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Zh. Vychisl. Mat. Mat. Fiz., 2020, Volume 60, Number 11, Pages 1843–1866 (Mi zvmmf11157)  

This article is cited in 1 scientific paper (total in 1 paper)

Optimal control

Accelerated methods for saddle-point problem

M. S. Alkousaab, A. V. Gasnikovabc, D. M. Dvinskikhcd, D. A. Kovaleve, F. S. Stonyakinf

a Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region
b National Research University "Higher School of Economics", Moscow
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
d Weierstrass Institute for Applied Analysis and Stochastics, Berlin
e King Abdullah University of Science and Technology
f Crimea Federal University, Simferopol

Abstract: Recently, it has been shown how, on the basis of the usual accelerated gradient method for solving problems of smooth convex optimization, accelerated methods for more complex problems (with a structure) and problems that are solved using various local information about the behavior of a function (stochastic gradient, Hessian, etc.) can be obtained. The term “accelerated methods” here means, on the one hand, the presence of some unified and fairly general way of acceleration. On the other hand, this also means the optimality of the methods, which can often be proved rigorously. In the present work, an attempt is made to construct in the same way a theory of accelerated methods for solving smooth convex-concave saddle-point problems with a structure. The main result of this article is the obtainment of in some sense necessary and sufficient conditions under which the complexity of solving nonlinear convex-concave saddle-point problems with a structure in the number of calculations of the gradients of composites in direct variables is equal in order of magnitude to the complexity of solving bilinear problems with a structure.

Key words: saddle problem, accelerated method, sliding, prox-friendly function.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 5-100
Russian Foundation for Basic Research 18-31-20005 мол-а-вед
Ministry of Science and Higher Education of the Russian Federation МК-15.2020.1
The research in Sections 1 and 2 was carried out within the Program of Fundamental Research of the National Research University Higher School of Economics and was supported by the program of the state support of the leading universities of the Russian Federation “5-100”. The research in Section 3 was supported by the Russian Foundation for Basic Research (project no. 18-31-20005 mol-a-ved) and, in Section 4, by the Russian Science Foundation (project no. 18-71-10108). The reach in Appendix 1 and partially Appendix 2 were supported by a Russian Federation Presidential grant for the state support of young Russian scientists: candidates of sciences (grant no. MK-15.2020.1).

DOI: https://doi.org/10.31857/S0044466920110022

English version:
Computational Mathematics and Mathematical Physics, 2020, 60:11, 1787–1809

Bibliographic databases:

UDC: 519.624
Received: 01.12.2019
Revised: 20.12.2019

Citation: M. S. Alkousa, A. V. Gasnikov, D. M. Dvinskikh, D. A. Kovalev, F. S. Stonyakin, “Accelerated methods for saddle-point problem”, Zh. Vychisl. Mat. Mat. Fiz., 60:11 (2020), 1843–1866; Comput. Math. Math. Phys., 60:11 (2020), 1787–1809

Citation in format AMSBIB
\by M.~S.~Alkousa, A.~V.~Gasnikov, D.~M.~Dvinskikh, D.~A.~Kovalev, F.~S.~Stonyakin
\paper Accelerated methods for saddle-point problem
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2020
\vol 60
\issue 11
\pages 1843--1866
\jour Comput. Math. Math. Phys.
\yr 2020
\vol 60
\issue 11
\pages 1787--1809

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    This publication is cited in the following articles:
    1. E. L. Gladin, M. Alkusa, A. V. Gasnikov, “O reshenii vypuklykh min-min zadach s gladkostyu i silnoi vypuklostyu po odnoi iz grupp peremennykh i maloi razmernostyu drugoi”, Avtomat. i telemekh., 2021, no. 10, 60–75  mathnet  crossref
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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