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Linear conditioning, weak sharpness and finite convergence for equilibrium problems

Author

Listed:
  • Luong Nguyen

    (Hong Duc University)

  • Qamrul Hasan Ansari

    (Aligarh Muslim University
    King Fahd University of Petroleum and Minerals)

  • Xiaolong Qin

    (Hangzhou Normal University)

Abstract

The present paper first provides sufficient conditions and characterizations for linearly conditioned bifunction associated with an equilibrium problem. It then introduces the notion of weak sharp solution for equilibrium problems which is analogous to the linear conditioning notion. This new notion generalizes and unifies the notion of weak sharp minima for optimization problems as well as the notion of weak sharp solutions for variational inequality problems. Some sufficient conditions and characterizations of weak sharpness are also presented. Finally, we study the finite convergence property of sequences generated by some algorithms for solving equilibrium problems under linear conditioning and weak shapness assumptions. An upper bound of the number of iterations by which the sequence generated by proximal point algorithm converges to a solution of equilibrium problems is also given.

Suggested Citation

  • Luong Nguyen & Qamrul Hasan Ansari & Xiaolong Qin, 2020. "Linear conditioning, weak sharpness and finite convergence for equilibrium problems," Journal of Global Optimization, Springer, vol. 77(2), pages 405-424, June.
    • Handle: RePEc:spr:jglopt:v:77:y:2020:i:2:d:10.1007_s10898-019-00869-9
    DOI: 10.1007/s10898-019-00869-9
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    References listed on IDEAS

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    1. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, June.
    2. Shin-ya Matsushita & Li Xu, 2014. "On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 701-715, June.
    3. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
    4. Suliman Al-Homidan & Qamrul Hasan Ansari & Luong Nguyen, 2017. "Weak Sharp Solutions for Nonsmooth Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 683-701, December.
    5. 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.
    6. Wu, Zili, 2018. "Characterizations of weakly sharp solutions for a variational inequality with a pseudomonotone mapping," European Journal of Operational Research, Elsevier, vol. 265(2), pages 448-453.
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    Cited by:

    1. Bing Tan & Shanshan Xu & Songxiao Li, 2020. "Modified Inertial Hybrid and Shrinking Projection Algorithms for Solving Fixed Point Problems," Mathematics, MDPI, vol. 8(2), pages 1-12, February.

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