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Connectedness and Path Connectedness of Weak Efficient Solution Sets of Vector Optimization Problems via Nonlinear Scalarization Methods

Author

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  • Xin Xu

    (College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China)

  • Yang Dong Xu

    (College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China)

Abstract

The connectedness and path connectedness of the solution sets to vector optimization problems is an important and interesting study in optimization theories and applications. Most papers involving the direction established the connectedness and connectedness for the solution sets of vector optimization problems or vector equilibrium problems by means of the linear scalarization method rather than the nonlinear scalarization method. The aim of the paper is to deal with the connectedness and the path connectedness for the weak efficient solution set to a vector optimization problem by using the nonlinear scalarization method. Firstly, the union relationship between the weak efficient solution set to the vector optimization problem and the solution sets to a series of parametric scalar minimization problems, is established. Then, some properties of the solution sets of scalar minimization problems are investigated. Finally, by using the union relationship, the connectedness and the path connectedness for the weak efficient solution set of the vector optimization problem are obtained.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:947-:d:275540
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    References listed on IDEAS

    as
    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. 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.
    3. Fioravante Patrone & Lucia Pusillo & Stef Tijs, 2007. "Multicriteria games and potentials," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(1), pages 138-145, July.
    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. 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.
    6. Patrone, F. & Pusillo, L. & Tijs, S.H., 2007. "Multicriteria games and potentials," Other publications TiSEM 47a4248e-bbe4-4037-99f1-f, Tilburg University, School of Economics and Management.
    7. Qiusheng Qiu & Xinmin Yang, 2010. "Some properties of approximate solutions for vector optimization problem with set-valued functions," Journal of Global Optimization, Springer, vol. 47(1), pages 1-12, May.
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