Goldfarb, Donald; Curtis, Frank E.
• Is Part Of: Optimization methods and software. Volume 35:Number 2(2020)
• Page Start: 257
• Page End: 303
• Publication Date: 2020-03-03
• Subjects: ADMM -- nonconvex optimization -- multilinear functions; 90C26 -- 90C30; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2019.1683553
• Languages: English
• ISSNs: 1055-6788

Abstract

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–Łojasiewicz (K–Ł) property holds, this is strengthened to convergence to a single constrained stationary point. Our analysis applies under assumptions that we have endeavoured 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.