DOI
10.34229/KCA2522-9664.25.4.9
UDC 517.988
The operator extrapolation method for variational inequalities and its application
Abstract. The study focuses on new iterative algorithms for solving monotone variational inequalities in Hilbert space. A variant of the operator extrapolation algorithm with a variable metric is proposed. Theorems on weak and strong convergence for variational inequalities with Lipschitz continuous and monotone operators are proved. Several applications of the operator extrapolation algorithm for saddle-type problems are considered.
Keywords: variational inequality, saddle point problem, Nash equilibrium, operator equation, operator extrapolation method, variable metric method, convergence.
full text
REFERENCES
- 1. Kinderler D., Stampacchia G. Introduction to variational inequalities and their applications. Moscow: Mir, 1983. 256 p.
- 2. Baiocchi K., Capelo A. Variational and quasi-variational inequalities. Moscow: Nauka, 1988. 448 p.
- 3. Alber Y., Ryazantseva I. Nonlinear ill-posed problems of monotone type. Dordrecht: Springer, 2006. 410 p. https://doi.org/10.1007/1-4020-4396-1.
- 4. Nagurney A. Network economics: A variational inequality approach. Dordrecht: Kluwer Academic Publishers, 1999. 325 p.
- 5. Gidel G., Berard H., Vignoud G., Vincent P., Lacoste-Julien S. A variational inequality perspective on generative adversarial networks. arXiv:1802.10551v5 [cs.LG] 28 Aug 2020. https://doi.org/10.48550/arXiv.1802.10551.
- 6. Nemirovski A. Prox-method with rate of convergence 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/.
- 7. Wang J.-K., Abernethy J., Levy K.Y. No-regret dynamics in the Fenchel game: A unified framework for algorithmic convex optimization. arXiv:2111.11309v3 [cs.LG] 18 Feb 2023. https://doi.org/10.48550/arXiv.2111.11309.
- 8. Cohen M.B., Sidford A., Tian K. Relative Lipschitzness in extragradient methods and a direct recipe for acceleration. arXiv:2011.06572v2 [math.OC] 15 Jul 2021. https://doi.org/10.48550/arXiv.2011.06572.
- 9. Ryu E.K., Yin W. Large-scale convex optimization. Cambridge, UK: Cambridge University Press, 2022. 303 p. https://doi.org/10.1017/9781009160865.
- 10. Korpelevich G.M. An extragradient method for finding saddle points and for other problems. Matecon. 1976. Vol. 12, N 4. P. 747–756. https://cs.uwaterloo.ca/~y328yu/classics/extragrad.pdf.
- 11. Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization. 2000. Vol. 38, Iss. 2. P. 431–446. https://doi.org/10.1137/S0363012998338806.
- 12. 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.
- 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. Nadezhkina N., Takahashi W. Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications. 2006. Vol. 128. P. 191–201. https://doi.org/10.1007/s10957-005-7564-z.
- 15. Beck A. First-order methods in optimization. Philadelphia, USA: Society for Industrial and Applied Mathematics, 2017. 475 p. https://doi.org/10.1137/1.9781611974997.
- 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. Semenov V.V., Kharkov O.S. Convergence rate of the extrapolation from the past and operator extrapolation algorithms. Cybernetics and Systems Analysis. 2024. Vol. 60, N 5. P. 783–791. https://doi.org/10.1007/s10559-024-00715-1.
- 18. 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. 2020. Vol. 56, N 6. P. 978–989. https://doi.org/10.1007/s10559-020-00318-6.
- 19. Vedel Ya.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, N 1. P. 91–100. https://doi.org/10.1007/s10559-021-00332-2.
- 20. Semenov V.V., Denisov S.V., Kravets A.V. Adaptive two-stage Bregman method for variational inequalities. Cybernetics and Systems Analysis. 2021. Vol. 57, N 6. P. 959–967. https://doi.org/10.1007/s10559-021-00421-2.
- 21. 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.
- 22. Kotsalis G., Lan G., Li T. Simple and optimal methods for stochastic variational inequalities, I: Operator extrapolation. arXiv:2011.02987v5 [math.OC] 19 Jun 2023. https://doi.org/10.48550/arXiv.2011.02987.
- 23. Semenov V.V., Kharkov O.S. Strong convergence of the regularized operator extrapolation algorithm for variational inequalities. Cybernetics and Systems Analysis. 2024. Vol. 60, N 3. P. 392–402. https://doi.org/10.1007/s10559-024-00680-9.
- 24. 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. 2022. Vol. 58, N 6. P. 740–753. https:/doi.org/10.1007/s10559-022-00507-5.
- 25. Davidon W.C. Variable metric method for minimization. Argonne National Laboratory research and development report 5990, 1959; reprinted in SIAM Journal on Optimization, 1991. Vol. 1. P. 1–17. https://www.math.mcgill.ca/dstephens/680/Papers/Davidon91.pdf.
- 26. Bonnans J.F., Gilbert J.Ch., Lemarechal C., Sagastizabal C.A. A family of variable metric proximal methods. Math. Programming. 1995. Vol. 68. P. 15–47. https://doi.org/10.1007/BF01585756.
- 27. Burke J.V., Qian M. A variable metric proximal point algorithm for monotone operators. SIAM Journal on Control and Optimization. 1999. Vol. 37. P. 353–375. https://doi.org/10.1137/S0363012992235547.
- 28. Burke J.V., Qian M. On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating. Math. Programming. 2000. Vol. 88. P. 157–181. https://doi.org/10.1007/PL00011373.
- 29. Lotito P.A., Parente L.A., Solodov M.V. A class of variable metric decomposition methods for monotone variational inclusions. J. Convex Anal. 2009. Vol. 16. P. 857–880. https://www.heldermann.de/JCA/JCA16/JCA163/jca16052.htm.
- 30. Combettes P.L., Vu B.C. Variable metric quasi-Fejer monotonicity. arXiv: 1206.5705v2 [math.OC] 31 Aug 2012. https://doi.org/10.48550/arXiv.1206.5705.
- 31. Combettes P.L., Vu B.C. Variable metric forward-backward splitting with applications to monotone inclusions in duality. arXiv:1206.6791v1 [math.OC] 28 Jun 2012. https://doi.org/10.48550/arXiv.1206.6791.
- 32. Vu B.C. A variable metric extension of the forward-backward-forward algorithm for monotone operators. arXiv:1210.2986v2 [math.OC] 31 Oct 2012. https://doi.org/10.48550/arXiv.1210.2986.
- 33. Uryasyev S.P. Adaptive algorithms of stochastic optimization and game theory. Moscow: Nauka, 1990. 184 p.
- 34. Bauschke H.H., Combettes P.L. Convex analysis and monotone operator theory in Hilbert spaces. Berlin-Heidelberg-New York: Springer, 2011. 408 p. https://doi.org/10.1007/978-1-4419-9467-7.
- 35. Dung N.V., Vu B.C. Convergence analysis of the stochastic reflected forward-backward splitting algorithm. arXiv:2102.08906 [math.OC] 7 Feb 2021. https://doi.org/10.48550/arXiv.2102.08906.
- 36. Csetnek E.R., Malitsky Y., Tam M.K. Shadow Douglas–Rachford splitting for monotone inclusions. arXiv:1903.03393 [math.OC] 8 Mar 2019. https://doi.org/10.48550/arXiv.1903.03393.
- 37. Alber Y.I. Metric and generalized projection operators in Banach spaces: properties and applications. arXiv:funct-an/9311001 24 Nov 1993. https://doi.org/10.48550/arXiv.funct-an/9311001.
- 38. Polyak B.T. Introduction to Optimization. Moscow: Nauka, 1983. 384 p.
- 39. Wright S.J., Recht B. Optimization for data analysis. Cambridge, UK: Cambridge University Press, 2022. 227 p. https://doi.org/10.1017/9781009004282.
- 40. Chakraborty A., Nedic A. Random methods for variational inequalities. arXiv: 2402. 05462v1 [math.OC] 8 Feb 2024. https://doi.org/10.48550/arXiv.2402.05462.
- 41. Semenov V.V. On the parallel proximal decomposition method for solving the problems of convex optimization. Journal of Automation and Information Sciences. 2010. Vol. 42, Iss. 4. P. 13–18. https://doi.org/10.1615/JAutomatInfScien.v42.i4.20.