Higher order optimality conditions in nonsmooth optimization

For an arbitrary function f: E --> (R) over bar defined on a Banach space E higher order lower directional derivatives and higher order optimality conditions are introduced. Three equivalent approaches are given, a direct definition applying an analogue of the Taylor expansion formula, a definition based on polynomial approximation and a definition based on divided differences. It is shown that the proposed sufficient optimality conditions are also necessary for a point x(0) to be an isolated minimizer off, which gives a characterization of the isolated minimizers.