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

Author

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  • 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
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    References listed on IDEAS

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    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. 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.
    3. 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.
    4. 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.
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    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.

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