Advertisement

On Time Dependent Vector Equilibrium Problems

  • A. Khan
  • F. Raciti
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

We consider the time dependent traffic equilibrium problem in the case of a vector valued cost operator. The motivation for this approach is that users can decide to choose a path according to several criteria. In fact, they may want to choose a minimum delay path as well as a minimum tax path. Other criteria can be introcuced in the model, depending on the particular problem under consideration. Thus, we are led to a multicriteria equilibrium problem which can be related to vector variational inequalities. The functional setting is the space L 2([0, T], R n). The extension of the definition of weak equilibria in such a space is not straightforward due to the fact that the cone made up of the non-negative functions has empty interior. We overcome this problem by using the notion of quasi interior of a closed convex set of a Hilbertspace and give sufficient conditions for the existence of weak equilibria.

Key words

Time Dependent Traffic Networks Vector Variational Inequalities Pareto optimization multicriteria equilibrium problems quasi interior 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Daniele, A. Maugeri and W. Oettli, Time-Dependent Traffic Equilibria, Jota, Vol. 103, No. 3 December 1999.Google Scholar
  2. [2]
    X.Q. Yang, C.J. Goh, On Vector Variational Inequalities: Application to Vector Equilibria, Journal of Optimization theory and Applications, Vol.95, No.2, pp.431–443, November 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    W. Oettli, Necessary and sufficient Conditions of Wardrop Type for vectorial traffic equilibria, in: Equilibrium Problems: Non smooth Optimization and Variational Inequality Models, Kluwer Academic Publishers, F. Giannessi, A. Maugeri and P. Pardalos (Eds.).Google Scholar
  4. [4]
    Dafermos S., Traffic Equilibrium and Variational Inequalities, Transp. Sc., 14, 42–54, 1980.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Smith M.J., The existence, uniqueness and stability of traffic equilibria, Transp. Res., 13B, 295–304. Smith M.J., A new Dinamic Traffic Model And the Existence and calculation of Dynamic User Equilibra on Congested Capacity-Constrained Road Networks, Transportation Research, Vol. 27B, pp.49–63, 1993.Google Scholar
  6. [6]
    F. Giannessi, Theorems of the Alternative, Quadratic Programs, and Complementary Problem, Variational Inequalities and Complementarity Problems, Ed. by R.W. Cottle, F. Giannessi and J.L. Lions, Wiley, New York, pp. 151–186, 1980.Google Scholar
  7. [7]
    Vector Variational Inequalities and Vector Equilibria, Ed. by F. Giannessi, Kluwer Academic Publishers, 2000.Google Scholar
  8. [8]
    Variational Inequalities and Equilibrium Models: NonSmooth Optimization, F. Giannessi, A. Maugeri and P. Pardalos Eds., Kluwer Academic Publishers 2001.Google Scholar
  9. [9]
    F. Raciti, Time Dependent Equilibrium in Traffic Networks with delay, in: Variational Inequalities and Equilibrium Models: NonSmooth Optimization, F. Giannessi, A. Maugeri and P. Pardalos Eds., Kluwer Academic Publishers, 2001.Google Scholar
  10. [10]
    A. Nagurnay and D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer, Boston, Dordrecht, 1996.Google Scholar
  11. [11]
    D. Zhang and A. Nagurney, On the Stability of Projected Dynamical Systems, Journal of Optimization Theory and Applications (1995), pp. 97–124.Google Scholar
  12. [12]
    Gwinner J., Time Dependent Variational Inequalities-Some Recent Trends, in Equilibrium Models and Variational Models, P. Daniele, F. Giannessi and A. Maugeri Eds., Kluwer Academic Publishers, 2002.Google Scholar
  13. [13]
    F. Raciti, Equilibria trajectories as stationary solutions of infinite dimensional dynamical systems, accepted for the publication in AML.Google Scholar
  14. [14]
    Chen Guang-Ya and Yang Xiao-Qi, The Vector Complementary Problem and Its Equivalences with the Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, pp. 136–158 (1990).Google Scholar
  15. [15]
    J.M. Borwein, A.S. Lewis, Partially finite convex programming, part I: quasi relative interiors and duality theory, Math. Programming B 57 (1992), pp. 15–48.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. Khan
    • 1
  • F. Raciti
    • 2
    • 3
  1. 1.Department of Mathematical Sciences, Fisher HallMichigan Technological UniversityHoughtonUSA
  2. 2.Dipartimento di Matematica e InformaticaUniversità di CataniaCataniaItaly
  3. 3.Facoltà di Ingegneria dell’ Università di CataniaCataniaItaly

Personalised recommendations