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Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems

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  • X. H. Gong

    (Nanchang University)

Abstract

In this paper, we introduce the concepts of globally efficient solution and cone-Benson efficient solution for a vector equilibrium problem; we give some scalarization results for Henig efficient solution sets, globally efficient solution sets, weak efficient solution sets, and cone-Benson efficient solution sets in locally convex spaces. Using the scalarization results, we show the connectedness and path connectedness of weak efficient solution sets and various proper efficient solution sets of vector equilibrium problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:133:y:2007:i:2:d:10.1007_s10957-007-9196-y
    DOI: 10.1007/s10957-007-9196-y
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    1. Jun-Yi Fu, 2000. "Generalized vector quasi-equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(1), pages 57-64, September.
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    Cited by:

    1. 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.
    2. X. H. Gong & J. C. Yao, 2008. "Connectedness of the Set of Efficient Solutions for Generalized Systems," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 189-196, August.
    3. Qiuying Li & Sanhua Wang, 2021. "Arcwise Connectedness of the Solution Sets for Generalized Vector Equilibrium Problems," Mathematics, MDPI, vol. 9(20), pages 1-9, October.
    4. Yu Han & Nan-jing Huang, 2018. "Existence and Connectedness of Solutions for Generalized Vector Quasi-Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 65-85, October.
    5. N. T. T. Huong & N. D. Yen, 2014. "The Pascoletti–Serafini Scalarization Scheme and Linear Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 559-576, August.
    6. 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.
    7. Zaiyun Peng & Ziyuan Wang & Xinmin Yang, 2020. "Connectedness of Solution Sets for Weak Generalized Symmetric Ky Fan Inequality Problems via Addition-Invariant Sets," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 188-206, April.
    8. Adela Capătă, 2011. "Existence results for proper efficient solutions of vector equilibrium problems and applications," Journal of Global Optimization, Springer, vol. 51(4), pages 657-675, December.
    9. Wensheng Jia & Xiaoling Qiu & Dingtao Peng, 2020. "An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality," Mathematics, MDPI, vol. 8(1), pages 1-9, January.
    10. Yangdong Xu & Pingping Zhang, 2018. "Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 131-152, July.
    11. X. H. Gong, 2008. "Continuity of the Solution Set to Parametric Weak Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 35-46, October.
    12. S. J. Li & Z. M. Fang, 2010. "Lower Semicontinuity of the Solution Mappings to a Parametric Generalized Ky Fan Inequality," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 507-515, December.
    13. X. Gong, 2011. "Chebyshev scalarization of solutions to the vector equilibrium problems," Journal of Global Optimization, Springer, vol. 49(4), pages 607-622, April.
    14. Xin Xu & Yang Dong Xu, 2019. "Connectedness and Path Connectedness of Weak Efficient Solution Sets of Vector Optimization Problems via Nonlinear Scalarization Methods," Mathematics, MDPI, vol. 7(10), pages 1-10, October.

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