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 $+\infty$ instead of $1$ to the outside elements.

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

Definition

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

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

taking values in the extended real number line defined by

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

Properties

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

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

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

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

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

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

References

^ R. T. Rockafellar, Convex Analysis, Princeton University Press, (1997) [1970], p.28.
^ J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization I, Springer-Verlag, 1993, p.152.
^ S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, (2009) [2004], p.68.
^ ^a ^b H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) [2011], p.12.
^ H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) [2011], p.139.
^ J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization II, Springer-Verlag, 1993, p.39.
^ H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) [2011], p.267.
^ J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization II, Springer-Verlag, 1993, p.65.

Bibliography

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.

[rocka1-1] R. T. Rockafellar, Convex Analysis, Princeton University Press, (1997) [1970], p.28.

[jbhu1-2] J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization I, Springer-Verlag, 1993, p.152.

[boyd1-3] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, (2009) [2004], p.68.

[bc1-4] H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) [2011], p.12.

[bc2-5] H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) [2011], p.139.

[jbhu2-6] J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization II, Springer-Verlag, 1993, p.39.

[bc3-7] H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) [2011], p.267.

[jbhu3-8] J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization II, Springer-Verlag, 1993, p.65.

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