Indicator function (convex analysis)

Definition

Let X be a set, and let A be a subset of X. The indicator function of A is the function

:\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.

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

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.

The subgradient of \iota_{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

Bibliography

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References

1. R. T. Rockafellar, '' Convex Analysis'', Princeton University Press, (1997) [1970], p.28.
2. J. B. Hiriart-Urruty, C. Lemaréchal, ''Convex Analysis and Optimization I'', Springer-Verlag, 1993, p.152.
3. S. Boyd, L. Vandenberghe, ''Convex Optimization'', Cambridge University Press, (2009) [2004], p.68.
4. H. H. Bauschke, P. L. Combettes, ''Convex Analysis and Monotone Operator Theory in Hilbert Spaces'', Springer (2017) [2011], p.12.
5. H. H. Bauschke, P. L. Combettes, ''Convex Analysis and Monotone Operator Theory in Hilbert Spaces'', Springer (2017) [2011], p.139.
6. J. B. Hiriart-Urruty, C. Lemaréchal, ''Convex Analysis and Optimization II'', Springer-Verlag, 1993, p.39.
7. H. H. Bauschke, P. L. Combettes, ''Convex Analysis and Monotone Operator Theory in Hilbert Spaces'', Springer (2017) [2011], p.267.
8. J. B. Hiriart-Urruty, C. Lemaréchal, ''Convex Analysis and Optimization II'', Springer-Verlag, 1993, p.65.