Subderivative

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

In mathematics, the concepts of subderivative, subgradient, and subdifferential arise in convex analysis, that is, in 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 may not necessarily be differentiable at all points, as for example, the absolute value, f(x)=|x|. However, as seen in the picture on the right (and which can be proved rigorously), 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).

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[edit] Definition

Rigorously, a subderivative of a convex 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 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.

[edit] 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}.

[edit] Properties

  • A convex function f:I→R is differentiable at x0 if and only if the subdifferential is made up of only one point, which is the derivative at x0.
  • A point x0 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 (x0, f(x0)). This last property is a generalization of the fact that the derivative of a function differentiable at a local minimum is zero.

[edit] 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.

[edit] References

  • Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, Fundamentals of Convex Analysis, Springer, 2001. ISBN 3-540-42205-6.