On the primal and dual formulations of traffic assignment problems with perception stochasticity and demand elasticity

This article reinvestigates the mathematical formulations of traffic assignment problems with perception stochasticity and demand elasticity in both the system optimum and user equilibrium principles. Our focus is given to a pair of new general formulations that pose a duality relationship to each other. In this primal-dual modeling framework, we found that the equilibrium or optimality conditions of a traffic assignment problem with perception stochasticity and demand elasticity 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. 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.