3.7 KB · ⇊ 3 parts · ⤓ Micron 3.7 KB · ⤓ HTML 30.4 KB

top

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

• Relationship with the indicator function

• Subgradient

• Bibliography

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 χ A : X → → R ∪ ∪ { + ∞ ∞ } {\displaystyle \chi _{A}:X\to \mathbb {R} \cup \{+\infty \}}

taking values in the extended real number line defined by

χ 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}}}

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 χ A ( x ) {\displaystyle \mathbf {1} _{A}(x)={\frac {1}{1+\chi _{A}(x)}}}

and

χ χ A ( x ) = ( + ∞ ∞ ) ( 1 − − 1 A ( x ) ) . {\displaystyle \chi _{A}(x)=(+\infty )\left(1-\mathbf {1} _{A}(x)\right).}

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

• citerefrockafellar1997Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.