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The Continuity and Convexity of a Nonlinear Scalarization Function with Applications in Set Optimization Problems Involving a Partial Order Relation

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  • Zi-Ru Zhang

    (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

In this paper, we deal with the properties and applications of a nonlinear scalarization function for sets by using the Minkowski difference. Under some suitable assumptions, the continuity and convexity concerned with the nonlinear scalarization function for sets are presented. As applications, the path connectedness of the solution sets to set optimization problems and the continuity of the solution mappings of parametric set optimization problems are established. The results achieved do not impose the monotonicity of the set-valued objective mapping, which are obviously different from the related ones in the literature.

Suggested Citation

  • Zi-Ru Zhang & Yang-Dong Xu, 2024. "The Continuity and Convexity of a Nonlinear Scalarization Function with Applications in Set Optimization Problems Involving a Partial Order Relation," Mathematics, MDPI, vol. 12(23), pages 1-22, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3839-:d:1536839
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    References listed on IDEAS

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    1. B. Jiménez & V. Novo & A. Vílchez, 2020. "Characterization of set relations through extensions of the oriented distance," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 89-115, February.
    2. 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.
    3. 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.
    4. Y. D. Xu & S. J. Li, 2016. "On the solution continuity of parametric set optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 223-237, August.
    5. 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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