UNIFORM CONVERGENCE OF THE NEWTON METHOD FOR AUBIN CONTINUOUS MAPS

Abstract

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. This paper is about the Newton method for solving equations involving setvalued maps and parameters. Such “equations”, commonly known as generalized equations, are of the form: (1) Find x ∈ X such that y ∈ f(x) + F(x), where y is a parameter, f is a function and F is a map, possibly set-valued. Throughout X and Y are Banach spaces, y ∈ Y, f: X ↦ → Y is C 1 on X and F: X ↦ → 2 Y has closed graph. The generalized equation (1) is an abstract model for various problems