Generalized second-order derivative in nonsmooth optimization

In this work a new notion of generalized second-order directional derivative and generalized Hessian for nonsmooth real-valued functions is studied. The general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation. Two applications of these derivatives in optimization theory are considered: first, to obtaining necessary and sufficient second-order optimality conditions for problems with or without constraints; and second, to extending the Newton method for the minimization of a c^1.1 function.