On the Primal and Dual Formulations of Stochastic Traffic Assignment with Elastic Demand

In a traffic assignment problem with elastic demand, the equilibrium conditions are achieved on two levels, namely, the stochastic user equilibrium on the path level and the supply-demand equilibrium on the origin-destination level. By recognizing that both equilibrium levels imply random individual perceptions and decision makings, this paper reinvestigates the mathematical formulations of this kind of problems in both the system optimum and user equilibrium principles. Unlike previous research devoted to the development of solution methods for some specific versions of elastic-demand traffic assignment problems, our focus is given to a pair of new general formulations that pose a duality relationship to each other. The primal formulation has a convex programming form with nonlinear constraints, while the dual one poses a concave programming problem. In this primal-dual modeling framework, we found that the equilibrium or optimality conditions of a traffic assignment problem with elastic demand can be redefined as a combination of three sets of equations and an arbitrary feasible solution of either the primal or dual formulation satisfies only two of them. We further rigorously proved the solution equivalency and uniqueness of both the primal and dual formulations, by using derivative-based techniques. We also found that when a problem of this type collapses to a logit-based case, both formulations can be conveniently written into a tractable (i.e., analytically evaluable) path flow-based form and the primal formulation becomes a convex programming problem with linear constraints. While the two formulations pose their respective modeling advantages and drawbacks, our preliminary algorithmic analysis and numerical test results indicate that the dual formulation-based algorithm, i.e., the Cauchy algorithm, can be more readily implemented for large-scale problems and converge evidently faster than the primal formulation-based one, i.e., the Frank-Wolfe algorithm, at least in the logit-based case