Доступ предоставлен для: Guest
Портал Begell Электронная Бибилиотека e-Книги Журналы Справочники и Сборники статей Коллекции
Journal of Automation and Information Sciences
SJR: 0.275 SNIP: 0.59 CiteScore™: 0.8

ISSN Печать: 1064-2315
ISSN Онлайн: 2163-9337

Том 52, 2020 Том 51, 2019 Том 50, 2018 Том 49, 2017 Том 48, 2016 Том 47, 2015 Том 46, 2014 Том 45, 2013 Том 44, 2012 Том 43, 2011 Том 42, 2010 Том 41, 2009 Том 40, 2008 Том 39, 2007 Том 38, 2006 Том 37, 2005 Том 36, 2004 Том 35, 2003 Том 34, 2002 Том 33, 2001 Том 32, 2000 Том 31, 1999 Том 30, 1998 Том 29, 1997 Том 28, 1996

Journal of Automation and Information Sciences

DOI: 10.1615/JAutomatInfScien.v51.i6.20
pages 12-24

Convergence of Extragradient Algorithm with Monotone Step Size Strategy for Variational Inequalities and Operator Equations

Sergey V. Denisov
Kiev National Taras Shevchenko University, Ukraine
Dmitriy A. Nomirovskii
Kiev National Taras Shevchenko University, Kiev
Bogdan V. Rublyov
Kiev National Taras Shevchenko University, Kiev
Vladimir V. Semenov
Kiev National Taras Shevchenko University, Kiev

Краткое описание

Variational inequalities and operator equations in an infinite dimensional Hilbert space with additional conditions in the terms of inclusion in the set of fixed points of a given operator are considered. For approximate solution of the problems, a new iterative algorithm that is a superposition of a modified Korpelevich extragradient algorithm with monotone step size strategy, which does not require knowledge of the Lipschitz operator constant, and the Krasnoselsky–Mann scheme for the approximation of fixed points, is proposed. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not perform additional calculations for the operator values and the projections mapping. The algorithm was investigated using the theory of iterative processes of the Fejer type. The weak convergence of the algorithm for problems with pseudomonotone, Lipschitz continuous and sequentially weakly continuous operators and quasi nonexpansive operators, which specify additional conditions, is proved. Previously, similar results on weak convergence were known only for variational inequalities with monotone, Lipschitz continuous operators and with nonexpansive operators, which specify additional conditions.


  1. Kinderlehrer D., Stampacchia G., An introduction to variational inequalities and their applications, Academic Press, New York, 1980, [Russian transl., Mir, Moscow, 1983. .

  2. Baiocchi C., Capello A., Variational and quasi-variational inequalities. Applications to free boundary problems, Wiley, New York, 1984, Russian transl., Nauka, Moscow 1988. .

  3. Nagurney A., Network economics: A variational inequality approach, Kluwer Academic Publishers, Dordrecht, 1999, DOI: https://doi.org/10.1007/978-l-4757-3005-0. .

  4. Konnov I.V., Combined relaxation methods for variational inequalities, Springer-Verlag, Berlin-Heidelberg-New York, 2001, DOI: https://doi.org/10.1007/978-3-642-56886-2. .

  5. Sandrakov G.V., Homogenization of variational inequalities for problems with regular obstacles, Doklady Akademii Nauk, 2004, 397, No. 2, 170-173. .

  6. Sandrakov G.V., Homogenization of variational inequalities for nonlinear diffusion problems in perforated domains, Izvestiya Mathematics, 2005, 69, No. 5, 1035-1059, DOI: http://dx.doi.org/10, 1070/IM2005v069n05ABEH002287. .

  7. Gidel G., Berard H., Vincent P., Lacoste-Julien S., A variational inequality perspective on generative adversarial networks, arXiv preprint arXiv: 1802.10551, 2018. .

  8. Korpelevich G.M., The extragradient method for finding saddle points and other problems, Ekonomika i Matematicheskie Melody, 1976, 12. No. 4, 747-756. .

  9. Khobotov E.N., Modification of the extra-gradient method for solving variational inequalities and certain optimization problems, USSR Computational Mathematics and Mathematical Physics, 1987, 27, No. 5, 120-127, DOI: https://doi.org/10.1016/0041-5553(87)90058-9. .

  10. Semenov V.V., Strongly convergent algorithms for variational inequality problem over the set of solutions the equilibrium problems, In: Zgurovsky M.Z., Sadovnichiy V.A. (eds.) Continuous and Distributed Systems. Solid Mechanics and Its Applications, Springer International Publishing Switzerland, 2014, 211, 131-146, DOI: https://doi.org/10.1007/978-3-319-03146-0_10. .

  11. Tseng P., A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal on Control and Optimization, 2000, 38, 431-446, DOI: https://doi.org/10.l137/ S0363012998338806. .

  12. Semenov V.V., A strongly convergent splitting method for systems of operator inclusions with monotone operators, Journal of Automation and Information Sciences, 2014, 46, No. 5, 45-56, DOI: https://doi.org/10.1615/JAutomatInfScien.v46.i5.40. .

  13. Semenov V.V., Hybrid splitting methods for the system of operator inclusions with monotone operators, Cybernetics and Systems Analysis, 2014, 50, No. 5, 741-749, DOI: https://doi.org/ 10.1007/s10559-014-9664-y. .

  14. Censor Y., GibaliA., Reich S., The subgradient extragradient method for solving variational inequalities in Hilbert space, Journal of Optimization Theory and Applications, 2011, 148, 318-335, DOI: https://doi.org/10.1007/sl0957-010-9757-3. .

  15. Lyashko S.I., Semenov V.V., VoitovaT.A., Low-cost modification of Korpelevich's methods for monotone equilibrium problems, Cybernetics and Systems Analysis, 2011, 47, No. 4, 631-639, DOI: https://doi.org/10.1007/sl0559-011-9343-1. .

  16. Denisov S.V., Semenov V.V., ChabakL.M., Convergence of the modified extragradient method for variational inequalities with Non-Lipschitz operators, Cybernetics and Systems Analysis, 2015, 51, No. 5, 757-765, DOI: https://doi.org/10.1007/sl0559-015-9768-z. .

  17. Verlan D.A., Semenov V.V., Chabak L.M., A strongly convergent modified extragradient method for variational inequalities with Non-Lipschitz operators, Journal of Automation and Information Sciences, 2015, 47, No. 7, 31-46, DOI: https://doi.org/10.1615/JAutomatlnfScien.v47.i7.40. .

  18. Nemirovski A., Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM Journal on Optimization, 2004, 15, 229-251, DOI: https://doi.org/10.1137/S1052623403425629. .

  19. Semcnov V.V., Modified extragradient method with Bregman divergence for variational inequalities, Journal of Automation and Information Sciences, 2018, 50, No. 8, 26-37, DOI: https://doi.org/ 10.1615/JAutomatInfScien.v50.i8.30. .

  20. Semenov V.V., A version of the mirror descent method to solve variational inequalities, Cybernetics and Systems Analysis, 2017, 53, No. 2, 234-243, DOI: https://doi.org/10.I007/-sl0559-017-9923-9. .

  21. Chabak L., Semenov V., Vedel Y., A new non-Euclidean proximal method for equilibrium problems, In: ChertovO., Mylovanov T., Kondratenko Y., KacprzykJ., Kreinovich V., StefanukV. (eds.), Recent Developments in Data Science and Intelligent Analysis of Information, ICDSIAI 2018, Advances in Intelligent Systems and Computing, Springer, Cham, 2019, 836, 50-58, DOI: https://doi.org/10.1007/978-3-319-97885-7_6. .

  22. Lyashko S.I., Semenov V.V., A new two-step proximal algorithm of solving the problem of equilibrium programming, In: GoldengorinB. (ed.), Optimization and Its Applications in Control and Data Sciences, Springer Optimization and Its Applications, Springer. Cham, 2016, 115, 315-325, DOI: https://doi.org/10.I007/978-3-3l9-42056-l_10. .

  23. Semenov V.V., On the parallel proximal decomposition method for solving the problems of convex optimization, Journal of Automation and Information Sciences, 2010, 42, No. 4, 13-18, DOI: https://doi.org/10.1615/JAutomatInfScien.v42.i4.20. .

  24. Vasin V.V., Iterative methods for solving ill-posed problems with a priori information in Hilbert spaces, USSR Computational Mathematics and Mathematical Physics, 1988, 28, No. 4, 6-13, DOI: https://doi.org/10.1016/0041-5553(88)90104-8. .

  25. Vasin V.V., Iterative processes of the Fejer type in ill-posed problems with a prori information, Russian Math. (Iz. VUZ), 2009, 53, No. 2, 1-20. DOI: https://doi.org/10.3103/-S1066369X09020017. .

  26. Vasin V.V., Eremin I.I., Operators and iterative processes of Fejer type. Theory and Applications, Walter de Gruyter, Berlin-New York, 2009. .

  27. HalpernB., Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 1967, 73, No. 6, 957-961, DOI: https://doi.org/10.1090/S0002-9904-1967-11864-0. .

  28. Bauschke H.H., Combettes P.L., Convex analysis and monotone operator theory in Hilbert spaces, Springer, Berlin-Heidelberg-New York, 2011, DOI: https://doi.org/10.1007/978-3-319- 48311-5. .

  29. Nakajo K., Takahashi W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 2003, 279, 372-379, DOI: https://doi.org/10.1016/ S0022-247X(02)00458-4. .

Articles with similar content:

A Strongly Convergent Modified Extragradient Method for Variational Inequalities with Non-Lipschitz Operators
Journal of Automation and Information Sciences, Vol.47, 2015, issue 7
Dmitriy A. Verlan , Lyubov M. Chabak , Vladimir V. Semenov
Necessary Optimality Conditions of Second Order in Classical Sense in Optimal Control Problems of Three-Point Conditions
Journal of Automation and Information Sciences, Vol.42, 2010, issue 3
Shamo Isakh ogly Djabrailov, Yagub Amiyar ogly Sharifov, Magomet Farman ogly Mekhtiyev
Modified Extragradient Method with Bregman Divergence for Variational Inequalities
Journal of Automation and Information Sciences, Vol.50, 2018, issue 8
Vladimir V. Semenov
Computational Method for Determination of Inertia Moments of Revolution Bodies with Respect to Geometrical Axes
Journal of Automation and Information Sciences, Vol.30, 1998, issue 4-5
I. F. Radchenko, Gennadiy M. Bakan
Stabilization of Non-Autonomous Systems with Respect to a Part of Variables by Means of Controlled Lyapunov Functions
Journal of Automation and Information Sciences, Vol.32, 2000, issue 10
Alexander L. Zuev