Uniform Convergence of the Newton Method for Aubin Continuous Maps

Abstract

In this paper we prove that the Newton method applied to the generalized equation y ∈ f(x) + F(x) with a C^1 function f and a set-valued map F acting in Banach spaces, is locally convergent uniformly in the parameter y if and only if the map (f +F)^(−1) is Aubin continuous at the reference point. We also show that the Aubin continuity actually implies uniform Q-quadratic convergence provided that the derivative of f is Lipschitz continuous. As an application, we give a characterization of the uniform local Q-quadratic convergence of the sequential quadratic programming method applied to a perturbed nonlinear program.