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Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization

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Abstract

In this paper, we propose the concept of a second-order composed contingent derivative for set-valued maps, discuss its relationship to the second-order contingent derivative and investigate some of its special properties. By virtue of the second-order composed contingent derivative, we extend the well-known Lagrange multiplier rule and the Kurcyusz–Robinson–Zowe regularity assumption to a constrained set-valued optimization problem in the second-order case. Simultaneously, we also establish some second-order Karush–Kuhn–Tucker necessary and sufficient optimality conditions for a set-valued optimization problem, whose feasible set is determined by a set-valued map, under a generalized second-order Kurcyusz–Robinson–Zowe regularity assumption.

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Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant numbers: 11171362 and 11201509) and the Fundamental Research Funds for the Central Universities (Grant number: CDJXS12100021). The authors are grateful to the two anonymous referees for their valuable comments and suggestions, which helped to improve the paper.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    S. K. Zhu & S. J. Li

  2. Mathematical Sciences Research Institute in Chongqing, Chongqing University, Chongqing, 401331, China

    S. J. Li

  3. Department of Mathematics and Statistics, Curtin University of Technology, G.P.O. Box U1987, Perth, WA, 6845, Australia

    K. L. Teo

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  1. S. K. Zhu
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  2. S. J. Li
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  3. K. L. Teo
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Correspondence to S. K. Zhu.

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Zhu, S.K., Li, S.J. & Teo, K.L. Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization. J Glob Optim 58, 673–692 (2014). https://doi.org/10.1007/s10898-013-0067-9

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  • DOI: https://doi.org/10.1007/s10898-013-0067-9

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