Approximation of the steepest descent direction for the origin–destination matrix adjustment problem

In this paper, a method to approximate the directions of Clarke's generalized gradient of the upper level function for the demand adjustment problem on traffic networks is presented. Its consistency is analyzed in detail. The theoretical background on which this method relies is the known property of proximal subgradients of approximating subgradients of proximal bounded and lower semicontinuous functions using the Moreau envelopes. A double penalty approach is employed to approximate the proximal subgradients provided by these envelopes. An algorithm based on partial linearization is used to solve the resulting nonconvex problem that approximates the Moreau envelopes, and a method to verify the accuracy of the approximation to the steepest descent direction at points of differentiability is developed, so it may be used as a suitable stopping criterion. Finally, a set of experiments with test problems are presented, illustrating the approximation of the solutions to a steepest descent direction evaluated numerically.