On the convergence of the method of successive averages for calculating equilibrium in traffic networks

The traffic assignment problem aims to calculate an equilibrium route flow vector, generally by seeking a zero of an appropriate objective function. If a continuous dynamical system follows a descent direction for this objective function at each nonequilibrium route flow vector, the system converges to equilibrium. It is shown that when this dynamical system is discretized with a fixed step length, the system eventually approaches close to equilibrium provided that the objective function is continuously differentiable and that the rate of descent is bounded below. The method of successive averages is widely used in traffic assignment; it has a decreasing step size at each iteration. With the same conditions as above, it is shown that the resulting dynamical system converges to equilibrium. In the steady-state model, the necessary conditions are shown to be satisfied, provided that the route cost vector is a continuously differentiable monotone function of the route flow vector. However, continuous differentiability of the cost function is shown not to hold in the dynamic queueing model.