Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems

Date
2016-01-01
Authors
Hong, Mingyi
Luo, Zhi-Quan
Razaviyayn, Mesiam
Journal Title
Journal ISSN
Volume Title
Publisher
Altmetrics
Authors
Research Projects
Organizational Units
Journal Issue
Series
Abstract

The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems. We show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. For the sharing problems, we show that the ADMM is convergent regardless of the number of variable blocks. Our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many ADMM variants involving proximal update rules and various flexible block selection rules.

Description

This is an article from SIAM Journal on Optimization 26 (2016): 337, doi:10.1137/140990309 Posted with permission.

Keywords
nonconvex, optimization, ADMM, consensus, sharing
Citation
DOI
Collections