Differential complementarity systems and dynamic traffic equilibria

Jong-Shi Pang
This lecture reports on the research to date that the speaker has undertaken jointly with his collaborators on the subject of differential complementarity systems (DCS) and how they can be used to model continuous-time dynamic user equilibria (DUE). In essence, the DCS is a mathematical model consisting of an ordinary differential equation (ODE) containing an auxiliary variable that is required to be a solution to a finite-dimensional complementarity problem parameterized by the state variable of the ODE. Aiming to predict short-term traffic patterns by assuming that travelers follows certain rational choice behaviors, such as route and/or departure time choices, the DUE problem in traffic analysis leads to a delay DCS whose solution remains a major challenge in both the practical application of the DUE concept and the analysis and computation of its mathematical formulation. Several simplified models of the DUE problem are presented, which lead to linear complementarity systems (LCS) without delay. A well-known cell model approximation of the DUE problem gives rise to a partial differential complementarity system. The delay DCS and the partial DCS are two novel mathematical paradigms that have never been studied in a systematic manner. It is hoped that through the applications presented here in the domain of traffic equilibria and possibly similar applications in nonsmooth frictional contact mechanics, this kind of challenging of differential-variational systems will motivate sufficient interest among the School participants and inspire them to undertake rigorous research on these mathematical systems.

Lecture slides

  • X. Ban, J.S. Pang, H. Liu, and R. Ma. Continuous-time dynamic user equilibrium: Network loading using a modified point-queue model. Transportation Research, Series B: Methodological. Volume 46 (2012) 360--380.
  • X. Ban, J.S. Pang, H. Liu, and R. Ma. Modeling and solving continuous-time instantaneous dynamic user equilibria: A differential complementarity systems approach Transportation Research, Series B: Methodological. Volume 46 (2012) 389--408.
  • L. Han and J.S. Pang. Time-stepping methods for linear complementarity systems. Proceedings of the Fifth International Conference of Chinese Mathematicians. December 2010. Accepted May 2011.
  • J.S. Pang, L. Han, G. Ramadurai, and S. Ukkusuri. A continuous-time dynamic equilibrium model for multi-user class single bottleneck traffic flows. Mathematical Programming, Series A. DOI 10.1007/s10107-010-0433-z.
  • J.S. Pang and D.E. Stewart. Differential variational inequalities. Mathematical Programming, Series A. Volume 113 (2008) 345--424.