  We tackle the network design problem for centralized traffic assignment,
which can be cast as a mixed-integer convex optimization (MICO) problem. For
this task, we propose different formulations and solution methods in both a
deterministic and a stochastic setting in which the demand is unknown in the
design phase. We leverage the recently proposed Boscia framework, which can
solve MICO problems when the main nonlinearity stems from a differentiable
objective function. Boscia tackles these problems by branch-and-bound with
continuous relaxations solved approximately with Frank-Wolfe algorithms. We
compare different linear relaxations and the corresponding subproblems solved
by Frank-Wolfe, and alternative problem formulations to identify the situations
in which each performs best. Our experiments evaluate the different approaches
on instances from the Transportation Networks library and highlight the
suitability of the mixed-integer Frank-Wolfe algorithm for this problem. In
particular, we find that the Boscia framework is particularly applicable to
this problem and that a mixed-integer linear Frank-Wolfe subproblem performs
well for the deterministic case, while a penalty-based approach, with decoupled
feasible regions for the design and flow variables, dominates other approaches
for stochastic instances with many scenarios.
