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Two Regularization Methods for the Variational Inequality Problem over the Set of Solutions of the Generalized Mixed 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

In this work, we consider bilevel problems: variational inequality problems over the set of solutions of the generalized mixed equilibrium problems. Two new inertial extragradient methods are proposed for solving these problems. Under appropriate conditions, we prove strong convergence theorems for the proposed methods by the regularization technique. Finally, some numerical examples are provided to show the efficiency of the proposed algorithms.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:16:p:2981-:d:891618
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    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. Abdellatif Moudafi, 2010. "Proximal methods for a class of bilevel monotone equilibrium problems," Journal of Global Optimization, Springer, vol. 47(2), pages 287-292, June.
    3. 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.
    4. Omar Bazighifan & Poom Kumam, 2020. "Oscillation Theorems for Advanced Differential Equations with p -Laplacian Like Operators," Mathematics, MDPI, vol. 8(5), pages 1-10, May.
    5. Yanlai Song, 2021. "Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems," Mathematics, MDPI, vol. 9(21), pages 1-19, October.
    6. Yanlai Song & Omar Bazighifan, 2022. "A New Alternative Regularization Method for Solving Generalized Equilibrium Problems," Mathematics, MDPI, vol. 10(8), pages 1-14, April.
    7. 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.
    Full references (including those not matched with items on IDEAS)

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