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Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems

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  • Phan Tu Vuong

    (Institute for Computational Science and Technology at Ho Chi Minh City (ICST HCMC))

  • Jean Jacques Strodiot

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

  • Van Hien Nguyen

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

Abstract

In this paper, we introduce some new iterative methods for finding a common element of the set of points satisfying a Ky Fan inequality, and the set of fixed points of a contraction mapping in a Hilbert space. The strong convergence of the iterates generated by each method is obtained thanks to a hybrid projection method, under the assumptions that the fixed-point mapping is a ξ-strict pseudocontraction, and the function associated with the Ky Fan inequality is pseudomonotone and weakly continuous. A Lipschitz-type condition is assumed to hold on this function when the basic iteration comes from the extragradient method. This assumption is unnecessary when an Armijo backtracking linesearch is incorporated in the extragradient method. The particular case of variational inequality problems is examined in a last section.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:2:d:10.1007_s10957-012-0085-7
    DOI: 10.1007/s10957-012-0085-7
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    References listed on IDEAS

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    1. 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.
    2. Xiaolong Qin & Sun Cho & Shin Kang, 2011. "An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings," Journal of Global Optimization, Springer, vol. 49(4), pages 679-693, April.
    3. 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.
    4. N. Nadezhkina & W. Takahashi, 2006. "Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 128(1), pages 191-201, January.
    5. Lu-Chuan Ceng & Nicolas Hadjisavvas & Ngai-Ching Wong, 2010. "Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems," Journal of Global Optimization, Springer, vol. 46(4), pages 635-646, April.
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    Cited by:

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    5. Dang Hieu, 2018. "An inertial-like proximal algorithm for equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 399-415, December.
    6. 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.
    7. 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.
    8. Javad Balooee & Shih-sen Chang & Jen-Chih Yao, 2023. "Generalized Set-valued Nonlinear Variational-like Inequalities and Fixed Point Problems: Existence and Approximation Solvability Results," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 891-938, June.

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