NEWTON DIFFERENTIABILITY OF CONVEX FUNCTIONS IN NORMED SPACES AND OF A CLASS OF OPERATORS

Newton differentiability is an important concept for analyzing generalized Newton methods for nonsmooth equations. In this work, for a convex function defined on an infinite-dimensional space, we discuss the relation between Newton and Bouligand differentiability and upper semicontinuity of its subdifferential. We also construct a Newton derivative of an operator of the form (Fx)(p) = f(x, p) for general nonlinear operators f that possess a Newton derivative with respect to x and also for the case where f is convex in x.