Potentialities of Nonsmooth Optimization

In this paper, we show that higher-order optimality conditions can be obtain for arbitrary nonsmooth function. We introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is consistent with the classical higher-order Fréchet directional derivative in the sense that both derivatives of the same order coincide if the last one exists. We obtain necessary and sufficient conditions of order n (n is a positive integer) for a local minimum and isolated local minimum of order n in terms of these derivatives and subdifferentials. We do not require any restrictions on the function in our results. A special class Fn of functions is defined and optimality conditions for isolated local minimum of order n for a function f ∈ Fn are derived. The derivative of order n does not appear in these characterizations. We prove necessary and sufficient criteria such that every stationary point of order n is a global minimizer. We compare our results with some previous ones.