Subderivative

In mathematics, subderivatives (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable.

The set of subderivatives at a point is called the subdifferential at that point.

[1] Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

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

in blue has non-differentiable kinks similar to the absolute value function), for any

in the domain of the function one can draw a line which goes through the point

and which is everywhere either touching or below the graph of f. The slope of such a line is called a subderivative.

Rigorously, a subderivative of a convex function

in the open interval

is a real number

By the converse of the mean value theorem, the set of subderivatives at

for a convex function is a nonempty closed interval

of all subderivatives is called the subdifferential of the function

is convex, then its subdifferential at any point is non-empty.

Then, the subdifferential at the origin is the interval

This is similar to the sign function, but is not single-valued at

, instead including all possible subderivatives.

The concepts of subderivative and subdifferential can be generalized to functions of several variables.

is a real-valued convex function defined on a convex open set in the Euclidean space

in that space is called a subgradient at

The subdifferential is always a nonempty convex compact set.

These concepts generalize further to convex functions

on a convex set in a locally convex space

in the dual space

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.

is continuous, the subdifferential is nonempty.

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 Francis H. Clarke and R. Tyrrell Rockafellar in the early 1980s.