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_ : X \to \mathbb \cup \$

taking values in the extended real number line defined by

:$\chi_ (x) := \begin 0, & x \in A; \\ + \infty, & x \not \in A. \end$

Relationship with the indicator function

Let $\mathbf_ : X \to \mathbb$ denote the usual indicator function:

:$\mathbf_ (x) := \begin 1, & x \in A; \\ 0, & x \not \in A. \end$

If one adopts the conventions that
* for any $a \in \mathbb \cup \$, $a + (+ \infty) = + \infty$ and $a (+\infty) = + \infty$, except $0(+\infty)=0$;
* $\frac = + \infty$; and
* $\frac = 0$;

then the indicator and characteristic functions are related by the equations

:$\mathbf_ (x) = \frac$

and

:$\chi_ (x) = (+ \infty) \left( 1 - \mathbf_ (x) \right).$

Subgradient

The subgradient of $\chi_ (x)$ for a set $A$ is the tangent cone of that set in $x$.

Bibliography

* {{cite book
, last = Rockafellar
, first = R. T.
, authorlink = R. Tyrrell Rockafellar
, title = Convex Analysis
, publisher = Princeton University Press
, location = Princeton, NJ
, year = 1997
, origyear = 1970
, isbn = 978-0-691-01586-6

Convex analysis