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Zh. Vychisl. Mat. Mat. Fiz., 2018, Volume 58, Number 11, Pages 1794–1803 (Mi zvmmf10852)  

This article is cited in 2 scientific papers (total in 2 papers)

Primal-dual mirror descent method for constraint stochastic optimization problems

A. S. Bayandinaa, A. V. Gasnikovab, E. V. Gasnikovaa, S. V. Matsievskiic

a Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia
b Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia
c Kant Baltic Federal University, Kaliningrad, Russia

Abstract: Extension of the mirror descent method developed for convex stochastic optimization problems to constrained convex stochastic optimization problems (subject to functional inequality constraints) is studied. A method that performs an ordinary mirror descent step if the constraints are insignificantly violated and performs a mirror descent step with respect to the violated constraint if this constraint is significantly violated is proposed. If the method parameters are chosen appropriately, a bound on the convergence rate (that is optimal for the given class of problems) is obtained and sharp bounds on the probability of large deviations are proved. For the deterministic case, the primal-dual property of the proposed method is proved. In other words, it is proved that, given the sequence of points (vectors) generated by the method, the solution of the dual method can be reconstructed up to the same accuracy with which the primal problem is solved. The efficiency of the method as applied for problems subject to a huge number of constraints is discussed. Note that the bound on the duality gap obtained in this paper does not include the unknown size of the solution to the dual problem.

Key words: Mirror descent method, convex stochastic optimization, constrained optimization, probability of large deviations, randomization.

Funding Agency Grant Number
Russian Science Foundation 14-50-00150
Russian Foundation for Basic Research 15-31-20571_мол_а_вед
Ministry of Education and Science of the Russian Federation МД-1320.2018.1

DOI: https://doi.org/10.31857/S004446690003533-7

References: PDF file   HTML file

English version:
Computational Mathematics and Mathematical Physics, 2018, 58:11, 1728–1736

Bibliographic databases:

UDC: 519.856
Received: 09.12.2016
Revised: 29.09.2017

Citation: A. S. Bayandina, A. V. Gasnikov, E. V. Gasnikova, S. V. Matsievskii, “Primal-dual mirror descent method for constraint stochastic optimization problems”, Zh. Vychisl. Mat. Mat. Fiz., 58:11 (2018), 1794–1803; Comput. Math. Math. Phys., 58:11 (2018), 1728–1736

Citation in format AMSBIB
\by A.~S.~Bayandina, A.~V.~Gasnikov, E.~V.~Gasnikova, S.~V.~Matsievskii
\paper Primal-dual mirror descent method for constraint stochastic optimization problems
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2018
\vol 58
\issue 11
\pages 1794--1803
\jour Comput. Math. Math. Phys.
\yr 2018
\vol 58
\issue 11
\pages 1728--1736

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Bayandina A., Dvurechensky P., Gasnikov A., Stonyakin F., Titov A., “Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints”, Large-Scale and Distributed Optimization, Lect. Notes Math., Lecture Notes in Mathematics, 2227, eds. Giselsson P., Rantzer A., Springer International Publishing Ag, 2018, 181–213  crossref  isi
    2. 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”, J. Appl. Industr. Math., 13:3 (2019), 557–574  mathnet  crossref  crossref
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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