Characteristic function (convex analysis)
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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 be a set, and let be a subset of . The characteristic function of is the function
taking values in the extended real number line defined by
Relationship with the indicator function[]
Let denote the usual indicator function:
If one adopts the conventions that
- for any , and , except ;
- ; and
- ;
then the indicator and characteristic functions are related by the equations
and
Subgradient[]
The subgradient of for a set is the tangent cone of that set in .
Bibliography[]
- Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.
- Convex analysis