Transportation Network Optimization Problems with Stochastic User Equilibrium Constraints

A comprehensive study of static transportation network optimization problems with stochastic user equilibrium constraints is presented. It is explicitly demonstrated that the formulation of the fixed-point problem—in terms of link flows for the general stochastic user equilibrium problem in which the Jacobian matrix of link travel cost functions may not be symmetric—possesses a unique solution with mild conditions. By developing a sensitivity analysis method for the stochastic user equilibrium problem, the study proves that the perturbed equilibrium link flows are continuously differentiable implicit functions with respect to perturbation parameters. Accordingly, it can be concluded that the proposed unified bilevel programming model, which can characterize transportation network optimization problems subject to stochastic user equilibrium constraints, is a smooth optimization problem. In addition, the study presents a single-level continuously differentiable optimization formulation that is equivalent to the unified bilevel programming model. Furthermore, as a unified solution method, a successive quadratic programming algorithm based on the sensitivity analysis method is used to solve the transportation network optimization problems with stochastic user equilibrium constraints. Finally, two examples are used to demonstrate the proposed models and algorithm.