# Subderivative

In mathematics, the **subderivative**, **subgradient**, and **subdifferential** generalize the derivative to functions which are not differentiable. The subdifferential of a function is set-valued. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

Let *f*:*I*→**R** be a real-valued convex function defined on an open interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value function *f*(*x*)=|*x*| is nondifferentiable when *x*=0. However, as seen in the picture on the right, for any *x*_{0} in the domain of the function one can draw a line which goes through the point (*x*_{0}, *f*(*x*_{0})) and which is everywhere either touching or below the graph of *f*. The slope of such a line is called a *subderivative* (because the line is under the graph of *f*).

## Definition

Rigorously, a *subderivative* of a function *f*:*I*→**R** at a point *x*_{0} in the open interval *I* is a real number *c* such that

for all *x* in *I*. One may show that the set of subderivatives at *x*_{0} for a convex function is a nonempty closed interval [*a*, *b*], where *a* and *b* are the one-sided limits

which are guaranteed to exist and satisfy *a* ≤ *b*.

The set [*a*, *b*] of all subderivatives is called the **subdifferential** of the function *f* at *x*_{0}. If *f* is convex and its subdifferential at contains exactly one subderivative, then *f* is differentiable at .^{[1]}

## Examples

Consider the function *f*(*x*)=|*x*| which is convex. Then, the subdifferential at the origin is the interval [−1, 1]. The subdifferential at any point *x*_{0}<0 is the singleton set {−1}, while the subdifferential at any point *x*_{0}>0 is the singleton set {1}.

## Properties

- A convex function
*f*:*I*→**R**is differentiable at*x*_{0}if and only if the subdifferential is made up of only one point, which is the derivative at*x*_{0}. - A point
*x*_{0}is a global minimum of a convex function*f*if and only if zero is contained in the subdifferential, that is, in the figure above, one may draw a horizontal "subtangent line" to the graph of*f*at (*x*_{0},*f*(*x*_{0})). This last property is a generalization of the fact that the derivative of a function differentiable at a local minimum is zero.

## The subgradient

The concepts of subderivative and subdifferential can be generalized to functions of several variables. If *f*:*U*→ **R** is a real-valued convex function defined on a convex open set in the Euclidean space **R**^{n}, a vector *v* in that space is called a **subgradient** at a point *x*_{0} in *U* if for any *x* in *U* one has

where the dot denotes the dot product.
The set of all subgradients at *x*_{0} is called the **subdifferential** at *x*_{0} and is denoted ∂*f*(*x*_{0}). The subdifferential is always a nonempty convex compact set.

These concepts generalize further to convex functions *f*:*U*→ **R** on a convex set in a locally convex space *V*. A functional *v*^{∗} in the dual space V^{∗} is called *subgradient* at *x*_{0} in *U* if

The set of all subgradients at *x*_{0} is called the subdifferential at *x*_{0} and is again denoted ∂*f*(*x*_{0}). The subdifferential is always a convex closed set. It can be an empty set; consider for example an unbounded operator, which is convex, but has no subgradient. If *f* is continuous, the subdifferential is nonempty.

## History

The subdifferential on convex functions was introduced by Jean Jacques Moreau and R. Tyrrell Rockafellar in the early 1960s. The *generalized subdifferential* for nonconvex functions was introduced by F.H. Clarke and R.T. Rockafellar in the early 1980s.^{[2]}

## See also

## References

- ↑ R. T. Rockafellar
*Convex analysis*1970. Theorem 25.1, p.242 - ↑ Clarke, Frank H. (1983).
*Optimization and nonsmooth analysis*. New York: John Wiley & Sons. pp. xiii+308. ISBN 0-471-87504-X. MR 0709590.

- Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal,
*Fundamentals of Convex Analysis*, Springer, 2001. ISBN 3-540-42205-6. - Zălinescu, C. (2002).
*Convex analysis in general vector spaces*. World Scientific Publishing Co., Inc. pp. xx+367. ISBN 981-238-067-1. MR 1921556.