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Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators

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  • Yonghong Yao

    (School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
    The Key Laboratory of Intelligent Information and Big Data Processing of NingXia Province, North Minzu University, Yinchuan 750021, China)

  • Naseer Shahzad

    (Department of Mathematics, King Abdulaziz University, P. O. B. 80203, Jeddah 21589, Saudi Arabia)

  • Jen-Chih Yao

    (Center for General Education, China Medical University, Taichung 40402, Taiwan)

Abstract

The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding a common element of the fixed point of pseudocontractive operators and the pseudomonotone equilibrium problem in Hilbert spaces. The suggested iterative algorithm is based on the projected method and subgradient method with a linearsearch technique. We show the strong convergence result for the iterative sequence generated by this algorithm. Some applications are also included. Our result improves and extends some existing results in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:461-:d:336956
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    References listed on IDEAS

    as
    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. Dang Hieu & Pham Ky Anh & Le Dung Muu, 2019. "Modified extragradient-like algorithms with new stepsizes for variational inequalities," Computational Optimization and Applications, Springer, vol. 73(3), pages 913-932, July.
    3. A. Tada & W. Takahashi, 2007. "Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 359-370, June.
    4. Jun Yang & Hongwei Liu, 2018. "A Modified Projected Gradient Method for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 197-211, October.
    5. 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.
    6. J. Bello Cruz & A. Iusem, 2010. "Convergence of direct methods for paramonotone variational inequalities," Computational Optimization and Applications, Springer, vol. 46(2), pages 247-263, June.
    Full references (including those not matched with items on IDEAS)

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