Solving a non-convex combined travel forecasting model by the method of successive averages with constant step sizes

The method of successive averages is often used in iterative algorithms for solving various mathematical problems, and travel forecasting models in particular. In each iteration of these algorithms the current solution is averaged with an alternative solution generated by the algorithm. If the problem has an equivalent convex optimization formulation with a computable objective function, a line search can be used to determine the weight applied to the alternative solution, referred to as the step size. In the absence of a convex optimization formulation, the typical approach is to set the step size according to a predetermined sequence that is decreasing towards zero, such as 1/k where k is the iteration index. In this paper, we examine the use of a constant step size in the method of successive averages. A theoretical derivation shows that if the alternative solution generated by the algorithm is a linear function of the current solution, then using a constant step size is advantageous and substantially superior to using a sequence of decreasing step sizes. We conjecture that similar results may be expected in nonlinear differentiable problems as well. Numerical results are presented for a travel forecasting model that combines user-equilibrium route choice with an origin-destination-mode (ODM) choice model. The proposed algorithm is based on origin-based assignment, in conjunction with successive averaging of constant step sizes for the differentiable ODM model. The results show that a properly chosen constant step size leads to excellent convergence for both convex and non-convex models. A general strategy for choosing step sizes without a-priori knowledge is presented as well.