Inertial Proximal ADMM for Linearly Constrained Separable Convex Optimization

Caihua CHEN, Raymond H. CHAN, Shiqian MA, Junfeng YANG*

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review

138 Citations (Scopus)

Abstract

The alternating direction method of multipliers (ADMM) is a popular and efficient first-order method that has recently found numerous applications, and the proximal ADMM is an important variant of it. The main contributions of this paper are the proposition and the analysis of a class of inertial proximal ADMMs, which unify the basic ideas of the inertial proximal point method and the proximal ADMM, for linearly constrained separable convex optimization. This class of methods are of inertial nature because at each iteration the proximal ADMM is applied to a point extrapolated at the current iterate in the direction of last movement. The recently proposed inertial primal-dual algorithm [A. Chambolle and T. Pock, On the ergodic convergence rates of a first-order primaldual algorithm, preprint, 2014, Algorithm 3] and the inertial linearized ADMM [C. Chen, S. Ma, and J. Yang, arXiv:1407.8238, eq. (3.23)] are covered as special cases. The proposed algorithmic framework is very general in the sense that the weighting matrices in the proximal terms are allowed to be only positive semidefinite, but not necessarily positive definite as required by existing methods of the same kind. By setting the two proximal terms to zero, we obtain an inertial variant of the classical ADMM, which is to the best of our knowledge new. We carry out a unified analysis for the entire class of methods under very mild assumptions. In particular, convergence, as well as asymptotic o(1/√k) and nonasymptotic O(1/√k) rates of convergence, are established for the best primal function value and feasibility residues, where k denotes the iteration counter. The global iterate convergence of the generated sequence is established under an additional assumption. We also present extensive experimental results on total variation–based image reconstruction problems to illustrate the profits gained by introducing the inertial extrapolation steps.

Original languageEnglish
Pages (from-to)2239-2267
Number of pages29
JournalSIAM Journal on Imaging Sciences
Volume8
Issue number4
DOIs
Publication statusPublished - Jan 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics.

Keywords

  • Alternating direction method of multipliers (ADMM)
  • Inertial PPM
  • Inertial proximal ADMM
  • Proximal ADMM
  • Proximal point method (PPM)

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