Nonlinear optimization under nonlinear constraints is usually difficult. However, standard ad-hoe search techniques may work successfully in some cases. Here, we consider an augmented Lagrangian formulation, and we develop a "projection heuristic" that "guides" the iterative search toward the optimum. We demonstrate the effectiveness of this approach by applying it to the problem of maximizing a circuit-switched communication network's throughput under quality-of-service (QoS) constraints by means of choosing the input offered load. This problem is useful for "sizing" the network capacity. Performance results using several versions of the algorithm demonstrate its robustness, in terms of its accuracy and convergence properties.