Characteristic function (convex analysis)

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]\displaystyle{ X }[/math] be a set, and let [math]\displaystyle{ A }[/math] be a subset of [math]\displaystyle{ X }[/math]. The characteristic function of [math]\displaystyle{ A }[/math] is the function

[math]\displaystyle{ \chi_{A} : X \to \mathbb{R} \cup \{ + \infty \} }[/math]

taking values in the extended real number line defined by

[math]\displaystyle{ \chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases} }[/math]

Relationship with the indicator function

Let [math]\displaystyle{ \mathbf{1}_{A} : X \to \mathbb{R} }[/math] denote the usual indicator function:

[math]\displaystyle{ \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]\displaystyle{ a \in \mathbb{R} \cup \{ + \infty \} }[/math], [math]\displaystyle{ a + (+ \infty) = + \infty }[/math] and [math]\displaystyle{ a (+\infty) = + \infty }[/math], except [math]\displaystyle{ 0(+\infty)=0 }[/math];
• [math]\displaystyle{ \frac{1}{0} = + \infty }[/math]; and
• [math]\displaystyle{ \frac{1}{+ \infty} = 0 }[/math];

then the indicator and characteristic functions are related by the equations

[math]\displaystyle{ \mathbf{1}_{A} (x) = \frac{1}{1 + \chi_{A} (x)} }[/math]

and

[math]\displaystyle{ \chi_{A} (x) = (+ \infty) \left( 1 - \mathbf{1}_{A} (x) \right). }[/math]

Subgradient

The subgradient of [math]\displaystyle{ \chi_{A} (x) }[/math] for a set [math]\displaystyle{ A }[/math] is the tangent cone of that set in [math]\displaystyle{ x }[/math].

Bibliography

• Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.