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Scalarization and Well-Posedness for Set Optimization Problems Involving General Set Less Relations

Author

Listed:
  • Yue Zeng

    (Sichuan University)

  • Zai-Yun Peng

    (Yunnan Normal University
    Chongqing Jiaotong University)

  • Christane Tammer

    (Martin-Luther-University Halle-Wittenberg)

  • Jen-Chih Yao

    (China Medical University)

  • Ke Deng

    (Sichuan University)

Abstract

The aim of this paper is to investigate the scalarization and well-posedness for set optimization problems where the solution concept is given by the lower set less relation induced by general sets. First, we propose three kinds of well-posedness for set optimization problems via general sets and study their relationships. Moreover, some sufficient conditions for these kinds of well-posedness for set optimization problems via free-disposal set are obtained under the Hausdorff P-continuity of the objective mapping of the set optimization problem, rather than the continuity in the sense of Berge. By employing a nonlinear scalarization technique by means of the oriented distance function, corresponding scalar optimization problems are established, and the relationship between solutions of the set optimization problem where the solution concept is given by the lower set less relation induced by co-radiant sets and solutions of the corresponding scalarized problem is also discussed. Furthermore, we also give some characterization of well-posedness for set optimization problem through two scalar optimization problems. Some examples are given to illustrate the main results.

Suggested Citation

  • Yue Zeng & Zai-Yun Peng & Christane Tammer & Jen-Chih Yao & Ke Deng, 2025. "Scalarization and Well-Posedness for Set Optimization Problems Involving General Set Less Relations," Journal of Optimization Theory and Applications, Springer, vol. 207(2), pages 1-23, November.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:2:d:10.1007_s10957-025-02784-4
    DOI: 10.1007/s10957-025-02784-4
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    References listed on IDEAS

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    1. G. P. Crespi & A. Guerraggio & M. Rocca, 2007. "Well Posedness in Vector Optimization Problems and Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 213-226, January.
    2. 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.
    3. 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.
    4. Giovanni Crespi & Ivan Ginchev & Matteo Rocca, 2006. "First-order optimality conditions in set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 87-106, February.
    5. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    6. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
    7. 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.
    8. 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.
    9. Khushboo & C. S. Lalitha, 2023. "Characterizations of set order relations and nonlinear scalarizations via generalized oriented distance function in set optimization," Journal of Global Optimization, Springer, vol. 85(1), pages 235-249, January.
    10. 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.
    11. E. Miglierina & E. Molho & M. Rocca, 2005. "Well-Posedness and Scalarization in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 126(2), pages 391-409, August.
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