Advertisement

Journal of Optimization Theory and Applications

, Volume 130, Issue 2, pp 185–207 | Cite as

Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization

  • F. Flores-Bazán
  • C. Vera
Article

Abstract

As a consequence of an abstract theorem proved elsewhere, a vector Weierstrass theorem for the existence of a weakly efficient solution without any convexity assumption is established. By using the notion (recently introduced in an earlier paper) of semistrict quasiconvexity for vector functions and assuming additional structure on the space, new existence results encompassing many results appearing in the literature are derived. Also, when the cone defining the preference relation satisfies some mild assumptions (but including the polyhedral and icecream cones), various characterizations for the nonemptiness and compactness of the weakly efficient solution set to convex vector optimization problems are given. Similar results for a class of nonconvex problems on the real line are established as well.

Keywords

Nonconvex vector optimization quasiconvex vector functions weakly efficient solutions efficient solutions asymptotic functions asymptotic cones 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    ANSARI, Q.H., Vector Equilibrium Problems and Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 1–16, 2000.Google Scholar
  2. 2.
    BIANCHI, M., HADJISAVVAS, N., and SCHAIBLE, S., Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 527–542, 1997.CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    CHEN, G.Y., and CRAVEN, B.D., Existence and Continuity of Solutions for Vector Optimization, Journal of Optimization Theory and Applications, Vol. 81, pp. 459–468, 1994.CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    DENG, S., Characterizations of the Nonemptiness and Compactness of Solutions Sets in Convex Vector Optimization, Journal of Optimization Theory and Applications, Vol. 96, pp. 123–131, 1998.CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    FLORES-BAZÁN, F., Ideal, Weakly Efficient Solutions for Vector Optimization Problems, Mathematical Programming, Vol. 93 A, pp. 453–475, 2002.CrossRefGoogle Scholar
  6. 6.
    FLORES-BAZÁN, F., Semistrictly Quasiconvex Mappings and Nonconvex Vector Optimization, Mathematical Methods of Operations Research, Vol. 59, pp. 129–145, 2004.CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    HADJISAVVAS, N., and SCHAIBLE, S., From Scalar to Vector Equilibrium Problems in the Quasimonotone Case, Journal of Optimization Theory and Applications, Vol. 96, pp. 297–309, 1998.CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    FLORES-BAZÁN, F., Radial Epiderivatives and Asymptotic Functions in Nonconvex Vector Optimization, SIAM Journal on Optimization, Vol. 14, pp. 284–305, 2003.CrossRefMATHGoogle Scholar
  9. 9.
    FERRO, F., Minimax Type Theorems for n-Valued Functions, Annali di Matematica Pura ed Applicata, Vol. 32, pp. 113–130, 1982.CrossRefMathSciNetGoogle Scholar
  10. 10.
    LUC, D.T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, New York, NY, Vol. 319, 1989.Google Scholar
  11. 11.
    LUC, D.T., An Existence Theorem in Vector Optimization, Mathematics of Operations Research, Vol. 14, pp. 693–699, 1989.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    PENOT, J.P., and THÉRA, M., Polarité des Applications Convexes á Valeurs Vectorielles, Comptes Rendus de l’Académie des Sciences, Series 1, Mathematics, Vol. 288, pp. 419–422, 1979.MATHGoogle Scholar
  13. 13.
    ROCKAFELLAR, R.T., and WETS, J.B., Variational Analysis, Springer, Berlin, Germany, 1998.MATHGoogle Scholar
  14. 14.
    JAHN, J., and SACHS, E., Generalized Quasiconvex Mappings and Vector Optimization, SIAM Journal on Control and Optimization, Vol. 24, pp. 306–322, 1986.CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    CAMBINI, R., Generalized Concavity for Bicriteria Functions, Generalized Convexity, Generalized Monotonicity: Recent Results, Edited by J.P. Crouzeix et al., Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, Holland, Vol. 27, pp. 439–451, 1998.Google Scholar
  16. 16.
    LUC, D.T., and PENOT, J.P., Convergence of Asymptotic Directions, Transactions of the American Mathematical Society, Vol. 353, pp. 4095–4121, 2001.CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    AUBIN, J.P., Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, Holland, 1979.MATHGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • F. Flores-Bazán
    • 1
  • C. Vera
    • 1
  1. 1.Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y MatemáticasUniversidad de ConcepciónConcepciónChile

Personalised recommendations