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The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé–Kuratowski convergence

Author

Listed:
  • Yu Han

    (Jiangxi University of Finance and Economics)

  • Kai Zhang

    (Jiangxi University of Finance and Economics)

  • Nan-jing Huang

    (Sichuan University)

Abstract

In this paper, we obtain the Painlevé–Kuratowski upper convergence and the Painlevé–Kuratowski lower convergence of the approximate solution sets for set optimization problems with the continuity and convexity of objective mappings. Moreover, we discuss the extended well-posedness and the weak extended well-posedness for set optimization problems under some mild conditions. We also give some examples to illustrate our main results.

Suggested Citation

  • Yu Han & Kai Zhang & Nan-jing Huang, 2020. "The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé–Kuratowski convergence," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 175-196, February.
    • Handle: RePEc:spr:mathme:v:91:y:2020:i:1:d:10.1007_s00186-019-00695-5
    DOI: 10.1007/s00186-019-00695-5
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    References listed on IDEAS

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    1. G. P. Crespi & M. Papalia & M. Rocca, 2009. "Extended Well-Posedness of Quasiconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 285-297, May.
    2. César Gutiérrez & Enrico Miglierina & Elena Molho & Vicente Novo, 2016. "Convergence of Solutions of a Set Optimization Problem in the Image Space," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 358-371, August.
    3. Yu Han & Nan-jing Huang, 2018. "Continuity and Convexity of a Nonlinear Scalarizing Function in Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 679-695, June.
    4. Xian-Jun Long & Jian-Wen Peng & Zai-Yun Peng, 2015. "Scalarization and pointwise well-posedness for set optimization problems," Journal of Global Optimization, Springer, vol. 62(4), pages 763-773, August.
    5. Giovanni P. Crespi & Mansi Dhingra & C. S. Lalitha, 2018. "Pointwise and global well-posedness in set optimization: a direct approach," Annals of Operations Research, Springer, vol. 269(1), pages 149-166, October.
    6. 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.
    7. Giovanni P. Crespi & Daishi Kuroiwa & Matteo Rocca, 2017. "Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization," Annals of Operations Research, Springer, vol. 251(1), pages 89-104, April.
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