Journal of Optimization Theory and Applications

, Volume 136, Issue 1, pp 105–123 | Cite as

Duality Results for Generalized Vector Variational Inequalities with Set-Valued Maps

  • P. H. Sach
  • D. S. Kim
  • L. A. Tuan
  • G. M. LeeEmail author


In this paper, we introduce new dual problems of generalized vector variational inequality problems with set-valued maps and we discuss a link between the solution sets of the primal and dual problems. The notion of solutions in each of these problems is introduced via the concepts of efficiency, weak efficiency or Benson proper efficiency in vector optimization. We provide also examples showing that some earlier duality results for vector variational inequality may not be true.


Vector variational inequalities Set-valued maps Duality Conjugate maps Biconjugate maps 


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  1. 1.
    Harker, P.T., Pang, J.-S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. B 48, 101–120 (1990) MathSciNetGoogle Scholar
  2. 2.
    Giannessi, F.: Theorems of the alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980) Google Scholar
  3. 3.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Giannessi, F.: Vector Variational Inequalities and Vector Equilibria, Mathematical Theories. Kluwer Academic, Dordrecht (2000) zbMATHGoogle Scholar
  5. 5.
    Mosco, U.: Dual variational inequalities. J. Math. Anal. Appl. 40, 202–206 (1972) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dolcetta, I.C., Matzeu, M.: Duality for implicit variational problems and numerical applications. Numer. Funct. Anal. Optim. 2, 231–265 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ansari, Q.H., Yang, X.Q., Yao, J.C.: Existence and duality of implicit vector variational problems. Numer. Funct. Anal. Optim. 22, 815–829 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lee, G.M., Kim, D.S., Lee, B.S., Chen, G.Y.: Generalized vector variational inequality and its duality for set-valued maps. Appl. Math. Lett. 11, 21–26 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Yang, X.Q.: Vector variational inequality and its duality, nonlinear analysis. Theory, Methods Appl. 21, 869–877 (1993) zbMATHCrossRefGoogle Scholar
  10. 10.
    Goh, C.J., Yang, X.Q.: Duality in Optimization and Variational Inequalities. Taylor & Francis, London (2002) zbMATHGoogle Scholar
  11. 11.
    Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic, Orlando (1985) zbMATHGoogle Scholar
  13. 13.
    Guerraggio, A., Molho, E., Zaffaroni, A.: On the Notion of Proper Efficiency in Vector Optimization. J. Optim. Theory Appl. 82, 1–21 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dauer, J.P., Saleh, O.A.: A Characterization of proper minimal points as solution of sublinear optimization problems. J. Math. Anal. Appl. 178, 227–246 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Sach, P.H.: Nearly subconvexlike set-valued maps and vector optimization problems. J. Optim. Theory Appl. 119, 335–356 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • P. H. Sach
    • 1
  • D. S. Kim
    • 2
  • L. A. Tuan
    • 3
  • G. M. Lee
    • 2
    Email author
  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Department of Applied MathematicsPukyong National UniversityPusanRepublic of Korea
  3. 3.Ninh Thuan College of PedagogyNinh ThuanVietnam

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