Skip to main content
Log in

Vector optimization problems with nonconvex preferences

  • Original Paper
  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, some vector optimization problems are considered where pseudo-ordering relations are determined by nonconvex cones in Banach spaces. We give some characterizations of solution sets for vector complementarity problems and vector variational inequalities. When the nonconvex cone is the union of some convex cones, it is shown that the solution set of these problems is either an intersection or an union of the solution sets of all subproblems corresponding to each of these convex cones depending on whether these problems are defined by the nonconvex cone itself or its complement. Moreover, some relations of vector complementarity problems, vector variational inequalities, and minimal element problems are also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ansari Q.H., Lai T.C. and Yao J.C. (1999). On the equivalence of extended generalized complementarity and generalized least-element problems. J. Optim. Theory Appl. 102: 277–288

    Article  Google Scholar 

  2. Chen G.Y., Huang X.X. and Yang X.Q (2005). Vector Optimization: Set-valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems 541. Springer-Verlag, Berlin

    Google Scholar 

  3. Chen G.Y. and Yang X.Q. (1990). The vector complementarity problem and its equivalences with the weak minimal element in ordered Banach spaces. J. Math. Anal. Appl. 153: 136–158

    Article  Google Scholar 

  4. Cryer C.W. and Dempster A.H. (1980). Equivalence of linear complementarity problems and linear program in vector lattice Hilbert space. SIAM J. Control. Optim. 18: 76–90

    Article  Google Scholar 

  5. Giannessi F. (1980). Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F. and Lions, J.L. (eds) Variational Inequalities and Complementarity Problems, pp 151–186. Wiley, New York

    Google Scholar 

  6. (2000). Vector Variational Inequalities and Vector Equilibrium. Kluwer Academic Publishers, Dordrecht, Boston, London

    Google Scholar 

  7. Göpfert A., Riahi H., Tammer C. and Zâlinescu C. (2003). Variational Methods in Partially Ordered Spaces. Springer-Verlag, New York

    Google Scholar 

  8. Huang N.J. and Fang Y.P. (2005). Strong vector F-complementary problem and least element problem of feasible set. Nonlinear Anal. 61: 901–918

    Article  Google Scholar 

  9. Huang, N.J., Yang, X.Q., Chan, W.K. Vector complementarity problems with a variable ordering relation. Eur. J. Oper. Res. (in press)

  10. Isac G., Bulavsky V.A. and Kalashnikov V.V. (2002). Complementarity, Equilibrium, Efficiency and Economics. Kluwer Academic Publishers, Dordrecht, Boston, London

    Google Scholar 

  11. Riddell R.C. (1981). Equivalence of nonlinear complementarity problems and least element problems in Banach lattice. Math. Oper. Res. 6: 462–474

    Article  Google Scholar 

  12. Rubinov A.M. (2000). Abstract Convexity and Global Optimization. Kluwer Academic Publishers, Dordrecht, Boston, London

    Google Scholar 

  13. Rubinov A.M. and Gasimov R.N. (2004). Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation. J. Glob. Optim. 29: 455–477

    Article  Google Scholar 

  14. Yang X.Q. (1993). Vector complementarity and minimal element problems. J. Optim. Theory Appl. 77: 483–495

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to X. Q. Yang.

Additional information

While this paper was being revised in September 2006, Professor Alex Rubinov (the second author of the paper) left us due to the illness. This is a very sad news to us. We dedicate this paper to the memory of Professor Rubinov as a mathematician and truly friend.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, N.J., Rubinov, A.M. & Yang, X.Q. Vector optimization problems with nonconvex preferences. J Glob Optim 40, 765–777 (2008). https://doi.org/10.1007/s10898-006-9113-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-006-9113-1

Keywords

2000 Mathematics subject classsification

Navigation