A Lagrangian heuristic algorithm for the time-dependent combined network design and routing problem

During the planning of communication networks, the routing decision process (distributed and online) often remains decoupled from the network design process, i.e., resource installation and allocation planning process (centralized and offline). To reconcile both processes and take into account demand variability, we generalize the capacitated multi-commodity fixed charge network design class of problems by including different types of fixed costs (installation and maintenance costs) and variable costs (routing costs) but also variable traffic demands over multiple periods. However, conventional integer programming methods can typically solve only small to medium size instances of this problem. Two major difficulties are encountered when using commercial solvers to solve the associated mixed integer programs: (i) problems are large scale and even solving the linear relaxation of the problem can be challenging; and (ii) the solver hardly find good feasible solutions for medium to large scale instances. As an alternative, we propose a Lagrangian approach for computing a lower bound by relaxing the flow conservation constraints such that the Lagrangian subproblem itself decomposes by node. Though this approach yields one subproblem per network node, solving the Lagrangian dual by means of the bundle method remains a complex computational tasks. However, it always provides a lower bound on the optimal solution. Moreover, based on this relaxation, we propose a Lagrangian heuristic that makes the approach more robust than a black-box usage of a MIP solver.