Journal of Global Optimization

, Volume 58, Issue 4, pp 673–692 | Cite as

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

  • S. K. Zhu
  • S. J. Li
  • K. L. Teo


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.


Set-valued optimization Second-order composed contingent derivative Lagrange multiplier rule Karush–Kuhn–Tucker condition Regularity assumption Optimality conditions 

Mathematics Subject Classification (2010)

49J53 49K30 90C29 90C46 



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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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingChina
  2. 2.Mathematical Sciences Research Institute in ChongqingChongqing UniversityChongqingChina
  3. 3.Department of Mathematics and StatisticsCurtin University of TechnologyPerthAustralia

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