This paper is about design methodologies for packet networks, under the constraints of end-to-end quality of service (QoS) metrics. The network modeling also considers the dynamics of today’s packet networks. We are particularly considering the problem of capacity and flow assignment where the routing assignments and capacities are considered to be decision variables. An efficient Lagrangean relaxation-based heuristic procedure is developed to find bounds and solutions for a corporate virtual private network (VPN), where the traffic is mostly based on TCP connections. Numerical results for a variety of problem instances are reported.