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Well-Posedness of Set Optimization Problems with Set Order Defined by Minkowski Difference

Author

Listed:
  • Zhiang Zhou

    (Chongqing University of Technology)

  • Kehao Feng

    (Chongqing University of Technology)

  • Qamrul Hasan Ansari

    (Chongqing University of Technology
    King Fahd University of Petroleum and Minerals)

Abstract

In this paper, we consider set optimization problems whose solution concepts are defined by means of set less order relations involving Minkwoski difference. We propose well-posedness and generalized well-posedness for set optimization problems, and derive sufficient and necessary conditions for these well-posedness. A relation between well-posedness and generalized well-posedness for set optimization problems is also given. We use a nonlinear scalarization method to establish necessary and sufficient conditions for the well-posedness and generalized well-posedness of set optimization problems. Several examples are given to illustrate our results.

Suggested Citation

  • Zhiang Zhou & Kehao Feng & Qamrul Hasan Ansari, 2025. "Well-Posedness of Set Optimization Problems with Set Order Defined by Minkowski Difference," Journal of Optimization Theory and Applications, Springer, vol. 204(2), pages 1-24, February.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:2:d:10.1007_s10957-025-02608-5
    DOI: 10.1007/s10957-025-02608-5
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    References listed on IDEAS

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    1. X. J. Long & J. W. Peng, 2013. "Generalized B-Well-Posedness for Set Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 612-623, June.
    2. Meenakshi Gupta & Manjari Srivastava, 2020. "Approximate Solutions and Levitin–Polyak Well-Posedness for Set Optimization Using Weak Efficiency," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 191-208, July.
    3. 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.
    4. X. X. Huang, 2001. "Extended and strongly extended well-posedness of set-valued optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 53(1), pages 101-116, April.
    5. Kuntal Som & V. Vetrivel, 2023. "Global well-posedness of set-valued optimization with application to uncertain problems," Journal of Global Optimization, Springer, vol. 85(2), pages 511-539, February.
    6. Meenakshi Gupta & Manjari Srivastava, 2019. "Well-posedness and scalarization in set optimization involving ordering cones with possibly empty interior," Journal of Global Optimization, Springer, vol. 73(2), pages 447-463, February.
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