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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Daniele, A. Maugeri and W. Oettli, Time-Dependent Traffic Equilibria, Jota, Vol. 103, No. 3 December 1999.
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.
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.).
Dafermos S., Traffic Equilibrium and Variational Inequalities, Transp. Sc., 14, 42–54, 1980.
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.
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.
Vector Variational Inequalities and Vector Equilibria, Ed. by F. Giannessi, Kluwer Academic Publishers, 2000.
Variational Inequalities and Equilibrium Models: NonSmooth Optimization, F. Giannessi, A. Maugeri and P. Pardalos Eds., Kluwer Academic Publishers 2001.
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.
A. Nagurnay and D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer, Boston, Dordrecht, 1996.
D. Zhang and A. Nagurney, On the Stability of Projected Dynamical Systems, Journal of Optimization Theory and Applications (1995), pp. 97–124.
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.
F. Raciti, Equilibria trajectories as stationary solutions of infinite dimensional dynamical systems, accepted for the publication in AML.
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).
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Khan, A., Raciti, F. (2005). On Time Dependent Vector Equilibrium Problems. In: Giannessi, F., Maugeri, A. (eds) Variational Analysis and Applications. Nonconvex Optimization and Its Applications, vol 79. Springer, Boston, MA. https://doi.org/10.1007/0-387-24276-7_35
Download citation
DOI: https://doi.org/10.1007/0-387-24276-7_35
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-24209-5
Online ISBN: 978-0-387-24276-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)