For a
locally Lipschitz continuous function f: \mathbb{R}^{n} \rightarrow \mathbb{R}, the
Clarke generalized directional derivative of f at x \in \mathbb{R}^n in the direction v \in \mathbb{R}^n is defined as f^{\circ} (x, v)= \limsup_{y \rightarrow x, h \downarrow 0} \frac{f(y+ hv)-f(y)}{h}, where \limsup denotes the
limit supremum. Then, using the above definition of f^{\circ}, the
Clarke generalized gradient of f at x (also called the
Clarke subdifferential) is given as \partial^{\circ}\! f(x):=\{\xi \in \mathbb{R}^{n}: \langle\xi, v\rangle \leq f^{\circ}(x, v), \forall v \in \mathbb{R}^{n}\}, where \langle \cdot, \cdot\rangle represents an
inner product of vectors in \mathbb{R}. Note that the Clarke generalized gradient is set-valued—that is, at each x \in \mathbb{R}^n, the function value \partial^{\circ}\! f(x) is a set. More generally, given a Banach space X and a subset Y \subset X, the Clarke generalized directional derivative and generalized gradients are defined as above for a
locally Lipschitz continuous function f : Y \to \mathbb{R}. == See also ==