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New Higher-Order Strong Karush–Kuhn–Tucker Conditions for Proper Solutions in Nonsmooth Optimization

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Abstract

This paper considers higher-order necessary conditions for Henig-proper, positively proper and Benson-proper solutions. Under suitable constraint qualifications, our conditions are of the Karush–Kuhn–Tucker rule. The conditions include higher-order complementarity slackness for both the objective and the constraining maps. They are in a nonclassical form with a supremum expression on the right-hand side (instead of zero). Our results are new and improve the existing ones in the literature, even when applied to special cases of multiobjective single-valued optimization problems.

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Acknowledgements

This research was supported by Vietnam National University, Ho Chi Minh City by the Grant Number B2018-28-02. A part of this work was done during research stays of the author at Vietnam Institute for Advanced Study in Mathematics (VIASM). The author would like to thank VIASM for its hospitality and support. I am grateful to the Editors and the referee for their useful comments, remarks and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Computing, University of Science, Ho Chi Minh City, Vietnam

    Nguyen Minh Tung

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Nguyen Minh Tung

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  1. Nguyen Minh Tung
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Correspondence to Nguyen Minh Tung.

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Communicated by Alexandre Cabot.

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Tung, N.M. New Higher-Order Strong Karush–Kuhn–Tucker Conditions for Proper Solutions in Nonsmooth Optimization. J Optim Theory Appl 185, 448–475 (2020). https://doi.org/10.1007/s10957-020-01654-5

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