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Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space

Author

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  • Thi Thu Van Nguyen

    (Institute for Computational Science and Technology at Ho Chi Minh City (ICST)
    VNU-HCM)

  • Jean Jacques Strodiot

    (Institute for Computational Science and Technology at Ho Chi Minh City (ICST)
    University of Namur)

  • Van Hien Nguyen

    (Institute for Computational Science and Technology at Ho Chi Minh City (ICST)
    University of Namur)

Abstract

This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixed point iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:160:y:2014:i:3:d:10.1007_s10957-013-0400-y
    DOI: 10.1007/s10957-013-0400-y
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    References listed on IDEAS

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    1. Yonghong Yao & Mihai Postolache, 2012. "Iterative Methods for Pseudomonotone Variational Inequalities and Fixed-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 273-287, October.
    2. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
    3. Phan Tu Vuong & Jean Jacques Strodiot & Van Hien Nguyen, 2012. "Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 605-627, November.
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    Cited by:

    1. Yekini Shehu & Qiao-Li Dong & Lulu Liu & Jen-Chih Yao, 2023. "Alternated inertial subgradient extragradient method for equilibrium problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(1), pages 1-30, April.
    2. Yekini Shehu & Olaniyi S. Iyiola & Duong Viet Thong & Nguyen Thi Cam Van, 2021. "An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(2), pages 213-242, April.
    3. 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.
    4. Yonghong Yao & Naseer Shahzad & Jen-Chih Yao, 2020. "Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators," Mathematics, MDPI, vol. 8(4), pages 1-15, March.

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