The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent

Caihua Chen*, Bingsheng He, Yinyu Ye, Xiaoming YUAN

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

253 Citations (Scopus)

Abstract

The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are alternatively updated. It is strongly desirable and practically valuable to extend the ADMM directly to the case of a multi-block convex minimization problem where its objective function is the sum of more than two separable convex functions. However, the convergence of this extension has been missing for a long time—neither an affirmative convergence proof nor an example showing its divergence is known in the literature. In this paper we give a negative answer to this long-standing open question: The direct extension of ADMM is not necessarily convergent. We present a sufficient condition to ensure the convergence of the direct extension of ADMM, and give an example to show its divergence.

Original languageEnglish
Pages (from-to)57-79
Number of pages23
JournalMathematical Programming
Volume155
Issue number1-2
DOIs
Publication statusPublished - 1 Jan 2016

Scopus Subject Areas

  • Software
  • Mathematics(all)

User-Defined Keywords

  • Alternating direction method of multipliers
  • Convergence analysis
  • Convex programming
  • Splitting methods

Fingerprint

Dive into the research topics of 'The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent'. Together they form a unique fingerprint.

Cite this