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On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality

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  • Takashi Maeda

    (Kanazawa University)

Abstract

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets of n-dimensional Euclidean space. Second, by using these quasi orderings, we define the concepts of lower semi-continuity for set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and we give some conditions under which these optimal solutions exist to the problems and give necessary and sufficient conditions for optimality.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:153:y:2012:i:2:d:10.1007_s10957-011-9952-x
    DOI: 10.1007/s10957-011-9952-x
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    References listed on IDEAS

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    1. M. Chinaie & J. Zafarani, 2009. "Image Space Analysis and Scalarization of Multivalued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 142(3), pages 451-467, September.
    2. P. Khanh & D. Quy, 2011. "On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings," Journal of Global Optimization, Springer, vol. 49(3), pages 381-396, March.
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    Cited by:

    1. Masamichi Kon, 2020. "A scalarization method for fuzzy set optimization problems," Fuzzy Optimization and Decision Making, Springer, vol. 19(2), pages 135-152, June.
    2. 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.
    3. 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.
    4. Gabriele Eichfelder & Maria Pilecka, 2016. "Set Approach for Set Optimization with Variable Ordering Structures Part I: Set Relations and Relationship to Vector Approach," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 931-946, December.

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