Abstract

Many methods have been introduced previously to solve the equilibrium problem, of which the extragradient method is efficient. In this paper, we introduce a modified version of the extragradient algorithm to figure out equilibrium in a real Hilbert space. The proposed scheme based on an inertial scheme and explicit step size rule. The method uses a monotonic step size rule based on local bifunction information rather than of its Lipschitz-type parameters or other line search technique. We also prove the convergence theorem of the given algorithm and discuss its applications in particular classes of equilibrium classes. Finally, several computational tests are shown to demonstrate the efficiency of our proposed algorithm.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 6
Year: 2021

DOI: 10.12732/ijam.v34i6.2

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References

1. [3] K. Fan, A minimax inequality and applications, Inequalities, 3 (1972), 103-113.
2. [4] M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, Journal of Optimization Theory and Applications, 90, No 1 (1996), 31-43.
3. [5] G. Mastroeni, On auxiliary principle for equilibrium problems, In: Nonconvex Optimization and Its Applications, Springer, New York (2003), 289-298.
4. [6] G. Bigi, M. Castellani, M. Pappalardo and M. Passacantando, Existence and solution methods for equilibria, European Journal of Operational Research, 227, No 1 (2013), 1-11.
5. [7] I. Konnov, Equilibrium Models and Variational Inequalities, Elsevier, Amsterdam (2007).
6. [8] P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, Journal of Nonlinear Convex Analysis, 6, No 1 (2005), 117-136.
7. [9] A. N. Iusem, G. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Mathematical Programming, 116, No 1 (2007), 259-273.
8. [10] H. Rehman, P. Kumam, Y. J. Cho and P. Yordsorn, Weak convergence of explicit extragradient algorithms for solving equilibrium problems, Journal of Inequalities and Applications, 2019, No 1 (2019); doi: 10.1186/s13660019-2233-1.
9. [11] H. Rehman, P. Kumam, Y. J. Cho, Y. I. Suleiman and W. Kumam, Modified Popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems, Optimization Methods and Software, 36, No 1 (2020), 82-113.
10. [12] H. Rehman, P. Kumam, A. B. Abubakar and Y. J. Cho, The extragradient algorithm with inertial effects extended to equilibrium problems, Computational and Applied Mathematics, 39, No 2 (2020); doi: 10.1007/s40314020-1093-0.
11. [13] H. Rehman, P. Kumam, I. K. Argyros and N. A. Alreshidi, Modified proximal-like extragradient methods for two classes of equilibrium problems in Hilbert spaces with applications, Computational and Applied Mathematics, 40, No 2 (2021); doi: 10.1007/s40314-020-01385-3.
12. [14] H. Rehman, P. Kumam, W. Kumam, M. Shutaywi and W. Jirakitpuwapat, The inertial sub-gradient extra-gradient method for a class of pseudomonotone equilibrium problems, Symmetry, 12, No 3 (2020), 463.
13. [15] G. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12, No 1 (1976), 747-756.
14. [16] S. D. Fl°am and A. S. Antipin, Equilibrium programming using proximallike algorithms, Mathematical Programming, 78, No 1 (1996), 29-41.
15. [17] D. Q. Tran, M. L. Dung and V. H. Nguyen, Extragradient algorithms extended to equilibrium problems Optimization, 57, No 6 (2008), 749-776.
16. [18] D. V. Hieu, Halpern subgradient extragradient method extended to equilibrium problems, Revista de la Real Acad. de Ciencias Exactas, Fis. y Nat. Ser. A. Matem´aticas, 111, No 3 (2017), 823-840.
17. [19] S. I. Lyashko and V. V. Semenov, A new two-step proximal algorithm of solving the problem of equilibrium programming, In: Optimization and Its Applications in Control and Data Sciences, Springer Intern. Publ., Cham (2016), 315-325.
18. [20] N. T. Vinh and L. D. Muu, Inertial extragradient algorithms for solving equilibrium problems, Acta Mathematica Vietnamica, 44, No 3 (2019), 639-663.
19. [21] Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, Journal of Optimization Theory and Applications, 148, No 2 (2010), 318-335.
20. [22] D. V. Hieu, P. K. Quy, and L. V. Vy, Explicit iterative algorithms for solving equilibrium problems, Calcolo, 56, No 2 (2019), 1-21.
21. [23] E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, New York (1978).
22. [24] J. V. Tiel, Convex Analysis: an Introductory Text,Wiley, New York (1984).
23. [25] H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. of the Australian Math. Soc., 65, No 1 (2002), 109-113.
24. [26] P.-E. Maing´e, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Analysis, 16, No 7-8 (2008), 899-912.
25. [27] P. L. C. a. Heinz H. Bauschke, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2017).
26. [28] F. Browder and W. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20, No 2(1967), 197-228.
27. [29] S. Wang, Y. Zhang, P. Ping, Y. Cho and H. Guo, New extragradient methods with non-convex combination for pseudomonotone equilibrium problems with applications in Hilbert spaces, Filomat, 33, No 6 (2019), 1677-1693.
28. [30] G. Stampacchia, Formes bilinaires coercitives sur les ensembles convexes, Comptes Rendus Hebdomadaires des Sceances de l’Acad´emie des Sciences, 258, No 18 (1964), 4413.