Gradients on Sets

For a locally Lipschitz continuous function

the generalized gradient

\partial f(x) ∂ f ( x )

of Clarke is used to develop some (set-valued) gradient on a set

A\subset X A ⊂ X

. Existence, uniqueness and some approximation are considered for optimal descent directions on set

A A

. The results serve as basis for nonsmooth numerical descent algorithms that can be found in subsequent papers.