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 $0$ 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

ι_{A} : X \to ℝ \cup {+ \infty}

taking values in the extended real number line defined by

ι_{A} (x) := {\begin{matrix} 0, & x \in A; \\ + \infty, & x \in̸ A . \end{matrix}

Properties

This function is convex if and only if the set $A$ is convex.

This function is lower-semicontinuous if and only if the set $A$ is closed.

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

For an arbitrary non-empty set its Legendre transform is the support function.

The subgradient of $ι_{A} (x)$ for a set $A$ and $x \in A$ is the normal cone of that set at $x$ .

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

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.
1 2 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] 1 2 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.