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One-Step iterative method for bilevel equilibrium problem in Hilbert space

Author

Listed:
  • Dang Hieu

    (TIMAS - Thang Long University)

  • Pham Kim Quy

    (University of Air Force)

Abstract

The purpose of this paper is to introduce a simple iterative method for finding a solution of an equilibrium problem whose constraint is the solution set of another monotone equilibrium problem in a Hilbert space. Unlike the multi-step methods, the new method only requires to find one value of the proximal mapping associated with cost bifunctions at the current approximation over each iterative step. The strong convergence of the iterative sequence generated by the method is established by incorporating with a regularization technique. The numerical behavior of our method is also illustrated in comparison with several other methods.

Suggested Citation

  • Dang Hieu & Pham Kim Quy, 2023. "One-Step iterative method for bilevel equilibrium problem in Hilbert space," Journal of Global Optimization, Springer, vol. 85(2), pages 487-510, February.
    • Handle: RePEc:spr:jglopt:v:85:y:2023:i:2:d:10.1007_s10898-022-01207-2
    DOI: 10.1007/s10898-022-01207-2
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    References listed on IDEAS

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    1. Thi Thu Van Nguyen & Jean Jacques Strodiot & Van Hien Nguyen, 2014. "Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 809-831, March.
    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. Giancarlo Bigi & Mauro Passacantando, 2015. "Descent and Penalization Techniques for Equilibrium Problems with Nonlinear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 804-818, March.
    4. Hong-Kun Xu, 2011. "Averaged Mappings and the Gradient-Projection Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 360-378, August.
    5. Martin Seydenschwanz, 2015. "Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions," Computational Optimization and Applications, Springer, vol. 61(3), pages 731-760, July.
    6. I.V. Konnov, 2003. "Application of the Proximal Point Method to Nonmonotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 317-333, November.
    7. O. Chaldi & Z. Chbani & H. Riahi, 2000. "Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 105(2), pages 299-323, May.
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