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Diskretnyi Analiz i Issledovanie Operatsii, 2019, Volume 26, Issue 3, Pages 88–114
DOI: https://doi.org/10.33048/daio.2019.26.636
(Mi da932)
 

This article is cited in 3 scientific papers (total in 3 papers)

Adaptive mirror descent algorithms for convex and strongly convex optimization problems with functional constraints

F. S. Stonyakinab, M.  Alkousab, A. N. Stepanova, A. A. Titovb

a Vernadsky Crimean Federal University, 4 Vernadsky Avenue, 295007 Simferopol, Russia
b Moscow Institute of Physics and Technologies, 9 Institutskii Bystreet, 141701 Dolgoprudnyi, Russia
Full-text PDF (416 kB) Citations (3)
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Abstract: Under consideration are some adaptive mirror descent algorithms for the problems of minimization of a convex objective functional with several convex Lipschitz (generally, nonsmooth) functional constraints. It is demonstrated that the methods are applicable to the objective functionals of various levels of smoothness: The Lipschitz condition holds either for the objective functional itself or for its gradient or Hessian (while the functional itself can fail to satisfy the Lipschitz condition). The main idea is the adaptive adjustment of the method with respect to the Lipschitz constant of the objective functional (its gradient or Hessian), as well as the Lipschitz constant of the constraint. The two types of methods are considered: adaptive (not requiring the knowledge of the Lipschitz constants neither for the objective functional nor for constraints) and partially adaptive (requiring the knowledge of the Lipschitz constant for constraints). Using the restart technique, some methods are proposed for strongly convex minimization problems. Some estimates of the rate of convergence are obtained for all algorithms under consideration in dependence on the level of smoothness of the objective functional. Numerical experiments are presented to illustrate the advantages of the proposed methods for some examples. Tab. 3, bibliogr. 22.
Keywords: adaptive Mirror Descent, Lipschitz condition, Lipschitz gradient, Lipschitz Hessian, strongly convex function, the technique of restarts.
Funding agency Grant number
Russian Foundation for Basic Research 18-31-00219_мол_а
The research by F. S. Stonyakin (analysis of Algorithms 1 and 3) was supported by the Russian Foundation for Basic Research (Project 18–31–00219 mol_a).
Received: 17.10.2018
Revised: 24.02.2019
Accepted: 27.02.2019
English version:
Journal of Applied and Industrial Mathematics, 2019, Volume 13, Issue 3, Pages 557–574
DOI: https://doi.org/10.1134/S1990478919030165
Bibliographic databases:
Document Type: Article
UDC: 519.85
Language: Russian
Citation: F. S. Stonyakin, M.  Alkousa, A. N. Stepanov, A. A. Titov, “Adaptive mirror descent algorithms for convex and strongly convex optimization problems with functional constraints”, Diskretn. Anal. Issled. Oper., 26:3 (2019), 88–114; J. Appl. Industr. Math., 13:3 (2019), 557–574
Citation in format AMSBIB
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\by F.~S.~Stonyakin, M.~~Alkousa, A.~N.~Stepanov, A.~A.~Titov
\paper Adaptive mirror descent algorithms for~convex~and strongly convex optimization problems with~functional constraints
\jour Diskretn. Anal. Issled. Oper.
\yr 2019
\vol 26
\issue 3
\pages 88--114
\mathnet{http://mi.mathnet.ru/da932}
\crossref{https://doi.org/10.33048/daio.2019.26.636}
\transl
\jour J. Appl. Industr. Math.
\yr 2019
\vol 13
\issue 3
\pages 557--574
\crossref{https://doi.org/10.1134/S1990478919030165}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85071498601}
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Дискретный анализ и исследование операций
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    Full-text PDF :166
    References:57
    First page:3
     
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