Английская Википедия:Characteristic function (convex analysis)
Шаблон:No footnotes In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function, and one can freely convert between the two, but the characteristic function as defined below is better-suited to the methods of convex analysis.
Definition
Let <math>X</math> be a set, and let <math>A</math> be a subset of <math>X</math>. The characteristic function of <math>A</math> is the function
- <math>\chi_{A} : X \to \mathbb{R} \cup \{ + \infty \}</math>
taking values in the extended real number line defined by
- <math>\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}</math>
Relationship with the indicator function
Let <math>\mathbf{1}_{A} : X \to \mathbb{R}</math> denote the usual indicator function:
- <math>\mathbf{1}_{A} (x) := \begin{cases} 1, & x \in A; \\ 0, & x \not \in A. \end{cases}</math>
If one adopts the conventions that
- for any <math>a \in \mathbb{R} \cup \{ + \infty \}</math>, <math>a + (+ \infty) = + \infty</math> and <math>a (+\infty) = + \infty</math>, except <math>0(+\infty)=0</math>;
- <math>\frac{1}{0} = + \infty</math>; and
- <math>\frac{1}{+ \infty} = 0</math>;
then the indicator and characteristic functions are related by the equations
- <math>\mathbf{1}_{A} (x) = \frac{1}{1 + \chi_{A} (x)}</math>
and
- <math>\chi_{A} (x) = (+ \infty) \left( 1 - \mathbf{1}_{A} (x) \right).</math>
Subgradient
The subgradient of <math>\chi_{A} (x)</math> for a set <math>A</math> is the tangent cone of that set in <math>x</math>.
Bibliography