Convergence of the Frank-Wolfe method for certain bounded variable traffic assignment problems

In two recent papers published in Transportation Research, Daganzo presented a modification of the Frank-Wolfe algorithm to solve certain link capacitated traffic assignment problems satisfying certain conditions. In order to show convergence of the modified algorithm, the assumption was made that the integral of the volume delay formula for each link tends to infinity as the link flow approaches the link capacity. In this paper we give a Theorem which establishes convergence of the modified algorithm under much weaker conditions. This result is then used to show convergence if the objective function of the assignment model is sufficiently large (not necessarily infinite) when the link flows are at capacity. Thus the modified method is applicable to a broader class of assignment problems. Two numerical examples illustrate (a) when the method converges and when it does not, and (b) that our Theorem provides a weaker condition for convergence of the method.