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Generalized Polarity and Weakest Constraint Qualifications in Multiobjective Optimization

Author

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  • Oliver Stein

    (Karlsruhe Institute of Technology (KIT))

  • Maximilian Volk

    (Karlsruhe Institute of Technology (KIT))

Abstract

In Haeser and Ramos (J Optim Theory Appl, 187:469–487, 2020), a generalization of the normal cone from single objective to multiobjective optimization is introduced, along with a weakest constraint qualification such that any local weak Pareto optimal point is a weak Kuhn–Tucker point. We extend this approach to other generalizations of the normal cone and corresponding weakest constraint qualifications, such that local Pareto optimal points are weak Kuhn–Tucker points, local proper Pareto optimal points are weak and proper Kuhn–Tucker points, respectively, and strict local Pareto optimal points of order one are weak, proper and strong Kuhn–Tucker points, respectively. The constructions are based on an appropriate generalization of polarity to pairs of matrices and vectors.

Suggested Citation

  • Oliver Stein & Maximilian Volk, 2023. "Generalized Polarity and Weakest Constraint Qualifications in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1156-1190, September.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:3:d:10.1007_s10957-023-02256-7
    DOI: 10.1007/s10957-023-02256-7
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    References listed on IDEAS

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    1. White, D. J., 1983. "Concepts of proper efficiency," European Journal of Operational Research, Elsevier, vol. 13(2), pages 180-188, June.
    2. Markus Hartikainen & Kaisa Miettinen & Margaret Wiecek, 2012. "PAINT: Pareto front interpolation for nonlinear multiobjective optimization," Computational Optimization and Applications, Springer, vol. 52(3), pages 845-867, July.
    3. Gabriel Haeser & Alberto Ramos, 2020. "Constraint Qualifications for Karush–Kuhn–Tucker Conditions in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 469-487, November.
    4. Mirjam Dür & Bolor Jargalsaikhan & Georg Still, 2015. "First order solutions in conic programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(2), pages 123-142, October.
    5. Allen Klinger, 1967. "Letter to the Editor—Improper Solutions of the Vector Maximum Problem," Operations Research, INFORMS, vol. 15(3), pages 570-572, June.
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