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Scalarization Functionals with Uniform Level Sets in Set Optimization

Author

Listed:
  • Truong Quang Bao

    (Northern Michigan University)

  • Christiane Tammer

    (Martin-Luther-University Halle-Wittenberg)

Abstract

We use the original form of Gerstewitz’s nonlinear scalarization functional to characterize upper and lower set-less minimizers of set-valued maps acting from a nonempty set into a real linear space with respect to the lower (resp. upper) set-less relation introduced by Kuroiwa. Our main results are as follows: An upper set-less minimizer to a set-valued map (with respect to the image space) is an upper set-less minimal solution to a scalarization of the set-valued map (with respect to the space of real numbers), where the hypergraphical multifunction is involved in the scalarization and vice versa, a lower set-less minimizer to a set-valued map (with respect to the image space) is an upper set-less minimal solution to an appropriate scalarization of the set-valued map (in the space of real numbers), where the epigraphical multifunction is involved in the scalarization and vice versa, and a lower set-less minimizer to a set-valued map becomes a (Pareto) minimizer to the same map provided that the map enjoys a domination property.

Suggested Citation

  • Truong Quang Bao & Christiane Tammer, 2019. "Scalarization Functionals with Uniform Level Sets in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 310-335, July.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:1:d:10.1007_s10957-019-01504-z
    DOI: 10.1007/s10957-019-01504-z
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    References listed on IDEAS

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    1. Petra Weidner, 2017. "Gerstewitz Functionals on Linear Spaces and Functionals with Uniform Sublevel Sets," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 812-827, June.
    2. Takashi Maeda, 2012. "On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 263-279, May.
    3. Klamroth, Kathrin & Köbis, Elisabeth & Schöbel, Anita & Tammer, Christiane, 2017. "A unified approach to uncertain optimization," European Journal of Operational Research, Elsevier, vol. 260(2), pages 403-420.
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    Cited by:

    1. Eichfelder, Gabriele & Quintana, Ernest, 2024. "Set-based robust optimization of uncertain multiobjective problems via epigraphical reformulations," European Journal of Operational Research, Elsevier, vol. 313(3), pages 871-882.
    2. 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.

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