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Set Approach for Set Optimization with Variable Ordering Structures Part I: Set Relations and Relationship to Vector Approach

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  • Gabriele Eichfelder

    (Technische Universität Ilmenau)

  • Maria Pilecka

    (Technische Universität Bergakademie Freiberg)

Abstract

This paper aims at combining variable ordering structures with set relations in set optimization, which have been defined using the constant ordering cone before. We provide several new set relations in the context of variable ordering structures, discuss their usefulness, and give different examples from a practical point of view. After analyzing the properties of the introduced relations, we define solution notions for set-valued optimization problems equipped with variable ordering structures. We also relate these new notions to those ones obtained by the so-called vector approach.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:3:d:10.1007_s10957-016-0992-0
    DOI: 10.1007/s10957-016-0992-0
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    References listed on IDEAS

    as
    1. Johannes Jahn, 2015. "Vectorization in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 783-795, December.
    2. J. Y. Bello Cruz & G. Bouza Allende, 2014. "A Steepest Descent-Like Method for Variable Order Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 371-391, August.
    3. Luc, Dinh The & Soubeyran, Antoine, 2013. "Variable preference relations: Existence of maximal elements," Journal of Mathematical Economics, Elsevier, vol. 49(4), pages 251-262.
    4. 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.
    5. Truong Q. Bao & Boris S. Mordukhovich, 2014. "Necessary Nondomination Conditions in Set and Vector Optimization with Variable Ordering Structures," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 350-370, August.
    6. Behnam Soleimani, 2014. "Characterization of Approximate Solutions of Vector Optimization Problems with a Variable Order Structure," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 605-632, August.
    7. Gabriele Eichfelder & Maria Pilecka, 2016. "Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 947-963, December.
    8. Marius Durea & Radu Strugariu & Christiane Tammer, 2015. "On set-valued optimization problems with variable ordering structure," Journal of Global Optimization, Springer, vol. 61(4), pages 745-767, April.
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    Cited by:

    1. Ovidiu Bagdasar & Nicolae Popovici, 2018. "Unifying local–global type properties in vector optimization," Journal of Global Optimization, Springer, vol. 72(2), pages 155-179, October.
    2. Gabriele Eichfelder & Maria Pilecka, 2016. "Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 947-963, December.
    3. 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.

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