Generalized partial derivatives and optimality conditions for non-differentiable functions

In [9] a generalization of the concept of a classical gradient was presented, which exists for a broad class of non-differentiable functions. This concept differs from traditional "set-valued" generalizations by an expansion of a directional derivative into a special orthogonal series, Investigating the coefficients of this series and interpreting them as a kind of partial derivatives Formal analogies to properties of classical partial derivatives could be drawn. Moreover the new concept reveals relations between questions appearing within the theory of optimization and results in other branches of mathematics, especially to the theory of harmonic and subharmonic functions. The aim of this paper is to show that with the new tool of generalized partial derivatives necessary and sufficient conditions for optimality can be deduced in the case, that a function under consideration is not differentiable at the point of interest. In such a way criteria of optimality can be formulated using the language of these generalized partial derivatives. Numerical considerations show, that the generalized partial derivatives can (approximately) he calculated without solving integrals by hyperinterpolation on the sphere.