Advertisement

On Monotone and Strongly Monotone Vector Variational Inequalities

  • Nguyen Dong Yen
  • Gue Myung Lee
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 38)

Abstract

By constructing an example we show that the solution sets of a strongly monotone vector variational inequality and of its relaxed inequality can be different from each other. A sufficient condition for the coincidence of these solution sets is given for general vector variational inequalities; connectedness and path-connectedness of the solution sets for some kinds of monotone problems in Hilbert spaces are studied in detail.

Key Words

Vector variational inequality monotonicity strong monotonicity solution sets connectedness path-connectedness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Brézis H., “Analyse fonctionnelle”. Masson, Paris, 1983.Google Scholar
  2. [2]
    Chen G.-Y., “Existence of Solutions for a Vector Variational Inequality: An Extension of the Hartman-Stampacchia Theorem”. Jou. of Optimiz. Theory and Appls., Vol. 74, 1992, pp. 445–456.zbMATHCrossRefGoogle Scholar
  3. [3]
    Choo E. U. and Atkins D.R., “Connectedness in Multiple Linear Fractional Programming”. Manag. Science, Vol. 29, 1983, pp. 250–255.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Giannessi F., “Theorems of Alternative, Quadratic Programs and Complementarity Problems”. In “Variational Inequality and Complementarity Problems” (Edited by R. W. Cottle, F. Gian-nessi and J.-L. Lions ), Wiley, New York, 1980, pp. 151–186.Google Scholar
  5. [5]
    Kinderlehrer D. and G. Stampacchia, “An Introduction to Variational Inequalities and Their Appls.”. Academic Press, New York, 1980.Google Scholar
  6. [6]
    Lee G.M., Kim D.S., Lee B.S. and Yen N.D., “Vector Variational Inequalities as a Tool for Studying Vector Optimiz. Problems”. Nonlinear Analysis, Vol. 34, 1998, pp. 745–765.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Malivert C., “Multicriteria Fractional Programming”. ( Manuscript, September 1996 )Google Scholar
  8. [8]
    Rockafellar R.T., “Convex Analysis”. Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  9. [9]
    Steuer R.E., “Multiple Criteria Optimiz.: Theory, Computation and Application”. J. Wiley and Sons, New York, 1986.Google Scholar
  10. [10]
    Warburton A.R., “Quasiconcave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives”. Jou. of Optimiz. Theory and Appls., Vol. 40, 1983, pp. 537–557.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Yang X.Q., “Vector Variational Inequality and its Duality”, Nonlinear Analysis, Vol. 21, 1993, pp. 869–877.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Yen N.D., “Hölder Continuity of Solutions to a Parametric Variational Inequality”. Applied Mathem. and Optimiz., Vol. 31, 1995, pp. 245–255.zbMATHCrossRefGoogle Scholar
  13. [13]
    Yen N.D. and Phuong T.D., “Connectedness and Stability of the Solution Sets in Linear Fractional Vector Optimiz. Problems”. This Volume.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Nguyen Dong Yen
    • 1
  • Gue Myung Lee
    • 2
  1. 1.Hanoi Institute of MathematicsNational Centre for Natural Science and TechnologyHanoiVietnam
  2. 2.Department of Applied MathematicsPukyong National UniversityPusanKorea

Personalised recommendations