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 $X$ be a set, and let $A$ be a subset of $X$. The characteristic function of $A$ is the function

${\chi }_A:X\to \mathbb{ℝ}\cup \{+\mathrm{\infty }\}$

taking values in the extended real number line defined by

${\chi }_A(x):=\{\begin{array}{ll}0,& x\in A;\\ +\mathrm{\infty },& x\in ̸A.\end{array}$

Relationship with the indicator function

Let ${\mathbf{𝟏}}_A:X\to \mathbb{ℝ}$ denote the usual indicator function:

${\mathbf{𝟏}}_A(x):=\{\begin{array}{ll}1,& x\in A;\\ 0,& x\in ̸A.\end{array}$

If one adopts the conventions that

• for any $a\in \mathbb{ℝ}\cup \{+\mathrm{\infty }\}$, $a+(+\mathrm{\infty })=+\mathrm{\infty }$ and $a(+\mathrm{\infty })=+\mathrm{\infty }$, except $0(+\mathrm{\infty })=0$;
• $\frac{1}{0}=+\mathrm{\infty }$; and
• $\frac{1}{+\mathrm{\infty }}=0$;

then the indicator and characteristic functions are related by the equations

${\mathbf{𝟏}}_A(x)=\frac{1}{1+{\chi }_A(x)}$

and

${\chi }_A(x)=(+\mathrm{\infty })(1-{\mathbf{𝟏}}_A(x)).$

Subgradient

The subgradient of ${\chi }_A(x)$ for a set $A$ is the tangent cone of that set in $x$.

Bibliography

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