A generalization of secant methods for solving nonlinear systems of equations

Many problems in transportation can be formulated as fixed point problems, or equivalently as a systems of nonlinear equations, as for example multi-user equilibrium problems, planning problems, and consistent anticipatory route guidance generation. These fixed point problems are often characterized by the fact that there is no analytical form of the objective function, the computational cost of evaluating the function is generally high, the dimension is very large and stochasticity is present. Therefore classical algorithms to solve linear systems of equation based on derivatives information or involving line search are unlikely to be attractive in this context. In this paper we introduce a new class of methods for solving nonlinear systems of equations motivated by the constraints described above. The main idea is to generalize classical secant methods by building the secant model using more than two previous iterates. This approach allows to use all the available information collected through the iterations to construct the model. We prefer a least-square approach to calibrate the secant model, as exact interpolation requires a fixed number of iterates, and may be numerically problematic. We also propose an explicit control of the numerical stability of the method. We show that our approach can lead to an update formula à la Broyden. In that case, we proof the local convergence of the corresponding quasi-Newton method. Finally, computational comparisons with classical methods, based on performance profiles, highlight a significant improvement in term of robustness and number of function evaluations. We also present preliminary numerical tests showing the robust behavior of our methods in the presence of noisy nonlinear system of equations. Numerical results of this method apply to the consistent anticipatory route guidance will be provided in the other presentation (“Solving the anticipatory route guidance generation problem using a generalization of secant methods”.