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An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces

Author

Listed:
  • Elisabeth Köbis

    (Institute of Mathematics, Faculty of Natural Sciences II, Martin-Luther-University Halle-Wittenberg, 06120 Halle, Germany
    These authors contributed equally to this work.)

  • Markus A. Köbis

    (Department of Mathematics and Computer Science, Institute of Mathematics, Free University Berlin, 14195 Berlin, Germany
    These authors contributed equally to this work.)

  • Xiaolong Qin

    (General Education Center, National Yunlin University of Science and Technology, Douliou 64002, Taiwan
    These authors contributed equally to this work.)

Abstract

This paper explores new notions of approximate minimality in set optimization using a set approach. We propose characterizations of several approximate minimal elements of families of sets in real linear spaces by means of general functionals, which can be unified in an inequality approach. As particular cases, we investigate the use of the prominent Tammer–Weidner nonlinear scalarizing functionals, without assuming any topology, in our context. We also derive numerical methods to obtain approximate minimal elements of families of finitely many sets by means of our obtained results.

Suggested Citation

  • Elisabeth Köbis & Markus A. Köbis & Xiaolong Qin, 2020. "An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces," Mathematics, MDPI, vol. 8(1), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:143-:d:311038
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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. Miguel Adán & Vicente Novo, 2005. "Duality and saddle-points for convex-like vector optimization problems on real linear spaces," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 343-357, December.
    3. Z. A. Zhou & J. W. Peng, 2012. "Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 830-841, September.
    4. Zhi-Ang Zhou & Xin-Min Yang, 2014. "Scalarization of $$\epsilon $$ ϵ -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 680-693, August.
    5. M. Adán & V. Novo, 2004. "Proper Efficiency in Vector Optimization on Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 515-540, June.
    6. 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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