Lagrangian relaxation for the time-dependent combined network design and routing problem

In communication networks, the routing decision process (distributed and online) remains decoupled from the network design process, i.e., resource installation and allocation planning process (centralized and offline). To reconcile both processes, we ambition to design a distributed optimization technique aware of distributed nature of the routing process by decomposing the optimization problem along same dimensions as (distributed) routing decision process. For this purpose, we generalize the capacitated multi-commodity capacitated fixed charge network design (MCND) 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. 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, the approach is more robust than any LP solvers and it always returns some solutions. Instead, we proved that CPLEX, which uses the Dual Simplex algorithm, is not able to provide a solution for large instances.