An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem

The continuous network design problem (CNDP) is characterized by a bilevel programming model and recognized to be one of the most difficult and challenging problems in transportation. The main difficulty stems from the fact that the bilevel formulation for the CNDP is nonconvex and nondifferentiable, and indeed only some heuristic methods have been so far proposed. In this paper, the bilevel programming model for CNDPs is transferred into a single level optimization problem by virtue of a marginal function tool. By exploring the inherent nature of the CNDP, the marginal function for the lower-level user equilibrium problem is proved to be continuously differentiable and its functional value and derivative in link capacity enhancement can be obtained efficiently by implementing a user equilibrium assignment subroutine. Thus a continuously differentiable but still nonconvex optimization formulation of the CNDP is created and a locally convergent augmented Lagrangian method is applied to solve this equivalent problem. The descent direction in each step of the inner loop of the solution method can be found by doing an all or nothing assignment. These favorable characteristics indicate the potential of the algorithm to solve large CNDPs. Numerical examples are presented to compare the proposed method with some existing algorithms.