This talk presents an extension of the concept of dual cones and introduces the notion of augmented dual cones. We show that the supporting and separation philosophy based on hyperplanes can be extended to a nonconvex analysis by using elements of these cones. A special class of monotone sublinear functions is introduced with the help of elements of augmented dual cones. We introduce the separation property and present the separation theorem which enables to separate two cones (one of them is not necessarily convex, having only the vertex in common) by a level set of some monotonically increasing sublinear function. By using the new separation philosophy, we introduce the notions of radial epiderivatives and weak subgradients for nonconvex real-valued functions and study relations between them and the directional derivatives. The well-known necessary and sufficient optimality condition of nonsmooth convex optimization given in the form of variational inequality is generalized to the nonconvex case by using the notion of weak subdifferentials. The equivalent formulation of this condition in terms of weak subdifferentials and augmented normal cones is also presented.