Английская Википедия:Characteristic function (convex analysis)

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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

• Шаблон:Cite book