### An Accelerated Popov’s Subgradient Extragradient Method for Strongly Pseudomonotone Equilibrium Problems in a Real Hilbert Space With Applications

#### Abstract

In this paper, we introduce a subgradient extragradient method to find the numerical solution of strongly pseudomonotone equilibrium problems with the Lipschitz-type condition on a bifunction in a real Hilbert space. The strong convergence theorem for the proposed method is precisely established on the basis of the standard cost bifunction assumptions. The application of our convergence results is also considered in the context of variational inequalities. For numerical analysis, we consider the well-known Nash-Cournot oligopolistic equilibrium model to support our well-established convergence results.

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P. N. Anh and L. T. H. An, The subgradient extragradient method extended to equilibrium problems, Optimization 64 (2015), 225 – 248, DOI: 10.1080/02331934.2012.745528.

P. N. Anh, T. N. Hai and P. M. Tuan, On ergodic algorithms for equilibrium problems, Journal of Global Optimization 64 (2016), 179 – 195, DOI: 10.1007/s10898-015-0330-3.

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Analysis 9 (2001) 3 – 11, DOI: 10.1023/A:1011253113155.

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2011), DOI: 10.1007/978-1-4419-9467-7.

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, Journal of Optimization Theory and Applications 90 (1996), 31 – 43, DOI: 10.1007/bf02192244.

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student 63 (1994), 123 – 145, URL: http://www.indianmathsociety.org.in/ms1991-99contents.pdf.

F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York (2007), URL: https://www.springer.com/gp/book/9780387955803.

K. Fan, A minimax inequality and applications, in Inequalities III, O. Shisha (editor), Academic Press, New York (1972).

D. V. Hieu, Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems, Numerical Algorithms 77 (2018), 983 – 1001, DOI: 10.1007/s11075-017-0350-9.

D. V. Hieu, New extragradient method for a class of equilibrium problems in Hilbert spaces, Applicable Analysis 97 (2017), 811 – 824, DOI: 10.1080/00036811.2017.1292350.

D. V. Hieu, P. K. Quy and L. V. Vy, Explicit iterative algorithms for solving equilibrium problems, Calcolo 56 (2019), Article number: 11, DOI: 10.1007/s10092-019-0308-5.

I. Konnov, Equilibrium Models and Variational Inequalities, Elsevier, Amsterdam (2007).

L. D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Analysis: Theory, Methods & Applications 18 (1992), 1159 – 1166, DOI: 10.1016/0362-546x(92)90159-c.

E. Ofoedu, Strong convergence theorem for uniformly L-lipschitzian asymptotically pseudocontractive mapping in real Banach space, Journal of Mathematical Analysis and Applications 321 (2006), 722 – 728, DOI: 10.1016/j.jmaa.2005.08.076.

T. D. Quoc, P. N. Anh and L. D. Muu, Dual extragradient algorithms extended to equilibrium problems, Journal of Global Optimization 52 (2011), 139 – 159, DOI: 10.1007/s10898-011-9693-2.

D. Q. Tran, M. L. Dung and V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization 57 (2008), 749 – 776, DOI: 10.1080/02331930601122876.

L. D. Muu, V. H. Nguyen and N. V. Quy On Nash-Cournot oligopolistic market equilibrium models with concave cost functions, Journal of Global Optimization 41 (2005), 351 – 364, DOI: 10.1007/s10898-007-9243-0.

P. Santos and S. Scheimberg, An inexact subgradient algorithm for equilibrium problems, Computational & Applied Mathematics 30 (2011), 91 – 107, DOI: 10.1590/S1807-03022011000100005.

S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, Journal of Mathematical Analysis and Applications 331 (2007), 506 – 515, DOI: 10.1016/j.jmaa.2006.08.036.

J. V. Tiel, Convex Analysis: An Introductory Text, Wiley, New York (1984).

H. ur 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 (2020), Article number: 100, 1 – 26, DOI: 10.1007/s40314-020-1093-0.

H. ur Rehman, P. Kumam, I. K. Argyros, N. A. Alreshidi, W. Kumam and W. Jirakitpuwapat, A self-adaptive extra-gradient methods for a family of pseudomonotone equilibrium programming with application in different classes of variational inequality problems, Symmetry 12 (2020), 523, DOI: 10.3390/sym12040523.

H. ur Rehman, P. Kumam, I. K. Argyros, W. Deebani and W. Kumam, Inertial extragradient method for solving a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces with application in variational inequality problem, Symmetry 12 (2020), 503, DOI: 10.3390/sym12040503.

H. ur Rehman, P. Kumam, I. K. Argyros, M. Shutaywi and 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), 822, DOI: 10.3390/math8050822.

H. ur Rehman, P. Kumam, Y. J. Cho and P. Yordsorn, Weak convergence of explicit extragradient algorithms for solving equilibirum problems, Journal of Inequalities and Applications 2019 (2019), Article number: 282, 1 – 25, DOI: 10.1186/s13660-019-2233-1.

H. ur 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 (2020), 1 – 32, DOI: 10.1080/10556788.2020.1734805.

H. ur Rehman, P. Kumam, W. Kumam, M. Shutaywi and W. Jirakitpuwapat, The inertial subgradient extra-gradient method for a class of pseudo-monotone equilibrium problems, Symmetry 12 (2020), 463, DOI: 10.3390/sym12030463.

H. ur Rehman, P. Kumam, M. Shutaywi, N. A. Alreshidi and W. Kumam, Inertial optimization based two-step methods for solving equilibrium problems with applications in variational inequality problems and growth control equilibrium models, Energies 13 (2020), 3292, DOI: 10.3390/en13123292.

H. ur Rehman, N. Pakkaranang, A. Hussain and N. Wairojjana, A modified extra-gradient method for a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces, Journal of Mathematics and Computer Science 22 (2020), 38 – 48, DOI: 10.22436/jmcs.022.01.04.

N. Wairojjana, H. ur Rehman, I. K. Argyros and N. Pakkaranang, An accelerated extragradient method for solving pseudomonotone equilibrium problems with applications, Axioms 9 (2020), 99, DOI: 10.3390/axioms9030099.

N. Wairojjana, H. ur Rehman, M. D. la Sen and N. Pakkaranang, A general inertial projectiontype algorithm for solving equilibrium problem in Hilbert spaces with applications in fixed-point problems, Axioms 9 (2020), 101, DOI: 10.3390/axioms9030101.

DOI: http://dx.doi.org/10.26713%2Fcma.v11i4.1421

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