ABSTRACT We present a survey on nonconvex models and algorithms for multicommodity network design problems. We put in perspective the alternative modelling of these problems, traditionally represented by mixed-integer linear programs, by separable nonconvex arc cost functions. We show in particular that some problems take profit of a continuous modelling and explore the case of capacity expansion of communications networks.

This paper describes an exact algorithm to solve a nonlinear mixed-integer programming model due to capacity expansion and flow assignment in multicommodity networks. The model combines continuous multicommodity flow variables associated with nonlinear congestion costs and discrete decision variables associated with the arc expansion costs. After establishing precise correspondences between a mixed-integer model and a continuous but nonconvex model, an implicit enumeration approach is derived based on the convexification of the continuous objective function. Numerical experiments on medium size instances considering one level of expansion are presented. The results reported on the performance of the proposed algorithm show that the approach is efficient, as commercial solvers were not able to tackle the instances considered.