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New Order Relations in Set Optimization

Author

Listed:
  • Johannes Jahn

    (Universität Erlangen-Nürnberg)

  • Truong Xuan Duc Ha

    (Institute of Mathematics)

Abstract

In this paper we study a set optimization problem (SOP), i.e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a convex cone K. We introduce new order relations on the power set $\mathcal{P}(Y)$ of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set $(\mathcal{Q},\preccurlyeq)$ without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order $\preccurlyeq$ : minimal elements, semicompactness, completeness, domination property of a subset of $\mathcal{Q}$ , and semicontinuity of a set-valued map with values in $\mathcal{Q}$ in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on $(\mathcal{Q},\preccurlyeq)$ from which one can easily derive similar results for the case, when F takes values on $\mathcal{P}(Y)$ equipped with various order relations.

Suggested Citation

  • Johannes Jahn & Truong Xuan Duc Ha, 2011. "New Order Relations in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 209-236, February.
    • Handle: RePEc:spr:joptap:v:148:y:2011:i:2:d:10.1007_s10957-010-9752-8
    DOI: 10.1007/s10957-010-9752-8
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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.
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    Cited by:

    1. Gabriele Eichfelder & Julia Niebling & Stefan Rocktäschel, 2020. "An algorithmic approach to multiobjective optimization with decision uncertainty," Journal of Global Optimization, Springer, vol. 77(1), pages 3-25, May.
    2. Gabriele Eichfelder & Corinna Krüger & Anita Schöbel, 2017. "Decision uncertainty in multiobjective optimization," Journal of Global Optimization, Springer, vol. 69(2), pages 485-510, October.
    3. Giovanni Paolo Crespi & Andreas H. Hamel & Matteo Rocca & Carola Schrage, 2021. "Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 361-381, February.
    4. Jiang, Ling & Cao, Jinde & Xiong, Lianglin, 2019. "Generalized multiobjective robustness and relations to set-valued optimization," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 599-608.
    5. Masamichi Kon, 2020. "A scalarization method for fuzzy set optimization problems," Fuzzy Optimization and Decision Making, Springer, vol. 19(2), pages 135-152, June.
    6. L. Huerga & B. Jiménez & V. Novo & A. Vílchez, 2021. "Six set scalarizations based on the oriented distance: continuity, convexity and application to convex set optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(2), pages 413-436, April.
    7. 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.
    8. Q. Q. Song & G. Q. Tang & L. S. Wang, 2013. "On Essential Stable Sets of Solutions in Set Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 591-599, March.
    9. Qamrul Hasan Ansari & Andreas H Hamel & Pradeep Kumar Sharma, 2020. "Ekeland’s variational principle with weighted set order relations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 117-136, February.

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