The Convergence of Equilibrium Algorithms with Predetermined Step Sizes

The focus of this paper is on a certain class of equilibrium traffic assignment problems characterized by a path formulation of the associated mathematical programs. In such cases the equilibration iterations would require path enumeration, and are therefore prohibitively expensive. In this paper we prove that a predetermined sequence of step sizes (in a descent direction) would guarantee, under certain regularity conditions, convergence to the equilibrium solution. This algorithm was suggested in the literature without a proof of convergence, which we give here.