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

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...

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Vydané v:Computational optimization and applications Ročník 61; číslo 2; s. 275 - 319
Hlavní autori: He, Niao, Juditsky, Anatoli, Nemirovski, Arkadi
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.06.2015
Springer Nature B.V
Springer Verlag
Predmet:
Algorithms
Convex analysis
Convex and Discrete Geometry
Management Science
Mathematics
Mathematics and Statistics
Numerical analysis
Operations Research
Operations Research/Decision Theory
Optimization
Optimization and Control
Signal processing
Statistics
Studies
65K05
Proximal methods
90C25
90C47
Numerical algorithms for variational problems
90C06
Composite optimization
Minimization problems with multi-term penalty
65K10
composite optimization
minimization problems with multi-term penalty
proximal methods
numerical algorithms for variational problems
ISSN:0926-6003, 1573-2894
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Popis
Shrnutí: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) 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 ℓ 1 and nuclear norm regularization) and to problems of semi-separable structures considered in the alternating directions methods, implying in both cases methods with the complexity bounds.
Bibliografia:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-014-9723-3