Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications

Habib ur Rehman; Poom Kumam; Kanokwan Sitthithakerngkiet; Habib ur Rehman; Poom Kumam; Kanokwan Sitthithakerngkiet

doi:10.3934/math.2021093

AIMS Mathematics

2021, Volume 6, Issue 2: 1538-1560. doi: 10.3934/math.2021093

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Research article

Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications

1.
KMUTT Fixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
2.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
3.
Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok (KMUTNB), Bangkok, Thailand

Received: 14 September 2020 Accepted: 13 November 2020 Published: 23 November 2020
MSC : 47H05, 47H10

The aim of this article is to introduce a new algorithm by integrating a viscosity-type method with the subgradient extragradient algorithm to solve the equilibrium problems involving pseudomonotone and Lipschitz-type continuous bifunction in a real Hilbert space. A strong convergence theorem is proved by the use of certain mild conditions on the bifunction as well as some restrictions on the iterative control parameters. Applications of the main results are also presented to address variational inequalities and fixed-point problems. The computational behaviour of the proposed algorithm on various test problems is described in comparison to other existing algorithms.
- strong convergence,
- equilibrium problem,
- Lipschitz-like conditions,
- fixed point problems,
- variational inequality problem
Citation: Habib ur Rehman, Poom Kumam, Kanokwan Sitthithakerngkiet. Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications[J]. AIMS Mathematics, 2021, 6(2): 1538-1560. doi: 10.3934/math.2021093

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Abstract

The aim of this article is to introduce a new algorithm by integrating a viscosity-type method with the subgradient extragradient algorithm to solve the equilibrium problems involving pseudomonotone and Lipschitz-type continuous bifunction in a real Hilbert space. A strong convergence theorem is proved by the use of certain mild conditions on the bifunction as well as some restrictions on the iterative control parameters. Applications of the main results are also presented to address variational inequalities and fixed-point problems. The computational behaviour of the proposed algorithm on various test problems is described in comparison to other existing algorithms.

References

[1]	P. N. Anh, L. T. H. An, The subgradient extragradient method extended to equilibrium problems, Optimization, 64 (2012), 225-248.
[2]	A. Antipin, Equilibrium programming: proximal methods, Comput. Math. Math. Phys., 37 (1997), 1285-1296.
[3]	M. Bianchi, S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optimiz. Theory App., 9 (1996), 31-43.
[4]	G. Bigi, M. Castellani, M. Pappalardo, M. Passacantando, Existence and solution methods for equilibria, Eur. J. Oper. Res., 227 (2013), 1-11. doi: 10.1016/j.ejor.2012.11.037
[5]	G. Bigi, M. Castellani, M. Pappalardo, M. Passacantando, Nonlinear programming techniques for equilibria, Springer International Publishing, 2019.
[6]	E. Blum, From optimization and variational inequalities to equilibrium problems, Mathematics Students, 63 (1994), 123-145.
[7]	F. Browder, W. Petryshyn, Construction of fixed points of nonlinear mappings in hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228. doi: 10.1016/0022-247X(67)90085-6
[8]	Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in hilbert space, J. Optimiz. Theory App., 148 (2010), 318-335.
[9]	C. E. Chidume, A. Adamu, L. C. Okereke, A krasnoselskii-type algorithm for approximating solutions of variational inequality problems and convex feasibility problems, J. Nonlinear Var. Anal., 2 (2018), 203-218. doi: 10.23952/jnva.2.2018.2.07
[10]	S. Dafermos, Traffic equilibrium and variational inequalities, Transport. Sci., 14 (1980), 42-54. doi: 10.1287/trsc.14.1.42
[11]	K. Fan, A minimax inequality and applications, Inequalities Ⅲ, New York: Academic Press, 1972.
[12]	M. Farid, The subgradient extragradient method for solving mixed equilibrium problems and fixed point problems in hilbert spaces, J. Appl. Numer. Optim., 1 (2019), 335-345.
[13]	S. D. Flåm, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program., 78 (1996), 29-41. doi: 10.1007/BF02614504
[14]	F. Giannessi, A. Maugeri, P. M. Pardalos, Equilibrium problems: Nonsmooth optimization and variational inequality models, Springer Science & Business Media, 2006.
[15]	H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, New York: Springer, 2017.
[16]	D. V. Hieu, Halpern subgradient extragradient method extended to equilibrium problems, RACSAM, 111 (2016), 823-840.
[17]	D. V. Hieu, P. K. Quy, L. V. Vy, Explicit iterative algorithms for solving equilibrium problems, Calcolo, 56 (2019), 11. doi: 10.1007/s10092-019-0308-5
[18]	G. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), 747-756.
[19]	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 International Publishing, 2016,315-325.
[20]	P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912. doi: 10.1007/s11228-008-0102-z
[21]	G. Mastroeni, On auxiliary principle for equilibrium problems, In: Nonconvex optimization and its applications, Springer, 2003,289-298.
[22]	A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.
[23]	A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55. doi: 10.1006/jmaa.1999.6615
[24]	L. Muu, W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal. Theor., 18 (1992), 1159-1166. doi: 10.1016/0362-546X(92)90159-C
[25]	F. U. Ogbuisi, Y. Shehu, A projected subgradient-proximal method for split equality equilibrium problems of pseudomonotone bifunctions in banach spaces, J. Nonlinear Var. Anal., 3 (2019), 205-224.
[26]	T. D. Quoc, P. N. Anh, L. D. Muu, Dual extragradient algorithms extended to equilibrium problems, J. Global Optim., 52 (2011), 139-159.
[27]	T. D. Quoc, M. N. V. H. Le Dung, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749-776. doi: 10.1080/02331930601122876
[28]	R. T. Rockafellar, Convex analysis, Princeton University Press, 1970.
[29]	P. Santos, S. Scheimberg, An inexact subgradient algorithm for equilibrium problems, Comput. Appl. Math., 30 (2011), 91-107.
[30]	T. M. M. Sow, An iterative algorithm for solving equilibrium problems, variational inequalities and fixed point problems of multivalued quasi-nonexpansive mappings, Appl. Set-Valued Anal. Optim., 1 (2019), 171-185.
[31]	G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences, 258 (1964), 4413.
[32]	S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces, J. Math. Anal. Appl., 331 (2007), 506-515. doi: 10.1016/j.jmaa.2006.08.036
[33]	L. Q. Thuy, C. F. Wen, J. C. Yao, T. N. Hai, An extragradient-like parallel method for pseudomonotone equilibrium problems and semigroup of nonexpansive mappings, Miskolc Math. Notes, 19 (2018), 1185. doi: 10.18514/MMN.2018.2114
[34]	D. Q. Tran, M. L. Dung, V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749-776. doi: 10.1080/02331930601122876
[35]	H. ur Rehman, P. Kumam, A. B. Abubakar, Y. J. Cho, The extragradient algorithm with inertial effects extended to equilibrium problems, Comp. Appl. Math., 39 (2010), 100.
[36]	H. ur Rehman, P. Kumam, Y. J. Cho, P. Yordsorn, Weak convergence of explicit extragradient algorithms for solving equilibirum problems. J. Inequal. Appl., 2019 (2019), 282. doi: 10.1186/s13660-019-2233-1
[37]	H. ur Rehman, P. Kumam, Y. Je Cho, Y.I. Suleiman, W. Kumam, Modified popov's explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optimization Methods and Software, 2020, 1-32.
[38]	H. ur 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), 463. doi: 10.3390/sym12030463
[39]	N. T. Vinh, L. D. Muu, Inertial extragradient algorithms for solving equilibrium problems, Acta Mathematica Vietnamica, 44 (2019), 639-663. doi: 10.1007/s40306-019-00338-1
[40]	H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, B. Aust. Math. Soc., 65 (2002), 109-113. doi: 10.1017/S0004972700020116
[41]	J. C. Yao, Variational inequalities with generalized monotone operators, Math. Oper. Res., 19 (1994), 691-705. doi: 10.1287/moor.19.3.691

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