Towards an Optimal Algorithm for Computing Fixed Points: Dynamical Systems Approach, with Applications to Transportation Engineering

In many practical problems, it is desirable to nd an equilibrium. For example, equilibria are important in transportation engineering. Many urban areas suffer from traf c congestion. Intuitively, it may seem that a road expansion (e.g., the opening of a new road) should always improve the trafc conditions. However, in reality, a new road can actually worsen traf c congestion. It is therefore extremely important that before we start a road expansion project, we rst predict the effect of this project on traf c congestion. When a new road is built, some traf c moves to this road to avoid congestion on the other roads; this causes congestion on the new road, which, in its turn, leads drivers to go back to their previous routes, etc. What we want to estimate is the resulting equilibrium. In many problems – e.g., in many transportation problems – natural iterations do not converge. It turns out that the convergence of the corresponding xed point iterations can be improved if we consider these iterations as an approximation to the appropriate dynamical system.