Explicit Halpern-type iterative algorithm for solving equilibrium problems with applications

Volume 25, Issue 2, pp 115--132 http://dx.doi.org/10.22436/jmcs.025.02.02

Publication Date: May 26, 2021 Submission Date: November 08, 2020 Revision Date: January 29, 2021 Accteptance Date: April 15, 2021

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Authors

Kanikar Muangchoo - Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon (RMUTP), 1381 Pracharat 1 Road, Wongsawang, Bang Sue, Bangkok 10800, Thailand.

Abstract

A number of iterative algorithms have been established to solve equilibrium problems, and one of the most effective methods is a two-step extragradient method. The main objective of this study is to introduce a modified algorithm that is constructed around two methods; Halpern-type method and extragradient method with a new size rule to solve the equilibrium problems accompanied with pseudo-monotone and Lipschitz-type continuous bi-function in a real Hilbert space. Using certain mild conditions on the bi-function, as well as certain conditions on the iterative control parameters, proves a strong convergence theorem. The proposed algorithm uses a monotonic step size rule depending on local bi-function information. The main results are also used to solve variational inequalities and fixed-point problems. The numerical behavior of the proposed algorithm on different test problems is provided compared to other existing algorithms.

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Keywords

• Equilibrium problem
• Lipschitz-type continuity
• strong convergence
• fixed point problem
• variational inequality problem

MSC

•  47J25
•  47H09
•  47H06
•  47J05

References

• [1] M. Abbas, H. Iqbal, Two inertial extragradient viscosity algorithms for solving variational inequality and fixed point problems, J. Nonlinear Var. Anal., 4 (2020), 377--398
  • View Article


• [2] A. S. Antipin, Equilibrium programming: proximal methods, Comput. Math. Math. Phys., 37 (1997), 1285--1296
  • View Article


• [3] H. H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl., 202 (1996), 150--159
  • View Article


• [4] H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, Cham (2017)
  • View Article


• [5] M. Bianchi, S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31--43
  • View Article


• [6] G. Bigi, M. Castellani, M. Pappalardo, M. Passacantando, Existence and solution methods for equilibria, European J. Oper. Res.,, 227 (2013), 1--11
  • View Article


• [7] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123--145
  • View Article


• [8] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197--228
  • View Article


• [9] Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl.,, 148 (2011), 318--335
  • View Article


• [10] Y. Censor, A. Gibali, S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 26 (2011), 827--845
  • View Article


• [11] K. Fan, A minimax inequality and applications, Academic Press, New York (1972)
  • Google Scholar


• [12] S. D. Flam, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program., 78 (1997), 29--41
  • View Article


• [13] F. Giannessi, A. Maugeri, P. M. Pardalos, Equilibrium problems: nonsmooth optimization and variational inequality models, Springer Science & Business Media, New York (2006)
  • View Article


• [14] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957--961
  • View Article


• [15] D. V. Hieu, Halpern subgradient extragradient method extended to equilibrium problems, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 111 (2017), 823--840
  • View Article


• [16] D. V. Hieu, Y. J. Cho, Y.-B. Xiao, Modified extragradient algorithms for solving equilibrium problems, Optimization, 67 (2018), 2003--2029
  • View Article


• [17] D. V. Hieu, Y. J. Cho, Y.-B. Xiao, P. Kumam, Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math., 2020 (2020), 1--19
  • View Article


• [18] D. V. Hieu, A. Gibali, Strong convergence of inertial algorithms for solving equilibrium problems, Optim. Lett., 14 (2020), 1817--1843
  • View Article


• [19] D. V. Hieu, P. K. Quy, L. T. Hong, L. V. Vy, Accelerated hybrid methods for solving pseudomonotone equilibrium problems, Adv. Comput. Math., 46 (2020), 24 pages
  • View Article


• [20] D. V. Hieu, P. K. Quy, L. V. Vy, Explicit iterative algorithms for solving equilibrium problems, Calcolo, 56 (2019), 21 pages
  • View Article


• [21] D. V. Hieu, J. J. Strodiot, Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces, J. Fixed Point Theory Appl., 20 (2018), 32 pages
  • View Article


• [22] D. V. Hieu, J. J. Strodiot, L. D. Muu, Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems, J. Comput. Appl. Math., 376 (2020), 21 pages
  • View Article


• [23] G. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747--756
  • View Article


• [24] R. Kraikaew, S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl.,, 163 (2014), 399--412
  • View Article


• [25] S. I. Lyashko, 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, Cham, 2016 (2016), 315--325
  • View Article


• [26] P.-E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899--912
  • View Article


• [27] G. Mastroeni, On auxiliary principle for equilibrium problems, in Nonconvex Optimization and Its Applications, Kluwer Acad. Publ., Norwell, MA, 2003 (2003), 289--298
  • View Article


• [28] L. D. Muu, W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal., 18 (1992), 1159--1166
  • View Article


• [29] S. Regmi, I. K. Argyros, S. Georg, Direct comparison between two third convergence order schemes for solving equations, Symmetry, 12 (2020), 10 pages
  • View Article


• [30] H. U. Rehman, P. Kumam, A. B. Abubakar, Y. J. Cho, The extragradient algorithm with inertial effects extended to equilibrium problems, Comput. Appl. Math., 39 (2020), 26 pages
  • View Article


• [31] H. U. Rehman, P. Kumam, I. K. Argyros, N. A. Alreshidi, W. Kumam, W. Jirakitpuwapat, A self-adaptive extragradient methods for a family of pseudomonotone equilibrium programming with application in different classes of variational inequality problems, Symmetry,, 12 (2020), 27 pages
  • View Article


• [32] H. U. Rehman, P. Kumam, I. K. Argyros, W. Deebani, W. Kumam, Inertial extra-gradient method for solving a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces with application in variational inequality problem, Symmetry, 12 (2020), 24 pages
  • View Article


• [33] H. U. Rehman, P. Kumam, I. K. Argyros, M. Shutaywi, Z. Shah, Optimization based methods for solving the equilibrium problems with applications in variational inequality problems and solution of nash equilibrium models, Mathematics, 8 (2020), 28 pages
  • View Article


• [34] H. U. Rehman, P. Kumam, Y. J. Cho, Y. I. Suleiman, W. Kumam, Modified Popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems, Optim. Methods Softw., 36 (2021), 82--113
  • View Article


• [35] H. U. Rehman, P. Kumam, Y. J. Cho, P. Yordsorn, Weak convergence of explicit extragradient algorithms for solving equilibirum problems, J. Inequal. Appl., 2019 (2019), 1--25
  • View Article


• [36] H. U. Rehman, P. Kumam, W. Kumam, M. Shutaywi, W. Jirakitpuwapat, The inertial sub-gradient extra-gradient method for a class of pseudo-monotone equilibrium problems, Symmetry, 12 (2020), 1--25
  • View Article


• [37] R. T. Rockafellar, Convex analysis, Princeton University Press, Princeton (1970)
  • View Article


• [38] G. Stampacchia, Formes bilin´eaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413--4416
  • Google Scholar


• [39] D. Q. Tran, M. L. Dung, V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749--776
  • View Article


• [40] L. Wang, L. Yu, T. Li, Parallel extragradient algorithms for a family of pseudomonotone equilibrium problems and fixed point problems of nonself-nonexpansive mappings in Hilbert space, J. Nonlinear Funct. Anal., 2020 (2020), 1--16
  • View Article


• [41] H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109--113
  • View Article