Stochastic traffic assignment, Lagrangian dual, and unconstrained convex optimization

In this paper, traffic assignment problems with stochastic travel cost perceptions are reformulated and investigated in a new unconstrained nonlinear programming formulation. The objective function of the unconstrained formulation consists of two terms, in which the first term specifies the routing principle of the target problem through a satisfaction function and the sum of the first and second terms denotes the system cost or optimization objective. This formulation proves to be the Lagrangian dual of a generic primal formulation proposed by Maher et al. (2005) for the stochastic system-optimal problem. The primal–dual modeling framework presents such a common functional form that can accommodate a wide range of different traffic assignment problems. Our particular attention is given to the dual formulation in that its unconstrained feature opens the door of applying unconstrained optimization algorithms for its embraced traffic assignment problems. Numerical examples are provided to support the insights and facts derived from applying the primal and dual formulations to model stochastic system-optimal and user-equilibrium problems and justify the conjugate relationship between the primal and dual models.