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
{\displaystyle X}
be a set , and let
A
{\displaystyle A}
be a subset of
X
{\displaystyle X}
. The characteristic function of
A
{\displaystyle A}
is the function
χ
A
:
X
→
R
∪
{
+
∞
}
{\displaystyle \chi _{A}:X\to \mathbb {R} \cup \{+\infty \}
taking values in the extended real number line defined by
χ
A
(
x
)
:=
{
0
,
x
∈
A
;
+
∞
,
x
∉
A
.
{\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}
Relationship with the indicator function
Let
1
A
:
X
→
R
{\displaystyle \mathbf {1} _{A}:X\to \mathbb {R} }
denote the usual indicator function:
1
A
(
x
)
:=
{
1
,
x
∈
A
;
0
,
x
∉
A
.
{\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1,&x\in A;\\0,&x\not \in A.\end{cases}
If one adopts the conventions that
for any
a
∈
R
∪
{
+
∞
}
{\displaystyle a\in \mathbb {R} \cup \{+\infty \}
,
a
+
(
+
∞
)
=
+
∞
{\displaystyle a+(+\infty )=+\infty }
and
a
(
+
∞
)
=
+
∞
{\displaystyle a(+\infty )=+\infty }
, except
0
(
+
∞
)
=
0
{\displaystyle 0(+\infty )=0}
;
1
0
=
+
∞
{\displaystyle {\frac {1}{0}=+\infty }
; and
1
+
∞
=
0
{\displaystyle {\frac {1}{+\infty }=0}
;
then the indicator and characteristic functions are related by the equations
1
A
(
x
)
=
1
1
+
χ
A
(
x
)
{\displaystyle \mathbf {1} _{A}(x)={\frac {1}{1+\chi _{A}(x)}
and
χ
A
(
x
)
=
(
+
∞
)
(
1
−
1
A
(
x
)
)
.
{\displaystyle \chi _{A}(x)=(+\infty )\left(1-\mathbf {1} _{A}(x)\right).}
Subgradient
The subgradient of
χ
A
(
x
)
{\displaystyle \chi _{A}(x)}
for a set
A
{\displaystyle A}
is the tangent cone of that set in
x
{\displaystyle x}
.
Bibliography
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