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Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems

Author

Listed:
  • L. P. Hai

    (University of Science, Vietnam National University-Ho Chi-Minh City)

  • L. Huerga

    (Universidad Nacional de Educación a Distancia)

  • P. Q. Khanh

    (International University, Vietnam National University-Ho Chi Minh City)

  • V. Novo

    (Universidad Nacional de Educación a Distancia)

Abstract

In this paper, we provide variants of the Ekeland variational principle for a type of approximate proper solutions of a vector equilibrium problem, whose final space is finite dimensional and partially ordered by a polyhedral cone. Depending on the choice of an approximation set that defines these solutions, we prove that they approximate suitably exact weak efficient/proper efficient/efficient solutions of the problem. The variants of the Ekeland variational principle are obtained for an unconstrained and also for a cone-constrained vector equilibrium problem, through a nonlinear scalarization, and expressed by means of the matrix that defines the ordering cone, which makes them easier to handle. At the end, the results are applied to multiobjective optimization problems, for which a related vector variational inequality problem is defined.

Suggested Citation

  • L. P. Hai & L. Huerga & P. Q. Khanh & V. Novo, 2019. "Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems," Journal of Global Optimization, Springer, vol. 74(2), pages 361-382, June.
    • Handle: RePEc:spr:jglopt:v:74:y:2019:i:2:d:10.1007_s10898-019-00772-3
    DOI: 10.1007/s10898-019-00772-3
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    References listed on IDEAS

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    1. X. H. Gong, 2007. "Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 151-161, May.
    2. X. Q. Yang & C. J. Goh, 1997. "On Vector Variational Inequalities: Application to Vector Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 95(2), pages 431-443, November.
    3. T. Q. Bao & S. Cobzaş & A. Soubeyran, 2018. "Variational principles, completeness and the existence of traps in behavioral sciences," Annals of Operations Research, Springer, vol. 269(1), pages 53-79, October.
    4. P. Khanh & D. Quy, 2011. "On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings," Journal of Global Optimization, Springer, vol. 49(3), pages 381-396, March.
    5. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    6. X. H. Gong, 2001. "Efficiency and Henig Efficiency for Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 108(1), pages 139-154, January.
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    Cited by:

    1. Majid Fakhar & Mohammadreza Khodakhah & Ali Mazyaki & Antoine Soubeyran & Jafar Zafarani, 2022. "Variational rationality, variational principles and the existence of traps in a changing environment," Journal of Global Optimization, Springer, vol. 82(1), pages 161-177, January.
    2. Nguyen Van Hung & Vicente Novo & Vo Minh Tam, 2022. "Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone," Journal of Global Optimization, Springer, vol. 82(1), pages 139-159, January.
    3. Le Phuoc Hai, 2021. "Ekeland variational principles involving set perturbations in vector equilibrium problems," Journal of Global Optimization, Springer, vol. 79(3), pages 733-756, March.
    4. Chuang-Liang Zhang & Nan-jing Huang, 2021. "Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 894-914, September.

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