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International Theoretical Science Journal
Volume 58 >>> № 4 JULY — AUGUST 2022
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UDC 517.988
V.V. Semenov1, S.V. Denisov2


1 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

 semenov.volodya@gmail.com

2 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

 sireukr@gmail.com

CONVERGENCE OF EXTRAPOLATION FROM THE PAST METHOD
FOR VARIATIONAL INEQUALITIES IN UNIFORMLY CONVEX BANACH SPACES

Abstract. The authors analyze two new algorithms for solving variational inequalities in Banach spaces. The first algorithm is a modification of Popov’s two-stage method that uses the Albert generalized projection instead of the metric one. The second algorithm is an adaptive version of the first one, where the step size update rule is used, which does not require knowledge of the Lipschitz constants and calculation of the operator values at additional points. For variational inequalities with monotone, Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, theorems on the weak convergence of the methods are proved.

Keywords: variational inequality, monotone operator, Albert generalized projection, 2-uniformly convex Banach space, uniformly smooth Banach space, algorithm, convergence.


FULL TEXT

REFERENCES

  1. Kinderlehrer D. Stampacchia G. Introduction to variational inequalities and their applications [Russian translation]. Moscow: Mir, 1983. 256 p.

  2. Lions J.-L., Magenes E. Inhomogeneous boundary-value problems and their applications [Russian translation]. Moscow: Mir, 1971. 371 p.

  3. Nagurney A. Network economics: A variational inequality approach. Dordrecht: Kluwer Academic Publishers, 1999. 325 p.

  4. Lyashko S.I., Klyushin D.A., Nomirovsky D.A., Semenov V.V. Identification of age-structured contamination sources in ground water. Boucekkine R., Hritonenko N., Yatsenko Y. (Еds.). Optimal Control of Age-Structured Populations in Economy, Demography, and the Environment. London; New York: Routledge, 2013. P. 277–292.

  5. Facchinei F., Pang J.-S. Finite-dimensional variational inequalities and complementarily problem. New York: Springer, 2003. Vol. 2. 666 p.

  6. 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. Vol. 15. P. 229–251. https://doi.org/10.1137/ S1052623403425629.

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

  8. Korpelevich G.M. An extragradient method for finding saddle points and for other problems. Matecon. 1976. Vol. 12, N 4. P. 747–756.

  9. Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization. 2000. Vol. 38. P. 431–446. https://doi.org10.1137/S0363012998338806.

  10. Censor Y., Gibali A., Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. Journal of Optimization Theory and Applications. 2011. Vol. 148. P. 318–335. https://doi.org/10.1007/s10957-010-9757-3.

  11. Semenov V.V. Modified extragradient method with Bregman divergence for variational inequalities. Journal of Automation and Information Sciences. 2018. Vol. 50, Iss. 8. P. 26–37. https://doi.org/10.1615/JAutomatInfScien.v50.i8.30.

  12. Nesterov Yu. Dual extrapolation and its applications to solving variational inequalities and related problems. Mathematical Programming. 2007. Vol. 109, Iss. 2–3. P. 319–344. https://doi.org/10.1007/s10107-006-0034-z .

  13. Denisov S.V., Nomirovskii D.A., Rublyov B.V., Semenov V.V. Convergence of extragradient algorithm with monotone step size strategy for variational inequalities and operator equations. Journal of Automation and Information Sciences. 2019. Vol. 51, Iss. 6. P. 12–24. https://doi.org/10.1615/JAutomatInfScien.v51.i6.20.

  14. Bach F., Levy K.Y. A universal algorithm for variational inequalities adaptive to smoothness and noise. arXiv:1902.01637. 2019.

  15. Vedel Y., Semenov V. Adaptive extraproximal algorithm for the equilibrium problem in Hadamard spaces. Olenev N., Evtushenko Y., Khachay M., Malkova V. (Eds.). Optimization and Applications. OPTIMA 2020. Lecture Notes in Computer Science. Cham: Springer, 2020. Vol. 12422. P. 287–300. https://doi.org/10.1007/978-3-030-62867-3_21.

  16. Popov L.D. A modification of the Arrow–Hurwicz method for search of saddle points. Mathematical notes of the Academy of Sciences of the USSR. 1980. Vol. 28, Iss. 5. P. 845–848. https://doi.org/10.1007/BF01141092.

  17. Malitsky Yu.V., Semenov V.V. An extragradient algorithm for monotone variational inequalities. Cybernetics and Systems Analysis. 2014. Vol. 50, Iss. 2. P. 271–277. https://doi.org/10.1007/s10559-014-9614-8.

  18. Lyashko S.I., Semenov V.V. A new two-step proximal algorithm of solving the problem of equilibrium programming. Goldengorin B. (Ed.). Optimization and Its Applications in Control and Data Sciences. Springer Optimization and Its Applications. Cham: Springer, 2016. Vol. 115. P. 315–325. https://doi.org/10.1007/978-3-319-42056-1_10.

  19. Chabak L., Semenov V., Vedel Y. A new non-euclidean proximal method for equilibrium problems. Chertov O., Mylovanov T., Kondratenko Y., Kacprzyk J., Kreinovich V., Stefanuk V. (Eds.). Recent Developments in Data Science and Intelligent Analysis of Information. ICDSIAI 2018. Advances in Intelligent Systems and Computing. Cham: Springer, 2019. Vol. 6. P. 50–58. https://doi.org/10.1007/978-3-319-97885-7_6.

  20. Nomirovskii D.A., Rublyov B.V., Semenov V.V. Convergence of two-stage method with Bregman divergence for solving variational inequalities. Cybernetics and Systems Analysis. 2019. Vol. 55, Iss. 3. P. 359–368. https://doi.org/10.1007/s10559-019-00142-7.

  21. Vedel Yа.I., Denisov S.V., Semenov V.V. An adaptive algorithm for the variational inequality over the set of solutions of the equilibrium problem. Cybernetics and Systems Analysis. 2021. Vol. 57, Iss. 1. P. 91–100. https://doi.org/10.1007/s10559-021-00332-2.

  22. Semenov V.V., Denisov S.V., Kravets A.V. Adaptive two-stage Bregman method for variational inequalities. Cybernetics and Systems Analysis. 2021. Vol. 57, Iss. 6. P. 959–967. https://doi.org/10.1007/s10559-021-00421-2.

  23. Vedel Yа.I., Sandrakov G.V., Semenov V.V. An adaptive two-stage proximal algorithm for equilibrium problems in Hadamard spaces. Cybernetics and Systems Analysis. 2020. Vol. 56, Iss. 6. P. 978–989. https://doi.org/10.1007/s10559-020-00318-6.

  24. Malitsky Y., Tam M.K. A forward-backward splitting method for monotone inclusions without cocoercivity. SIAM Journal on Optimization. 2020. Vol. 30. P. 1451–1472. https://doi.org/10.1137/18M1207260.

  25. Alber Y., Ryazantseva I. Nonlinear ill posed problems of monotone type. Dordrecht: Springer, 2006. 410 p.

  26. Alber Y.I. Metric and generalized projection operators in Banach spaces: properties and applications. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. New York: Dekker, 1996. Vol. 178. P. 15–50.

  27. Iiduka H., Takahashi W. Weak convergence of a projection algorithm for variational inequalities in a Banach space. Journal of Mathematical Analysis and Applications. 2008. Vol. 339, N 1. P. 668–679. https://doi.org/10.1016/j.jmaa.2007.07.019.

  28. Cholamjiak P., Shehu Y. Inertial forward-backward splitting method in Banach spaces with application to compressed sensing. Appl. Math. 2019. Vol. 64, N 4. P. 409–435. https://doi.org/10.21136/AM.2019.0323-18.

  29. Shehu Y. Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces. Results in Mathematics. 2019. Vol. 74, N 138. https://doi.org/10.1007/ s00025-019-1061-4.

  30. Shehu Y. Single projection algorithm for variational inequalities in Banach spaces with application to contact problem. Acta Math. Sci. 2020. Vol. 40. P. 1045–1063. https://doi.org/10.1007/s10473-020-0412-2.

  31. Yang J., Cholamjiak P., Sunthrayuth P. Modified Tseng’s splitting algorithms for the sum of two monotone operators in Banach spaces. AIMS Mathematics. 2021. Vol. 6, Iss. 5. P. 4873–4900. https://doi.org/10.3934/math.2021286.

  32. Vedel Y., Semenov V., Denisov S. A novel algorithm with self-adaptive technique for solving variational inequalities in Banach spaces. Olenev N. N., Evtushenko Y. G., Jaimovi M., Khachay M., Malkova V. (Eds.). Advances in Optimization and Applications. OPTIMA 2021. Communications in Computer and Information Science. Cham: Springer, 2021. Vol. 1514. P. 50–64. https://doi.org/10.1007/978-3-030-92711-0_4.

  33. Distel J. Geometry of Banach spaces [in Russian]. Kyiv: Vishcha shkola, 1980. 215 p.

  34. Beauzamy B. Introduction to Banach spaces and their geometry. Amsterdam: North-Holland, 1985. 307 p.

  35. Aoyama K., Kohsaka F. Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings. Fixed Point Theory Appl. 2014. P. 95. https://doi.org/10.1186/ 1687-1812-2014-95.

  36. Xu H.K. Inequalities in Banach spaces with applications. Nonlinear Anal. 1991. Vol. 16, Iss. 12. P. 1127–1138.




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