Skip to main content
SpringerLink
Log in

Modified projected subgradient method for solving pseudomonotone equilibrium and fixed point problems in Banach spaces

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

Abstract

Motivated by the work of D.V. Hieu and J.-J. Strodiot [Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces, J. Fixed Point Theory Appl., (2018), 20:131], we introduce a new projected subgradient method for solving pseudomonotone equilibrium and fixed point problem in Banach spaces. The main iterative steps in the proposed method use a projection method and do not require any Lipschitz-like condition on the equilibrium bifunction. A strong convergence result is proved under mild conditions and we applied our algorithm to solving pseudomonotone variational inequalities in Banach spaces. Also, we provide some numerical examples to illustrate the performance of the proposed method and compare it with other methods in the literature.

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

  • Alakoya TO, Jolaoso LO, Mewomo OT (2020) Modified inertial subgradient extragradient method with self-adaptive stepsize for solving monotone variational inequality and fixed point problems. Optimization. https://doi.org/10.1080/02331934.2020.1723586

    Article  MATH  Google Scholar 

  • Alber YI (1996) Metric and generalized projections in Banach spaces: properties and applications. In: Kartsatos AG (ed) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp 15–50

  • Alber YI, Ryazantseva I (2006) Nonlinear ill-posed problems of monotone type. Spinger, Dordrecht

    MATH  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 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, Muu LD (2014) A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problem. Optim Lett 8:727–738

    Article  MathSciNet  Google Scholar 

  • Anh PN, Tu HP (2020) Subgradient projection methods extended to monotone bilevel equilibrium problems in Hilbert spaces. Numer Algorithm. https://doi.org/10.1007/s11075-020-00878-w

    Article  MATH  Google Scholar 

  • Anh PN, Hien ND, Phuong NX, Ngocd VT (2020) A parallel subgradient method extended to variational inequalities involving nonexpansive mappings. Appl Anal. https://doi.org/10.1080/00036811.2019.1584288

    Article  MathSciNet  Google Scholar 

  • 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 

  • Bigi G, Castellani M, Pappalardo M (2009) A new solution method for equilibrium problems. Optim Methods Softw 24:895–911

    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 

  • Brézis H, Nirenberg L, Stampacchia G (1972) A remark on Ky Fan’s minimax principle, Bollettino U. M. I., (III), VI , 129-132

  • Combettes PL (2001) Quasi-Fejérian analysis of some optimization algorithms. In: Butnariu D, Reich S, Censor Y (Eds) Inherently parallel algorithms in feasibility and optimization and their applications. Amsterdam: North-Holland; pp 115–152 (Studies in computational mathematics)

  • Fan K (1972) A Minimax Inequality and Applications. In: Shisha O (ed) Inequality III. Academic Press, New York, pp 103–113

    Google Scholar 

  • Gabrie AG, Wangkeeree R (2018) Hybrid projected subgradient-proximal algorithms for solving split equilibrium problems and split common fixed point problems of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl 2018:5

    Article  MathSciNet  Google Scholar 

  • Hieu DV, Cholamjiak P (2020) Modified extragradient method with Bregman distance for variational inequalities. Appl Anal. https://doi.org/10.1080/00036811.2020.1757078

  • Hieu DV (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 (2018) New extragradient method for a class of equilibrium problems in Hilbert spaces. Appl Anal 97(5):811–824. https://doi.org/10.1080/00036811.2017.1292350

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Hieu DV, Strodiot J-J (2018) Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces. J Fixed Point Theory Appl 20:131

    Article  MathSciNet  Google Scholar 

  • Hieu DV, Cho YJ, Xiao YB (2018) Modified extragradient algorithms for solving equilibrium problems. Optimization 67:2003–2029

    Article  MathSciNet  Google Scholar 

  • Huang YY, Jeng JC, Kuo TY, Hong CC (2011) Fixed point and weak convergence theorems for point-dependent \(\lambda \)-hybrid mappings in Banach spaces. Fixed Point Theory and Appl 2011:105

    Article  MathSciNet  Google Scholar 

  • Iusem AN, Svaiter BF, Teboulle M (1994) Entropy-like proximal methods in convex programming. Math Oper Res 19:790–814

    Article  MathSciNet  Google Scholar 

  • Jolaoso LO, Karahan I (2020) 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. https://doi.org/10.1007/s40314-020-01178-8.

  • Jolaoso LO, Taiwo A, Alakoya TO, Mewomo OT (2019) A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem. Comput Appl Math 39. https://doi.org/10.1007/s40314-019-1014-2

  • 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 LO, Taiwo A, Alakoya TO, Mewomo OT (2019) A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces. Demonstr Math 52:183–203

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • 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. https://doi.org/10.1007/s10957-020-01672-3

    Article  MathSciNet  MATH  Google Scholar 

  • Kamimura S, Takahashi W (2002) Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim 13:938–945

    Article  MathSciNet  Google Scholar 

  • Khan MAA, Cholamjiak P (2020) A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces. J Fixed Point Theory Appl 22 , Article 62. https://doi.org/10.1007/s11784-020-00796-3

  • 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 

  • Lyashko SI, Semenov VV (2016) A new two-step proximal algorithm of solving the problem of equilibrium programming, In: Goldengorin B (ed) Optimization and Applications in Control and Data Sciences, Springer Optimization and Its Applications, 115:315-326

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

    Article  MathSciNet  Google Scholar 

  • Nakajo K (2015) Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces. Appl Math Comput 271:251–258

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Raeisi M, Eskandani GZ (2019) A hybrid extragradient method for a general split equality problem involving resolvents and pseudomonotone bifunctions in Banach spaces. Calcolo, 56 , Article no. 4

  • Rehman H, Kumam P, Cho YJ, Yordsorn P (2019) Weak convergence of explicit extragradient algorithms for solving equilibrium problems. J Inequal Appl 1:1–25

    MathSciNet  Google Scholar 

  • Santos P, Scheimberg S (2011) An inexact subgradient algorithm for equilibrium problem. Comput Appl Math 30(1):91–107

    MathSciNet  MATH  Google Scholar 

  • Suantai S, Shehu Y, Cholamjiak P (2019) Nonlinear iterative methods for solving the split common null point problems in Banach spaces. Optim Methods Softw 34:853–874

    Article  MathSciNet  Google Scholar 

  • Sunthrayuth P, Cholamjiak P (2018) Iterative methods for solving quasi-variational inclusion and fixed point problem in q-uniformly smooth Banach spaces. Numer Algor 78:1019–1044

    Article  MathSciNet  Google Scholar 

  • Sunthrayuth P, Cholamjiak P (2019) A modified extragradient method for variational inclusion and fixed point problems in Banach spaces. Appl Anal https://doi.org/10.1080/00036811.2019.1673374

  • Tada A, Takahashi W (2006) Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In: Takahashi W, Tanaka T (eds) Nonlinear Analysis and Convex Analysis. Yokohama Publishers, Yokohama

    Google Scholar 

  • Takahashi W, Zembayashi K (2009) Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal 70:45–57

    Article  MathSciNet  Google Scholar 

  • Takahashi Y, Hashimoto K, Kato M (2002) On sharp uniform convexity, smoothness, and strong type, cotype inequalities. J Nonlinear Convex Anal 3:267–281

    MathSciNet  MATH  Google Scholar 

  • Thong LQ, Hai TN (2017) A projected subgradient algorithm for bilevel equilibrium problems and applications. J Optim Theory Appl 175:411–431

    Article  MathSciNet  Google Scholar 

  • Tiel JV (1984) Convex Analysis: An introductory text. Wiley, New York

    MATH  Google Scholar 

  • Vuong PT (2018) On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J Optim Theory Appl 176(2):399–409

    Article  MathSciNet  Google Scholar 

  • Vuong PT, Strodiot JJ, Nguyen VH (2013) Extraradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J Optim Theory Appl 155:605–627

    Article  Google Scholar 

  • Xu HK (1991) Inequalities in Banach spaces with applications. Nonlinear Anal 16:1127–1138

    Article  MathSciNet  Google Scholar 

  • Yao, Y, Shahzad N, Yao JC (2020) Projected subgradient algorithms for pseudomonotone equilibrium problems and fixed points of pseudocontractive operators. Mathematics 8(4) , 2020:461. https://doi.org/10.3390/math.804046

Download references

Acknowledgements

The author 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 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, P.O. Box 94, Medunsa, 0204, Pretoria, South Africa

    Lateef Olakunle Jolaoso

Authors
  1. Lateef Olakunle Jolaoso
    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 Joerg Fliege.

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. Modified projected subgradient method for solving pseudomonotone equilibrium and fixed point problems in Banach spaces. Comp. Appl. Math. 40, 101 (2021). https://doi.org/10.1007/s40314-021-01490-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-021-01490-x

Keywords

Mathematics Subject Classification

Search

Navigation