Journal of Optimization Theory and Applications

, Volume 139, Issue 2, pp 243–261 | Cite as

Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization

  • P. Q. Khanh
  • N. D. Tuan


Higher-order variational sets are proposed for set-valued mappings, which are shown to be more convenient than generalized derivatives in approximating mappings at a considered point. Both higher-order necessary and sufficient conditions for local Henig-proper efficiency, local strong Henig-proper efficiency and local λ-proper efficiency in set-valued nonsmooth vector optimization are established using these sets. The technique is simple and the results help to unify first and higher-order conditions. As consequences, recent existing results are derived. Examples are provided to show some advantages of our notions and results.


Higher-order variational sets Higher-order optimality conditions Set-valued nonsmooth vector optimization Local Henig-proper efficiency Local strong Henig-proper efficiency Local λ-proper efficiency 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsInternational University of Hochiminh CityThu Duc, Hochiminh CityVietnam
  2. 2.Department of MathematicsUniversity of Natural Sciences of Hochiminh CityHochiminh CityVietnam

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