Optimization Formulations and Static Equilibrium in Congested Transportation Networks

In this paper we study the concepts of equilibrium and optimum in static transportation networks with elastic and non-elastic demands. The main mathematical tool of our paper is the theory of variational inequalities. We demonstrate that this theory is useful for proving the existence theorems. It also can justify Beckmann’s formulation of the equilibrium problem. The main contribution of this paper is to propose a new definition of equilibrium, the normal equilibrium, which exists under very general assumptions. This concept can be used, in particular, when the travel costs are discontinuous and unbounded. As examples we consider the models of signalized intersections, traffic lights and unbounded travel-time relationships. For some of those cases, the standard concepts of user and Wardrop equilibria cannot be used. ∗We would like to thank FNRS Suisse, Modèle de congestion urbain for partial financial support †THEMA, Université de Cergy-Pontoise, France ‡CORE, Université Catholique de Louvain, Belgium