IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i14p2443-d861926.html
   My bibliography  Save this article

Regularization Method for the Variational Inequality Problem over the Set of Solutions to the Generalized Equilibrium Problem

Author

Listed:
  • Yanlai Song

    (College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
    These authors contributed equally to this work.)

  • Omar Bazighifan

    (Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
    Department of Mathematics, Faculty of Science, Hadhramout University, Mukalla 50512, Yemen
    These authors contributed equally to this work.)

Abstract

The paper is devoted to bilevel problems: variational inequality problems over the set of solutions to the generalized equilibrium problems in a Hilbert space. To solve these problems, an iterative algorithm is proposed that combines the ideas of the Tseng’s extragradient method, the inertial idea and iterative regularization. The proposed method adopts a non-monotonic stepsize rule without any line search procedure. Under suitable conditions, the strong convergence of the resulting method is obtained. Several numerical experiments are also provided to illustrate the efficiency of the proposed method with respect to certain existing ones.

Suggested Citation

  • Yanlai Song & Omar Bazighifan, 2022. "Regularization Method for the Variational Inequality Problem over the Set of Solutions to the Generalized Equilibrium Problem," Mathematics, MDPI, vol. 10(14), pages 1-20, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2443-:d:861926
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/14/2443/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/14/2443/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Heinz H. Bauschke & Patrick L. Combettes, 2001. "A Weak-to-Strong Convergence Principle for Fejér-Monotone Methods in Hilbert Spaces," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 248-264, May.
    2. Yao, Yonghong & Cho, Yeol Je & Liou, Yeong-Cheng, 2011. "Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems," European Journal of Operational Research, Elsevier, vol. 212(2), pages 242-250, July.
    3. N. Aliev & S. Mohammad Hosseini, 2001. "A Regularization of Fredholm type singular integral equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 26, pages 1-6, January.
    4. J. Glackin & J. G. Ecker & M. Kupferschmid, 2009. "Solving Bilevel Linear Programs Using Multiple Objective Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 197-212, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Yanlai Song & Omar Bazighifan, 2022. "Two Regularization Methods for the Variational Inequality Problem over the Set of Solutions of the Generalized Mixed Equilibrium Problem," Mathematics, MDPI, vol. 10(16), pages 1-20, August.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yanlai Song & Omar Bazighifan, 2022. "Two Regularization Methods for the Variational Inequality Problem over the Set of Solutions of the Generalized Mixed Equilibrium Problem," Mathematics, MDPI, vol. 10(16), pages 1-20, August.
    2. Yanlai Song & Omar Bazighifan, 2022. "A New Alternative Regularization Method for Solving Generalized Equilibrium Problems," Mathematics, MDPI, vol. 10(8), pages 1-14, April.
    3. Yanlai Song & Omar Bazighifan, 2022. "Modified Inertial Subgradient Extragradient Method with Regularization for Variational Inequality and Null Point Problems," Mathematics, MDPI, vol. 10(14), pages 1-17, July.
    4. Yonghong Yao & Yeong-Cheng Liou & Ngai-Ching Wong, 2013. "Superimposed optimization methods for the mixed equilibrium problem and variational inclusion," Journal of Global Optimization, Springer, vol. 57(3), pages 935-950, November.
    5. Hecheng Li, 2015. "A genetic algorithm using a finite search space for solving nonlinear/linear fractional bilevel programming problems," Annals of Operations Research, Springer, vol. 235(1), pages 543-558, December.
    6. Cholamjiak, Watcharaporn & Dutta, Hemen, 2022. "Viscosity modification with parallel inertial two steps forward-backward splitting methods for inclusion problems applied to signal recovery," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    7. Yanlai Song, 2021. "Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems," Mathematics, MDPI, vol. 9(21), pages 1-19, October.
    8. Ferdinard U. Ogbuisi & Yekini Shehu & Jen-Chih Yao, 2023. "Relaxed Single Projection Methods for Solving Bilevel Variational Inequality Problems in Hilbert Spaces," Networks and Spatial Economics, Springer, vol. 23(3), pages 641-678, September.
    9. Yanlai Song & Mihai Postolache, 2021. "Modified Inertial Forward–Backward Algorithm in Banach Spaces and Its Application," Mathematics, MDPI, vol. 9(12), pages 1-17, June.
    10. A. Moudafi, 2011. "Split Monotone Variational Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 275-283, August.
    11. Renli Liang & Yanqin Bai & Hai Xiang Lin, 2019. "An inexact splitting method for the subspace segmentation from incomplete and noisy observations," Journal of Global Optimization, Springer, vol. 73(2), pages 411-429, February.
    12. Sauli Ruuska & Kaisa Miettinen & Margaret M. Wiecek, 2012. "Connections Between Single-Level and Bilevel Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 60-74, April.
    13. Yan Tang & Yeol Je Cho, 2019. "Convergence Theorems for Common Solutions of Split Variational Inclusion and Systems of Equilibrium Problems," Mathematics, MDPI, vol. 7(3), pages 1-25, March.
    14. Farajzadeh, A.P. & Plubtieng, S. & Ungchittrakool, K. & Kumtaeng, D., 2015. "Generalized mixed equilibrium problems with generalized α -η -monotone bifunction in topological vector spaces," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 313-319.
    15. Nakajo, Kazuhide, 2015. "Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 251-258.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2443-:d:861926. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.