An efficient method for nonlinearly constrained networks

Many nonlinear network flow problems (in addition to the balance constraints in the nodes and capacity constraints on the arc flows) have nonlinear side constraints, which specify a flow relationship between several of the arcs in the network flow model. The short-term hydrothermal coordination of electric power generation is an example of this type. In this work we solve this kind of problem using an approach in which the efficiency of the well-known techniques for network flow can be preserved. It lies in relaxing the side constraints in an augmented Lagrangian function, and minimizing a sequence of these functions subject only to the network constraints for different estimates of the Lagrange multipliers of the side constraints. This method gives rise to an algorithm, which combines first- and superlinear-order multiplier methods to estimate these multipliers. When the number of free variables is very high we can obtain a superlinear-order estimate by means of the limited memory BFGS method fitted to our problem. An extensive computational comparison with other methods has been performed. The numerical results reported indicate that the algorithm described may be employed advantageously to solve large-scale network flow problems with nonlinear side constraints.