Mirror Prox algorithm for multi-term composite minimization and semi-separable problems

Niao He, Anatoli Juditsky, Arkadi Nemirovski

Research output: Contribution to journalArticlepeer-review

Abstract

In the paper, we develop a composite version of Mirror Prox algorithm for solving convex–concave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called “composite minimization” (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions “as if” there were no nonsmooth component at all). We demonstrate that the composite Mirror Prox inherits the favourable (and unimprovable already in the large-scale bilinear saddle point case) [InlineEquation not available: see fulltext.] efficiency estimate of its prototype. We demonstrate that the proposed approach can be successfully applied to Lasso-type problems with several penalizing terms (e.g. acting together $$\ell _1$$ℓ1 and nuclear norm regularization) and to problems of semi-separable structures considered in the alternating directions methods, implying in both cases methods with the [InlineEquation not available: see fulltext.] complexity bounds.

Original languageEnglish (US)
Article number9723
Pages (from-to)275-319
Number of pages45
JournalComputational Optimization and Applications
Volume61
Issue number2
DOIs
StatePublished - Jun 26 2015
Externally publishedYes

Keywords

  • Composite optimization
  • Minimization problems with multi-term penalty
  • Numerical algorithms for variational problems
  • Proximal methods

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

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