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Journal of Automation and Information Sciences

Publicado 12 números por año

ISSN Imprimir: 1064-2315

ISSN En Línea: 2163-9337

SJR: 0.173 SNIP: 0.588 CiteScore™:: 2

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

Volumen 51, Edición 6, 2019, pp. 12-24
DOI: 10.1615/JAutomatInfScien.v51.i6.20
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SINOPSIS

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.

REFERENCIAS
  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. .

CITADO POR
  1. Vedel Ya. I., Sandrakov G. V., Semenov V. V., An Adaptive Two-Stage Proximal Algorithm for Equilibrium Problems in Hadamard Spaces, Cybernetics and Systems Analysis, 56, 6, 2020. Crossref

  2. Vedel Yana, Semenov Vladimir, Adaptive Extraproximal Algorithm for the Equilibrium Problem in Hadamard Spaces, in Optimization and Applications, 12422, 2020. Crossref

  3. Semenov V. V., Denisov S. V., Kravets A. V., Adaptive Two-Stage Bregman Method for Variational Inequalities, Cybernetics and Systems Analysis, 57, 6, 2021. Crossref

  4. Vedel Yana, Semenov Vladimir, Denisov Sergey, A Novel Algorithm with Self-adaptive Technique for Solving Variational Inequalities in Banach Spaces, in Advances in Optimization and Applications, 1514, 2021. Crossref

  5. Vedel Ya. I., Sandrakov G. V., Semenov V. V., Chabak L. M., Convergence of a Two-Stage Proximal Algorithm for the Equilibrium Problem in Hadamard Spaces, Cybernetics and Systems Analysis, 56, 5, 2020. Crossref

  6. Semenov V. V., Vedel Ya. I., Denisov S. V., CONVERGENCE OF ADAPTIVE EXTRA-PROXIMAL ALGORITHMS FOR EQUILIBRIUM PROBLEMS IN HADAMARD SPACES, Journal of Numerical and Applied Mathematics, 1, 2022. Crossref

  7. Semenov V. V., Denisov S. V., Convergence of the Method of Extrapolation from the Past for Variational Inequalities in Uniformly Convex Banach Spaces*, Cybernetics and Systems Analysis, 58, 4, 2022. Crossref

  8. Semenov V. V., Denisov S. V., Sandrakov G. V., Kharkov O. S., Convergence of the Operator Extrapolation Method for Variational Inequalities in Banach Spaces*, Cybernetics and Systems Analysis, 58, 5, 2022. Crossref

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