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A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications

Author

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  • Le Quang Thuy

    (Hanoi University of Science and Technology)

  • Trinh Ngoc Hai

    (Hanoi University of Science and Technology)

Abstract

In this paper, we propose a new algorithm for solving a bilevel equilibrium problem in a real Hilbert space. In contrast to most other projection-type algorithms, which require to solve subproblems at each iteration, the subgradient method proposed in this paper requires only to calculate, at each iteration, two subgradients of convex functions and one projection onto a convex set. Hence, our algorithm has a low computational cost. We prove a strong convergence theorem for the proposed algorithm and apply it for solving the equilibrium problem over the fixed point set of a nonexpansive mapping. Some numerical experiments and comparisons are given to illustrate our results. Also, an application to Nash–Cournot equilibrium models of a semioligopolistic market is presented.

Suggested Citation

  • Le Quang Thuy & Trinh Ngoc Hai, 2017. "A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 411-431, November.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:2:d:10.1007_s10957-017-1176-2
    DOI: 10.1007/s10957-017-1176-2
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    References listed on IDEAS

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    1. Phan Vuong & Jean Strodiot & Van Nguyen, 2014. "Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces," Journal of Global Optimization, Springer, vol. 59(1), pages 173-190, May.
    2. Dang Hieu & Pham Ky Anh & Le Dung Muu, 2017. "Modified hybrid projection methods for finding common solutions to variational inequality problems," Computational Optimization and Applications, Springer, vol. 66(1), pages 75-96, January.
    3. P. Anh & T. Hai & P. Tuan, 2016. "On ergodic algorithms for equilibrium problems," Journal of Global Optimization, Springer, vol. 64(1), pages 179-195, January.
    4. H. Iiduka, 2009. "Hybrid Conjugate Gradient Method for a Convex Optimization Problem over the Fixed-Point Set of a Nonexpansive Mapping," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 463-475, March.
    5. Tran Quoc & Pham Anh & Le Muu, 2012. "Dual extragradient algorithms extended to equilibrium problems," Journal of Global Optimization, Springer, vol. 52(1), pages 139-159, January.
    6. L. D. Muu & T. D. Quoc, 2009. "Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 185-204, July.
    7. 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.
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    Cited by:

    1. Pham Ky Anh & Trinh Ngoc Hai, 2019. "Novel self-adaptive algorithms for non-Lipschitz equilibrium problems with applications," Journal of Global Optimization, Springer, vol. 73(3), pages 637-657, March.
    2. Pham Ky Anh & Trinh Ngoc Hai, 2021. "Dynamical system for solving bilevel variational inequalities," Journal of Global Optimization, Springer, vol. 80(4), pages 945-963, August.

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