Indicator function (convex analysis)

In the field of mathematics known as convex analysis, the indicator 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 indicator function used in probability, but assigns ${\displaystyle +\infty }${\displaystyle +\infty } instead of ${\displaystyle 1}${\displaystyle 1} to the outside elements.

Each field seems to have its own meaning of an "indicator function", as in complex analysis for instance.

Let ${\displaystyle X}${\displaystyle X} be a set, and let ${\displaystyle A}${\displaystyle A} be a subset of ${\displaystyle X}${\displaystyle X}. The indicator function of ${\displaystyle A}${\displaystyle A} is the function ^{[1] [2] [3] [4]}

${\displaystyle \iota _{A}:X\to \mathbb {R} \cup \{+\infty \}}${\displaystyle \iota _{A}:X\to \mathbb {R} \cup \{+\infty \}}

taking values in the extended real number line defined by

${\displaystyle \iota _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}${\displaystyle \iota _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}

This function is convex if and only if the set ${\displaystyle A}${\displaystyle A} is convex.^[5]

This function is lower-semicontinuous if and only if the set ${\displaystyle A}${\displaystyle A} is closed.^[4]

For any arbitrary sets ${\displaystyle A}${\displaystyle A} and ${\displaystyle B}${\displaystyle B}, it is that ${\displaystyle \iota _{A}+\iota _{B}=\iota _{A\cap B}}${\displaystyle \iota _{A}+\iota _{B}=\iota _{A\cap B}}.

For an arbitrary non-empty set its Legendre transform is the support function.^[6]

The subgradient of ${\displaystyle \iota _{A}(x)}${\displaystyle \iota _{A}(x)} for a set ${\displaystyle A}${\displaystyle A} and ${\displaystyle x\in A}${\displaystyle x\in A} is the normal cone of that set at ${\displaystyle x}${\displaystyle x}.^[7]

Its infimal convolution with the Euclidean norm ${\displaystyle ||\cdot ||_{2}}${\displaystyle ||\cdot ||_{2}} is the Euclidean distance to that set.^[8]

• Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.
• Hiriart-Urruty, J. B.; Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms I & II. Springer-Verlag.
• Boyd, S. P.; Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
• Bauschke, H. H.; Combettes, P. L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer.

• Convex analysis