New gradient method for the capacitated network flow model

The capacitated network flow model is formulated in general nonlinear optimization problem with explicit link capacity constraints. The model is expected to describe a congested network due to the link capacity constraints. For utilizing properties of the conventional user equilibrium model, the capacitated flow model can be turned into the form of the conventional user equilibrium model. A new gradient which is equal to the average cost of all the used paths within the given origin–destination pair is defined. The Newton method with the new gradient is combined with the interior penalty method to obtain the capacitated network flow solution. Numerical experiment demonstrates the excellent performances of the proposed methods over the small network.