A global optimization method for continuous network design problems

The continuous network design problem (CNDP) is generally formulated as a mathematical program with equilibrium constraints (MPEC). It aims to optimize the network performance via expansion of existing links subject to the Wardrop user equilibrium constraint. As one of the extremely challenging problems in the transportation research field, various solution methods have been proposed for solving the CNDP. However, most of the algorithms developed up to date can only find a local optimum due to inherent nonconvexity of the MPEC. This paper proposes a viable global optimization method for the CNDP. Based on the concepts of gap function and penalty, the CNDP is transferred into a sequence of single level concave programs, which is amenable to a global solution. It is proved that any accumulation of the solutions to the sequence of concave programs is a globally optimal solution to the original CNDP. Owing to their special structure, all concave programs can be solved by a multicutting plane method. The penalty term in each step of the inner subproblem can be calculated by simply executing an all-or-nothing assignment.