Convergence of fixed-point algorithms for elastic demand dynamic user equilibrium. (August 2021) - Friesz, Terry L.
Highlights: Dynamic user equilibrium (DUE) path delays may be realistically described as weakly monotone increasing (WMI), and inverse travel demand functions may be realistically described as strongly monotone decreasing (SMD). We establish convergence of fixed-point DUE algorithms under the conditions of Lipschitz continuity, WMI path delays, and SMD inverse travel demand functions. We provide a detailed numerical example showing that the assumptions of Lipschitz continuity, WMI path delays, and SMD inverse travel demand functions hold along the feasible trajectory of fixed-point iterates en route to a DUE solution. Abstract: In this paper we present sufficient conditions for convergence of projection and fixed-point algorithms used to compute dynamic user equilibrium with elastic travel demand (E-DUE). The assumption of strongly monotone increasing path delay operators is not needed. In its place, we assume path delay operators are merely weakly monotone increasing, a property assured by Lipschitz continuity, while inverse demand functions are strongly monotone decreasing. Lipschitz continuity of path delay is a very mild regularity condition. As such, nonmonotone delay operators may be weakly monotone increasing and satisfy our convergence criteria, provided inverse demand functions are strongly monotone decreasing. We illustrate convergence for nonmonotone path delays via a numerical example.