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Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application

Author

Listed:
  • Yangdong Xu

    (Chongqing University of Posts and Telecommunications)

  • Pingping Zhang

    (Chongqing University of Posts and Telecommunications)

Abstract

In this paper, the connectedness of solution set of a strong vector equilibrium problem in a finite dimensional space, is investigated. Firstly, a nonconvex separation theorem is given, that is, a neither open nor closed set and a compact subset in a finite dimensional space can be strictly separated by a sublinear and strongly monotone function. Secondly, in terms of the nonconvex separation theorem, the union relation between the solution set of the strong vector equilibrium problem and the solution sets of a series of nonlinear scalar problems, is established. Under suitable assumptions, the connectedness and the path connectedness of the solution set of the strong vector equilibrium problem are obtained. In particular, we solve partly the question related to the path connectedness of the solution set of the strong vector equilibrium problem. The question is proposed by Han and Huang in the reference (J Optim Theory Appl, 2016. https://doi.org/10.1007/s10957-016-1032-9 ). Finally, as an application, we apply the main results to derive the connectedness of the solution set of a linear vector optimization problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:178:y:2018:i:1:d:10.1007_s10957-018-1244-2
    DOI: 10.1007/s10957-018-1244-2
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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. 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.
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    8. S. Li & K. Teo & X. Yang, 2005. "Generalized vector quasi-equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 61(3), pages 385-397, July.
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    Cited by:

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    2. 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.
    3. 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.

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