Continuous approximations, codifferentiable functions and minimization methods

Our starting point relies on the observation that, for a nondifferentiable function, the classical directional derivative fails to be continuous with respect to the initial point; this is also related to the lack of continuity properties of the quasidifferential or other differential objects obtained as linearization of the directional derivative. In this paper we describe the notion of codifferentiability as a mean to obtain a continuous approximation for a nonsmooth function. Particular emphasis is given to applications to optimization theory: necessary optimality conditions, minimization methods, extensions of the Newton method for a system of nonsmooth equations.We also describe how the main ideas behind codifferentiability can be extended to mappings between Banach spaces. In the last section we discuss the concept of continuous approximation without linearization and show how the conceptual study of a number of topics in nonsmooth optimization can satisfactorily be treated in this more general setting.