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Painlevé-Kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets

Author

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  • Zai-Yun Peng

    (Chongqing JiaoTong University)

  • Xue-Jing Chen

    (Chongqing JiaoTong University)

  • Yun-Bin Zhao

    (Chinese University of Hong Kong)

  • Xiao-Bing Li

    (Chongqing JiaoTong University)

Abstract

The aim of this paper is to explore the stability of (weak)-minimal solutions for set-valued optimization problems via improvement sets. Firstly, the optimality and closedness of solution sets for the set-valued optimization problem under the upper order relation are discussed. Then, a new convergence concept for set-valued mapping sequences is introduced, and some properties of the set-valued mapping sequences are shown under the new convergence assumption. Moreover, by means of upper level sets, Painlevé-Kuratowski convergences of (weak) E-u-solutions to set-valued optimization problems with respect to the perturbations of feasible sets and objective mappings are established under mild conditions. The order that we use to establish the result depends on the improvement set, which is not necessarily a cone order. Our results can be seen as the extension of the related work established recently in this field.

Suggested Citation

  • Zai-Yun Peng & Xue-Jing Chen & Yun-Bin Zhao & Xiao-Bing Li, 2023. "Painlevé-Kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets," Journal of Global Optimization, Springer, vol. 87(2), pages 759-781, November.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01166-8
    DOI: 10.1007/s10898-022-01166-8
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    References listed on IDEAS

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    1. 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.
    2. Zai-Yun Peng & Jian-Wen Peng & Xian-Jun Long & Jen-Chih Yao, 2018. "On the stability of solutions for semi-infinite vector optimization problems," Journal of Global Optimization, Springer, vol. 70(1), pages 55-69, January.
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    4. C. S. Lalitha & Prashanto Chatterjee, 2015. "Stability and Scalarization in Vector Optimization Using Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 825-843, September.
    5. Bonnisseau, Jean-Marc & Cornet, Bernard, 1990. "Existence of Marginal Cost Pricing Equilibria in Economies with Several Nonconvex Firms," Econometrica, Econometric Society, vol. 58(3), pages 661-682, May.
    6. Lam Quoc Anh & Tran Quoc Duy & Dinh Vinh Hien & Daishi Kuroiwa & Narin Petrot, 2020. "Convergence of Solutions to Set Optimization Problems with the Set Less Order Relation," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 416-432, May.
    7. Gutiérrez, C. & Jiménez, B. & Novo, V., 2012. "Improvement sets and vector optimization," European Journal of Operational Research, Elsevier, vol. 223(2), pages 304-311.
    8. C. S. Lalitha & Prashanto Chatterjee, 2012. "Stability and Scalarization of Weak Efficient, Efficient and Henig Proper Efficient Sets Using Generalized Quasiconvexities," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 941-961, December.
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