Second-Order Strong Karush/Kuhn–Tucker Conditions for Proper Efficiencies in Multiobjective Optimization

  • Min Feng
  • Shengjie LiEmail author


In this paper, strong Karush/Kuhn–Tucker conditions are studied for smooth multiobjective optimization with inequality constraints. We introduce a new second-order regularity condition of Abadie type in terms of the second-order directional derivatives and then obtain a second-order strong Karush/Kuhn–Tucker necessary condition at a Borwein-properly efficient solution. Simultaneously, we also use an example to show that, if the Abadie type regularity condition is weakened to the Guignard type one, the second-order strong Karush/Kuhn–Tucker necessary condition may not hold. Finally, then we also apply the second-order strong Karush/Kuhn–Tucker conditions to derive a sufficient result for local Geoffrion-proper efficiency.


Multiobjective optimization Strong Karush/Kuhn–Tucker conditions Second-order regularity conditions Second-order optimality conditions Borwein-properly efficient solutions Geoffrion-proper efficiency 

Mathematics Subject Classification

26A24 49K99 90C29 



We would like to thank Prof. Fabián Flores-Bazán and two anonymous reviewers whose comments led to an improvement of the paper. The research was supported by the National Natural Science Foundation of China (Grant Numbers: 11571055, 11601437) and the Fundamental Research Funds for the Central Universities (Grant Number: 106112017CDJZRPY0020).


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Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingChina

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