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Treatment of Set-Valued Robustness via Separation and Scalarization

Author

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  • Madhusudan Das

    (Indian Institute of Technology)

  • Chandal Nahak

    (Indian Institute of Technology)

  • Mahendra Prasad Biswal

    (Indian Institute of Technology)

Abstract

This paper aims to present alternative characterizations for different types of set-valued robustness concepts. Equivalent scalar representations for various set order relations are derived when the sets are the union of sets. Utilizing these findings in conjunction with image space analysis, specific isolated sets are defined for different notions of robust solutions. These isolated sets serve as the basis for deriving both necessary and sufficient robust optimality conditions. The validity of the results is demonstrated through several illustrative examples. Additionally, the paper concludes with an application of our present approach to two-player zero-sum matrix games.

Suggested Citation

  • Madhusudan Das & Chandal Nahak & Mahendra Prasad Biswal, 2024. "Treatment of Set-Valued Robustness via Separation and Scalarization," Journal of Optimization Theory and Applications, Springer, vol. 201(2), pages 843-865, May.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:2:d:10.1007_s10957-024-02423-4
    DOI: 10.1007/s10957-024-02423-4
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    References listed on IDEAS

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    1. Jiawei Chen & Elisabeth Köbis & Markus Köbis & Jen-Chih Yao, 2018. "Image Space Analysis for Constrained Inverse Vector Variational Inequalities via Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 816-834, June.
    2. Jiawei Chen & Shengjie Li & Zhongping Wan & Jen-Chih Yao, 2015. "Vector Variational-Like Inequalities with Constraints: Separation and Alternative," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 460-479, August.
    3. Jonas Ide & Elisabeth Köbis, 2014. "Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(1), pages 99-127, August.
    4. Andreas H. Hamel & Andreas Löhne, 2018. "A set optimization approach to zero-sum matrix games with multi-dimensional payoffs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 369-397, December.
    5. S. J. Li & Y. D. Xu & S. K. Zhu, 2012. "Nonlinear Separation Approach to Constrained Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 842-856, September.
    6. Kuntal Som & V. Vetrivel, 2021. "On robustness for set-valued optimization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 905-925, April.
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