Skip to main content
SpringerLink
Log in

A self-adaptive parallel subgradient extragradient method for finite family of pseudomonotone equilibrium and fixed point problems

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we introduce a self-adaptive parallel subgradient extragradient method for solving a finite family of pseudomonotone equilibrium problem and common fixed point problems in real Hilbert spaces. The algorithm is designed such that its stepsize is determined by a self-adaptive process and a convex combination method is used to approximate the sequences generated by the strongly convex optimization problems. This improves the convergence of the method and also avoids the need for choosing prior estimate of the Lipschitz-like constants of the bifunctions. Under suitable conditions, we prove the strong convergence of the sequence generated by our proposed method to the desired solution. We also provide some numerical experiments to illustrate the performance and efficiency of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Anh PN (2013) A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems. Bull. Malays. Math. Sci. Soc. 36(1):107–116

    MathSciNet  MATH  Google Scholar 

  • Anh PN, An LTH (2019) New subgradient extragradient methods for solving monotone bilevel equilibrium problems. Optimization 68(11):2097–2122

    Article  MathSciNet  Google Scholar 

  • Anh PN, An LTH (2015) The subgradient extragradient method extended to equilibrium problems. Optimization 64(2):225–248

    Article  MathSciNet  Google Scholar 

  • Anh PK, Hieu DV (2016) Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math. 44(2):351–374

    Article  MathSciNet  Google Scholar 

  • Anh TV, Muu LD, Son DX (2018) Parallel algorithms for solving a class of variational inequalities over the common fixed points set of a finite family of demicontractive mappings. Numer. Funct. Anal. Optim. 39:1477–1494

    Article  MathSciNet  Google Scholar 

  • Bigi G, Castellani M, Pappalardo M, Passacantando M (2019) Nonlinear programming technique for equilibria. Spinger Nature, Switzerland AG

  • Bigi G, Passacantando M (2015) Descent and penalization techniques for equilibrium problems with nonlinear constraints. J. Optim. Theory Appl. 164:804–818

    Article  MathSciNet  Google Scholar 

  • Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Stud. 63:123–145

    MathSciNet  MATH  Google Scholar 

  • Censor Y, Gibali A, Reich S, Sabach S (2012) Common solutions to variational inequalities. Set Valued Var. Anal. 20:229–247

    Article  MathSciNet  Google Scholar 

  • Chidume CE, Ezeora JN (2014) Krasnoselskii-type algorithm for family of multivalued strictly pseudocontractive mappings. Fixed Point Theory Appl. 2014:111

    Article  Google Scholar 

  • Cholamjiak P, Suantai S, Sunthrayuth P (2020) An explicit parallel algorithm for solving variational inclusion problem and fixed point problem in Banach spaces. Banach J. Math. Anal. 14(14):20–40

    Article  MathSciNet  Google Scholar 

  • Daniele P, Giannessi F, Maugeri A (2003) Equilibrium Problems and Variational Models. Kluwer Academic Publishers, Dordrecht, The Netherlands

  • Hieu DV (2016) Common solutions to pseudomonotone equilibrium problems. Bull. Iranian Math. Soc. 42(5):1207–1219

    MathSciNet  MATH  Google Scholar 

  • Hieu D.V,(2017) Halpern subgradient extragradient method extended to equilibrium problems. Rev. R. Acad. Cienc. Exactas F’i,s. Nat. Ser. A Math. RACSAM, 111, 823–840

  • Hieu DV (2017) New subgradient extragradient methods for common solutions to equilibrium problems. Comput Optim Appl. 6(7):571–594

    Article  MathSciNet  Google Scholar 

  • Hieu DV (2018) Modified subgradient extragradient algorithm for pseudomonotone equilibrium problems. Bul. Kor. Math. Soc. 55(5):1503–1521

    MathSciNet  MATH  Google Scholar 

  • Hieu DV, Cho YJ, Xiao YB, Kumam P (2021) Modified Extragradient Method for Pseudomonotone Variational Inequalities in Infinite Dimensional Hilbert Spaces. Vietnam J. Math. 4(9):1165–1183

    Article  MathSciNet  Google Scholar 

  • Hieu DV, Thai BH, Kumam P (2020) Parallel modified methods for pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings. Adv. Oper. Theory 5:1684–1717

    Article  MathSciNet  Google Scholar 

  • Hieu DV, Muu LD, Anh PK (2016) Parallel hybrid extragradient methods for pseudmonotone equilibrium problems and nonexpansive mappings. Numer. Algor. 7(3):197–217

    Article  Google Scholar 

  • Iiduka H (2010) A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping. Optimization 5(9):873–885

    Article  MathSciNet  Google Scholar 

  • Iiduka H, Yamada I (2009) A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 1(9):1881–1893

    Article  MathSciNet  Google Scholar 

  • Jolaoso L.O, Alakoya T.O, Taiwo A, Mewomo O.T,(2020) A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo, II. Ser., 69 , 711-735

  • Jolaoso LO, Alakoya TO, Taiwo A, Mewomo OT (2020) An inertial extragradient method via viscoscity approximation approach for solving equilibrium problem in Hilbert spaces. Optimization. https://doi.org/10.1080/02331934.2020.1716752

  • Jolaoso L.O, Karahan I, A general alternative regularization method with line search technique for solving split equilibrium and fixed point problems in Hilbert spaces, Comput. Appl. Math., 39, Article 150

  • Jolaoso L.O, Shehu Y. (2021) Single Bregman projection method for solving variational inequalities in reflexive Banach spaces, Appl. Analysis, https://doi.org/10.1080/00036811.2020.1869947

  • Jolaoso LO, Taiwo A, Alakoya TO, Mewomo OT (2020) A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods in a reflexive Banach space. J. Optim. Theory Appl. 185(3):744–766

    Article  MathSciNet  Google Scholar 

  • Jolaoso LO, Aphane M (2020) A self-adaptive inertial subgradient extragradient method for pseudomonotone equilibrium and common fixed point problems. Fixed Point Theory Appl. 2020:9. https://doi.org/10.1186/s13663-020-00676-y

    Article  MathSciNet  MATH  Google Scholar 

  • Jolaoso L.O, Aphane M,(2020) An efficient parallel extragradient method for systems of variational inequalities involving fixed points of demicontractive mappings, Symmetry, 12 (11) (2020), Article: 1915

  • Jolaoso LO, Shehu Y, Cho YJ (2021) Convergence analysis for variational inequalities and fixed point problems in reflexive Banach spaces. J Inequal Appl 2021:44. https://doi.org/10.1186/s13660-021-02570-6

    Article  MathSciNet  MATH  Google Scholar 

  • Kesornprom S, Cholamjiak P (2019) Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications. Optimization 68(6):2365–2391

    MathSciNet  MATH  Google Scholar 

  • Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody. 12:747–756 ((In Russian))

    MathSciNet  MATH  Google Scholar 

  • Maingé E (2010) Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur. J. Oper. Res. 205:501–506

    Article  MathSciNet  Google Scholar 

  • Maingé PE (2008) Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16:899–912

    Article  MathSciNet  Google Scholar 

  • Moudafi A (2000) Viscosity approximation method for fixed-points problems. J. Math. Anal. Appl. 241(1):46–55

    Article  MathSciNet  Google Scholar 

  • Muu LD, Oettli W (1992) Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18:1159–1166

    Article  MathSciNet  Google Scholar 

  • Quoc TD, Muu LD, Nguyen VH (2008) Extragradient algorithms extended to equilibrium problems. Optimization 57:749–776

    Article  MathSciNet  Google Scholar 

  • Rudin W (1991) Functional Analysis. McGraw-Hill Series in Higher Mathematics, New York

  • Suantai S, Peeyada P, Yambangwai D, Cholamjiak W,(2020) A parallel-viscosity-type subgradient extragradient-line method for finding the common solution of variational inequality problems applied to image restoration problems, Mathematics, 8 (2020), Article: 248

  • Tran QD, Muu LD, Nguyen VH (2008) Extragradient algorithms extended to equilibrium problems. Optimization 57:749–776

    Article  MathSciNet  Google Scholar 

  • Xu HK (2002) Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66:240–256

    Article  MathSciNet  Google Scholar 

  • Yang J, Liu H (2020) The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert space. Optim. Letters. 14:1803–1816

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors acknowledge with thanks, the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University for making their facilities available for the research. The author also thanks the anonymous reviewers for their valuable comments which improved the first draft of the paper.

Funding

The first author is supported by the Postdoctoral research grant from the Sefako Makgatho Health Sciences University, South Africa.

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, PO Box 94, Medunsa, Pretoria, 0204, South Africa

    Lateef Olakunle Jolaoso & Maggie Aphane

Authors
  1. Lateef Olakunle Jolaoso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maggie Aphane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lateef Olakunle Jolaoso.

Additional information

Communicated by Eduardo Souza de Cursi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jolaoso, L.O., Aphane, M. A self-adaptive parallel subgradient extragradient method for finite family of pseudomonotone equilibrium and fixed point problems. Comp. Appl. Math. 41, 111 (2022). https://doi.org/10.1007/s40314-022-01817-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01817-2

Keywords

Mathematics Subject Classification

Search

Navigation