Subderivative

A convex function (blue) and "subtangent lines" at x0 (red).

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 x0 in the domain of the function one can draw a line which goes through the point (x0, f(x0)) 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 x0 in the open interval I is a real number c such that

f(x)-f(x_0)\ge c(x-x_0)

for all x in I. One may show that the set of subderivatives at x0 for a convex function is a nonempty closed interval [a, b], where a and b are the one-sided limits

a=\lim_{x\to x_0^-}\frac{f(x)-f(x_0)}{x-x_0}
b=\lim_{x\to x_0^+}\frac{f(x)-f(x_0)}{x-x_0}

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 x0. If f is convex and its subdifferential at x_0 contains exactly one subderivative, then f is differentiable at x_0.[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 x0<0 is the singleton set {−1}, while the subdifferential at any point x0>0 is the singleton {1}.

Properties

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 Rn, a vector v in that space is called a subgradient at a point x0 in U if for any x in U one has

f(x)-f(x_0)\ge v\cdot (x-x_0)

where the dot denotes the dot product. The set of all subgradients at x0 is called the subdifferential at x0 and is denoted ∂f(x0). 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 x0 in U if

f(x)-f(x_0)\ge v^*(x-x_0).

The set of all subgradients at x0 is called the subdifferential at x0 and is again denoted ∂f(x0). 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

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