A Mathematical Model and Descent Algorithm for Bilevel Traac Management

We provide a new mathematical model for strategic traac management, formulated and analyzed as a mathematical program with equilibrium constraints (MPEC). The model includes two types of control (upper-level) variables, which may be used to describe such traac management actions as traac signal setting, network design, and congestion pricing. The lower-level problem of the MPEC describes a traac equilibrium model in the sense of Wardrop, in which the control variables enter as parameters in the travel costs. We consider a (small) variety of model settings, including xed or elastic demands, the possible presence of side constraints in the traac equilibrium system, and representations of traac ows and management actions in both link{route and link{node space. For this model, we also propose and analyze a descent algorithm. The algorithm utilizes a new reformulation of the MPEC into a constrained, locally Lipschitz minimization problem in the product space of controls and traac ows. The reformulation is based on the Minty parameterization ((Min67]) of the graph of the normal cone operator for the traac ow polyhedron. Two immediate advantages of making use of this reformulation are that the resulting descent algorithm can be operated and established to be convergent without requiring that the travel cost mapping is monotone, and without having to ever solve the lower-level equilibrium problem. We provide example realizations of the algorithm, establish their convergence, and interpret their workings in terms of the traac network.