A modified extra-gradient method for a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces

Volume 22, Issue 1, pp 38--48 http://dx.doi.org/10.22436/jmcs.022.01.04

Publication Date: June 05, 2020 Submission Date: March 19, 2020 Revision Date: April 10, 2020 Accteptance Date: April 27, 2020

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Authors

Habib ur Rehman - KMUTTFixed 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. Nuttapol Pakkaranang - KMUTTFixed 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. Azhar Hussain - Department of Mathematics, University of Sargodha, Sargodha-40100, Pakistan. Nopparat Wairojjana - Applied Mathematics Program, Faculty of Science and Technology, Valaya Alongkorn Rajabhat University under the Royal Patronage (VRU), 1 Moo 20 Phaholyothin Road, Klong Neung, Klong Luang, Pathumthani, 13180, Thailand.

Abstract

In this paper, we propose a modified extragradient method for solving a strongly pseudomonotone equilibrium problem in a real Hilbert space. A strong convergence theorem relative to our proposed method is proved and the proposed method has worked without having the information of a strongly pseudomonotone constant and the Lipschitz-type constants of a bifunction. We have carried out our numerical explanations to justify our well-established convergence results, and we can see that our proposed method has a substantial improvement over the time of execution and number iterations.

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Keywords

• Equilibrium problem
• strongly pseudomonotone bifunction
• strong convergence theorem
• Lipschitz-type conditions
• variational inequality problems

MSC

•  47J25
•  47H09
•  47H06
•  47J05

References

• [1] J. Abubakar, P. Kumam, H. ur Rehman, A. H. Ibrahim, Inertial iterative schemes with variable step sizes for variational inequality problem involving pseudomonotone operator, Mathematics, 8 (2020), 25 pages
  • View Article
  • Google Scholar


• [2] J. Abubakar, K. Sombut, H. ur Rehman, A. H. Ibrahim, An accelerated subgradient extragradient algorithm for strongly pseudomonotone variational inequality problems, Thai J. Math., 18 (2020), 166--187
  • View Article
  • MathSciNet
  • Google Scholar


• [3] M. Adeel, K. A. Khan, Ð. Pečarić, J. Pečarić, Generalization of the Levinson inequality with applications to information theorey, J. Inequal. Appl., 2019 (2019), 19 pages
  • View Article
  • Google Scholar


• [4] F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3--11
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2017)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183--202
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [8] E. Blum, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123--145
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117--136
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] S. Dafermos, Traffic equilibrium and variational inequalities, Transportation Sci., 14 (1980), 42--54
  • View Article
  • MathSciNet
  • Google Scholar


• [11] M. C. Ferris, J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669--713
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] S. D. Flam, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming, 78 (1996), 29--41
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] F. Giannessi, A. Maugeri, P. M. Pardalos, Equilibrium problems: nonsmooth optimization and variational inequality models, Springer, New York (2001)
  • View Article
  • Google Scholar
  • MATH


• [14] D. V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal., 21 (2016), 478--501
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] D. V. Hieu, New extragradient method for a class of equilibrium problems in Hilbert spaces, Appl. Anal., 97 (2017), 811--824
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] D. V. Hieu, Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems, Numer. Algorithms, 77 (2018), 983--1001
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] A. N. Iusem, W. Sosa, Iterative algorithms for equilibrium problems, Optimization, 52 (2003), 301--316
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] I. V. Konnov, Application of the proximal point method to nonmonotone equilibrium problems, J. Optim. Theory Appl., 119 (2003), 317--333
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] I. Konnov, Equilibrium models and variational inequalities, Elsevier B. V., Amsterdam (2007)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] A. Krylatov, V. Zakharov, T. Tuovinen, Optimization Models and Methods for Equilibrium Traffic Assignment, Springer, Cham (2020)
  • View Article
  • MathSciNet
  • Google Scholar


• [21] X. Li, A. Hussain, M. Adeel, E. Savas, Fixed point theorems for $Z_{\theta}$-contraction and applications to nonlinear integral equations, IEEE Access, 7 (2019), 120023--120029
  • View Article


• [22] E. Ofoedu, Strong convergence theorem for uniformly l-lipschitzian asymptotically pseudocontractive mapping in real Banach space, J. Math. Anal. Appl., 321 (2006), 722--728
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] B.T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R. Comput. Math. Math. Phys., 4 (1964), 1--17
  • View Article
  • Google Scholar
  • MATH


• [24] T. D. Quoc, P. N. Anh, L. D. Muu, Dual extragradient algorithms extended to equilibrium problems, J. Global Optim., 52 (2011), 139--159
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] 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
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] J. V. Tiel, Convex analysis: an introductory text, Wiley, New York (1984)
  • View Article
  • Google Scholar
  • MATH


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


• [28] H. ur Rehman, D. Gopal, P. Kumam, Generalizations of darbo's fixed point theorem for new condensing operators with application to a functional integral equation, Demonstr. Math., 52 (2019), 166--182
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] H. ur 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
  • MathSciNet
  • Google Scholar
  • MATH


• [30] H. ur Rehman, P. Kumam, I. K. Argyros, N. A. Alreshidi, W. Kumam, 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), 27 pages
  • View Article
  • Google Scholar


• [31] H. ur 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
  • Google Scholar


• [32] H. ur Rehman, P. Kumam, Y. J. Cho, Y. I. Suleiman, W. Kumam, Modified popov's explicit iterative algorithms for solving pseudomonotone equilibrium problems, Opti. Methods Soft., 2020 (2020), 1--32
  • View Article
  • Google Scholar


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


• [34] H. ur Rehman, P. Kumam, S. Dhompongsa, Existence of tripled fixed points and solution of functional integral equations through a measure of noncompactness, Carpathian J. Math., 35 (2019), 193--208
  • View Article
  • MathSciNet
  • Google Scholar


• [35] 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), 25 pages
  • View Article
  • Google Scholar