Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems

Mingyi Hong, Zhi Quan Luo, Meisam Razaviyayn

Research output: Contribution to journalArticle

242 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)337-364
Number of pages28
JournalSIAM Journal on Optimization
Volume26
Issue number1
DOIs
StatePublished - Jan 1 2016

Keywords

  • ADMM
  • Consensus
  • Nonconvex optimization
  • Sharing

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