The mathematical foundations of dynamic user equilibrium

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The mathematical foundations of dynamic user equilibrium

Friesz, Terry L. / Han, Ke

Highlights • The paper includes a primer on differential variational inequalities (DVI). • The theory of DVI is applied to the study of dynamic user equilibrium in continuous time. • The DUE problem is articulated as a DVI and fixed-point problem. • These formulations are explored using optimal control theory. • A continuous-time computational scheme is developed then applied to two example problems.

Abstract This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dynamic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE; then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples.