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Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 6, pp 1619–1650 | Cite as

Optimality Conditions for the Efficient Solutions of Vector Equilibrium Problems with Constraints in Terms of Directional Derivatives and Applications

  • Tran Van SuEmail author
  • Dinh Dieu Hang
Original Paper

Abstract

The aim of this paper is to establish the Kuhn–Tucker optimality conditions for the efficient solution types of constrained vector equilibrium problems in terms of directional derivatives in Banach spaces. Under the suitable assumptions on generalized convexity of objective and constraint functions, the Kuhn–Tucker necessary and sufficient optimality conditions for efficient solution, weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution of vector equilibrium problem with set and cone constraints are established. Some applications to the constrained vector variational inequality problem and the constrained vector optimization problem are also given. Besides, the Karush–Kuhn–Tucker necessary and sufficient optimality conditions for weakly efficient solutions to the model of transportation–production and Nash–Cournot equilibria problems are obtained. We also provide several examples to illustrate our results.

Keywords

Vector equilibrium problem with constraints Efficient solution Weakly efficient solution Henig efficient solution Globally efficient solution Superefficient solution Kuhn–Tucker optimality conditions Quasirelative interiors Directional derivatives Directional subdifferentials 

Mathematics Subject Classification

90C46 91B50 49J52 

Notes

Acknowledgements

The author thanks the referees for their valuable comments and suggestions which improve the paper. The first author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.301

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsQuang Nam UniversityTamkyVietnam
  2. 2.Graduate University of Science and Technology, VASTHanoiVietnam
  3. 3.Department of Basic SciencesThai Nguyen University of Information and Communication TechnologyThai NguyenVietnam

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