Clarke generalized derivative

Generalized derivatives, normals and tangent cones are used in non-smooth analysis, a body of theory concerned with the calculus of functions and sets that do not admit linear approximations in the sense of the customary derivative or the usual tangent space. The growth of the subject from the 1970s onwards reflects the essential need to grapple with non-smoothness (together with absence of convexity) in such topics as optimization, control, viscosity solutions, optimal design, variational methods, and in fact, in non-linear analysis generally. Certain characteristics have become standard in all types of non-smooth analysis: the presence of a calculus of subgradients and normal vectors, in tandem with constructs dual to these (directional derivatives, tangents), and especially the placement on an equal footing of sets and of functions, which lends to the subject a distinctly geometric flavour. This pattern was established in the early 1970s with Clarke's calculus of proximal normals and generalized gradients, and implies that either geometric or functional constructs can be chosen as the basic notions from which the others are derived.

The basic definition in a Hilbert space setting can be taken to be that of $\partial _ { P } f ( x )$, the set of proximal subgradients of a function $f : H \rightarrow \mathbf{R} \cup \{ \infty \}$ at a point $x$: $\zeta$ belongs to $\partial _ { P } f ( x )$ if there exists a $\sigma \geq 0$ such that

\begin{equation*} f ( y ) - f ( x ) + \sigma \| y - x \| ^ { 2 } \geq \langle \zeta , y - x \rangle, \ \forall y \text{ near } x. \end{equation*}

When $f$ is lower semi-continuous (cf. also Semi-continuous function), $\partial _ { P } f ( x )$ is non-empty for a dense set of $x$, and there is a useful calculus associated with it.

In the setting of a Banach space $X$, in the case of a locally Lipschitz function (cf. also Lipschitz condition), the consideration of the generalized directional derivative given by

\begin{equation*} f ^ { \circ } ( x ; v ) : = \liminf _ { y \rightarrow x , t \downarrow 0 } \frac { f ( y + t v ) - f ( y ) } { t } \end{equation*}

leads to the set $\partial f ( x )$ of generalized gradients:

\begin{equation*} \partial f ( x ) : = \{ \zeta : f ^ { \circ } ( x ; v ) \geq \langle \zeta , v \rangle , \forall v \in X \}, \end{equation*}

a set which is non-empty, convex and compact. The calculus rules for various generalized derivatives become more precise if the data are regular (for example, smooth or convex).

Regarding sets, a basic tool is the (Clarke) tangent cone to a set $S$ at $x$; it consists of those $v$ which satisfy: For all sequences $\{ x _ { i } \}$ in $S$ converging to $x$, for all sequences $\{ t _ { i } \}$ decreasing to $0$, there exists a sequence $\{ v _ { i } \}$ converging to $v$ such that $x _ { i } + t _ { i } v _ { i } \in S$.

Other constructs of interest include Dini subderivates, generalized Jacobians, viscosity subdifferentials, and various normal cones. The relationships between the different theories are well understood; together with a variety of applications and other references, they can be found in, e.g., [a5].

References

[a1]	J-P. Aubin, H. Frankowska, "Set-valued analysis" , Birkhäuser (1990)
[a2]	F. Clarke, "Generalized gradients and applications" Trans. Amer. Math. Soc. , 205 (1975) pp. 247–262
[a3]	F. Clarke, "Optimization and nonsmooth analysis" , Wiley/Interscience (1983) (Reprinted: SIAM Publications, Classics in Applied Mathematics 5, 1990)
[a4]	F. Clarke, "Methods of dynamic and nonsmooth optimization" , CBMS/NSF Regional Conf. Ser. Appl. Math. , 57 , SIAM (1989)
[a5]	F. Clarke, Yu. Ledyaev, R. Stern, P. Wolenski, "Nonsmooth analysis and control theory" , Graduate Texts in Math. , 178 , Springer (1998)
[a6]	R.T. Rockafellar, R. Wets, "Variational analysis" , Springer (1998)