Gap Function for Set-Valued Vector Variational-Like Inequalities



Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational-like inequalities (VVLI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVLI. We investigate the existence of a solution for the generalized VVLI with a set-valued mapping by exploiting the existence of a solution of the VVLI with a single-valued function and a continuous selection theorem.


Vector variational-like inequalities Set-valued mappings Gap functions Existence of a solution Semidefinite programming 


  1. 1.
    Yang, X.Q., Yao, J.C.: Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115, 407–417 (2002) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Lin, K.L., Yang, D.P., Yao, J.C.: Generalized vector variational inequalities. J. Optim. Theory Appl. 92, 117–125 (1997) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Daniilidis, A., Hadjisavvas, N.: Existence theorems for vector variational inequalities. Bull. Aust. Math. Soc. 54, 473–481 (1996) MATHMathSciNetGoogle Scholar
  4. 4.
    Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problems. J. Math. Anal. Appl. 206, 42–58 (1997) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, G.-Y.: Existence of solutions for a vector variational inequality: an extension of Hartmann-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen, G.-Y., Cheng, G.M.: Vector variational inequalities and vector optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 285, pp. 408–416. Springer, Berlin (1987) Google Scholar
  7. 7.
    Parida, I., Sahoo, M., Kumar, A.: A variational-like inequality problem. Bull. Aust. Math. Soc. 39, 225–231 (1989) MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Goh, C.J., Yang, X.Q.: Vector equilibrium problems and vector optimization. Eur. J. Oper. Res. 116, 615–628 (1999) MATHCrossRefGoogle Scholar
  9. 9.
    Ansari, Q.H., Yao, J.C.: On nondifferentiable and nonconvex vector optimization problems. J. Optim. Theory Appl. 106, 475–488 (2000) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen, G.-Y., Goh, C.J., Yang, X.Q.: On a gap function for vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 55–72. Kluwer Academic, Dordrecht (2000) Google Scholar
  11. 11.
    Ansari, Q.H., Yao, J.C.: Generalized variational-like inequalities and a gap function. Bull. Aust. Math. Soc. 59, 33–44 (1999) MATHMathSciNetGoogle Scholar
  12. 12.
    Ding, X.P., Tarafdar, E.: Generalized vector variational-like inequalities with C x-η-pseudo-monotone set-valued mapping. In: Giannessi, F. (ed.) Vector variational inequalities and vector equilibria, pp. 125–140. Kluwer Academic, Dordrecht (2000) Google Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, College of Basic Sciences and HumanitiesG.B. Pant University of Agriculture and TechnologyPantnagarIndia
  2. 2.City University of Hong KongDepartment of Management SciencesKowloonHong Kong
  3. 3.Academy of Mathematics and Systems SciencesChinese Academy of SciencesBeijingChina

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