Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 69, 2020 - Issue 4
374
Views
25
CrossRef citations to date
0
Altmetric
Articles

The subgradient extragradient method for pseudomonotone equilibrium problems

, &
Pages 901-923 | Received 24 Aug 2018, Accepted 18 May 2019, Published online: 18 Jun 2019

References

  • Fan K. A minimax inequality and applications. In: Inequalities III. New York: Academic Press; 1972. p. 103–113.
  • Fan K. Some properties of convex sets related to fixed point theorems. Math Ann. 1984;266:519–537.
  • Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Math Student. 1994;63(1):123–145.
  • Konnov IV. Combined relaxation methods for variational inequalities. Berlin: Springer; 2000.
  • Konnov IV. Equilibrium models and variational inequalities. Amsterdam: Elsevier; 2007.
  • Muu LD, Oettli W. Convergence of an adative penalty scheme for finding constrained equilibria. Nonlinear Anal. 1992;18:1159–1166.
  • Giannessi F, Maugeri A, Pardalos PM, editors. Equilibrium problems: nonsmooth optimization and variational inequality models. Dordrecht: Kluwer; 2001.
  • Iusem AN, Sosa W. Iterative algorithms for equilibrium problems. Optimization. 2003;52:301–316.
  • Mastroeni G. Gap functions for equilibrium problems. J Global Optim. 2003;27:411–426.
  • Muu LD. Stability property of a class of variational inequalities. Optimization. 1984;15:347–353.
  • Hieu DV. Halpern subgradient extragradient method extended to equilibrium problems. RACSAM. 2017;111(3):823–840.
  • Hieu DV. New extragradient method for a class of equilibrium problems in Hilbert spaces. Appl Anal. 2018;97(5):811–824.
  • Wang S, Zhao M, Kumam P, et al. A viscosity extragradient method for an equilibrium problem and fixed point problem in Hilbert space. J Fixed Point Theory Appl. 2018;20(1): Art. 19, 14 pp.
  • Anh PK, Hai TN. A splitting algorithm for equilibrium problem given by the difference of two bifunctions. J Fixed Point Theory Appl. 2018;20(1): Art. 53, 15 pp.
  • Hieu DV. Projected subgradient algorithms on systems of equilibrium problems. Optim Lett. 2018;12(3):551–566.
  • Muu LD, Quoc TD. Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model. J Optim Theory Appl. 2009;142:185–204.
  • Scheimberg S, Santos PSM. A relaxed projection method for finite dimensional equilibrium problems. Optimization. 2011;60:1193–1208.
  • Shehu Y. Iterative procedures for left Bregman strongly relatively nonexpansive mappings with application to equilibrium problems. Fixed Point Theory. 2016;17(1):173–188.
  • Shehu Y. Iterative approximation of countable family of relatively nonexpansive mappings and system of equilibrium problems in Banach spaces. Afr Mat. 2015;26(5–6):1049–1069.
  • Shehu Y. Strong convergence theorem for nonexpansive semigroups and systems of equilibrium problems. J Global Optim. 2013;56(4):1675–1688.
  • Shehu Y, Ogbuisi FU. An iterative algorithm for finding a common solution of fixed point problem and system of equilibrium problems. Dyn Contin Discrete Impuls Syst Ser B Appl Algorithms. 2016;23(4):251–267.
  • Dadashi V, Khatibzadeh H. On the weak and strong convergence of the proximal point algorithm in reflexive Banach spaces. Optimization. 2017;66(9):1487–1494.
  • Dadashi V, Postolache M. Hybrid proximal point algorithm and applications to equilibrium problems and convex programming. J Optim Theory Appl. 2017;174:518–529.
  • Iusem AN, Sosa W. On the proximal point method for equilibrium problem in Hilbert spaces. Optimization. 2010;59:1259–1274.
  • Santos PJS, Santos PSM, Scheimberg S. A proximal Newton-type method for equilibrium problems. Optim Lett. 2018;12(5):997–1009.
  • Kumam P, Katchang P. A viscosity of extragradient approximation method for finding equilibrium problems and fixed point problems for nonexpansive mappings. Nonlinear Anal Hybrid Syst. 2009;3:475–486.
  • Kumam W, Kumam P. Hybrid iterative scheme by relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization. Nonlinear Anal Hybrid Syst. 2009;3:640–656.
  • Quoc TD, Anh PN, Muu LD. Dual extragradient algorithms extended to equilibrium problems. J Global Optim. 2012;52:139–159.
  • Quoc TD, Muu LD, Hien NV. Extragradient algorithms extended to equilibrium problems. Optimization. 2008;57:749–776.
  • Van NTT, Strodiot JJ, Nguyen VH. The interior proximal extragradient method for solving equilibrium problems. J Global Optim. 2009;44:175–192.
  • Van NTT, Strodiot JJ, Nguyen VH, et al. An extragradient-type method for solving nonmonotone quasi-equilibrium problems. Optimization. 2018;67(5):651–664.
  • Kassay G, Reich S, Sabach S. Iterative methods for solving systems of variational inequalities in reexive Banach spaces. SIAM J Optim. 2011;21:1319–1344.
  • Reich S, Sabach S. Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemporary Math. 2012;568:225–240.
  • Bigi G, Passacantando M. Descent and penalization techniques for equilibrium problems with nonlinear constraints. J Optim Theory Appl. 2015;164:804–818.
  • Chadli O, Konnov IV, Yao JC. Descent methods for equilibrium problems in a Banach space. Comput Math Appl. 2004;48:609–616.
  • Di Lorenzo D, Passacantando M, Sciandrone M. A convergent inexact solution method for equilibrium problems. Optim Methods Softw. 2014;29:979–991.
  • Konnov IV, Pinyagina OV. D-gap functions for a class of equilibrium problems in Banach spaces. Comput Methods Appl Math. 2003;3:274–286.
  • Zhang L, Wu SY. An algorithm based on the generalized D-gap function for equilibrium problems. J Comput Appl Math. 2009;231:403–411.
  • Korpelevich GM. An extragradient method for finding saddle points and for other problems. Ékonom i Mat Metody. 1976;12:747–756.
  • Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148(2):318–335.
  • Censor Y, Gibali A, Reich S. Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space. Optimization. 2012;61(9):1119–1132.
  • Gibali A. A new non-Lipschitzian projection method for solving variational inequalites in Euclidean spaces. J Nonlinear Anal Optim. 2015;6(1):41–51.
  • Malitsky YV. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim. 2015;25(1):502–520.
  • Malitsky YV, Semenov VV. An extragradient algorithm for monotone variational inequalities. Cybern Syst Anal. 2014;50(2):271–277.
  • Hieu DV, Anh PK, Muu LD. Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput Optim Appl. 2017;66:75–96.
  • Tran DQ, Dung ML, Nguyen VH. Extragradient algorithms extended to equilibrium problem. Optimization. 2008;57:749–776.
  • Daniele P, Giannessi F, Maugeri A. Equilibrium problems and variational models. Dordrecht: Kluwer Academic; 2003.
  • Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementary problems. New York: Springer-Verlag; 2003.
  • van Tiel J. Convex analysis: an introductory text. New York: John Wiley & Sons, Inc.; 1984. viii+125 pp. ISBN: 0-471-90263-2.
  • Xu HK. Averaged mappings and the gradient-projection algorithm. J Optim Theory Appl. 2011;150:360–378.
  • Muu LD, Quy NV. On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J Math. 2015;43:229–238.
  • Cegielski A. Iterative methods for fixed point problems in Hilbert spaces. Berlin: Springer; 2012. (Lecture notes in mathematics; 2057).
  • Contreras J, Klusch M, Krawczyk JB. Numerical solution to Nash Cournot equilibria in coupled constraint electricity markets. IEEE Trans Power Syst. 2004;19:195–206.
  • Shehu Y, Dong Q-L, Jiang D. Single projection method for pseudo-monotone variational inequality in Hilbert Spaces. Optimization. In press. DOI:10.1080/02331934.2018.1522636

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.