Uniform Convergence of the Newton Method for Aubin Continuous Maps

In this paper we prove that the Newton method applied to the generalized equation y ∈ f(x) + F (x) with a C 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 ) 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: Find x ∈ X such that y ∈ f(x) + F (x), (1) 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 7→ Y is C on X and F : X 7→ 2 has closed graph. The generalized equation (1) is an abstract model for various problems 1991 Mathematics Subject Classification: 90C30, 47H04, 49M37, 65K10