Closed convex function
In mathematics, a function [math]\displaystyle{ f: \mathbb{R}^n \rightarrow \mathbb{R} }[/math] is said to be closed if for each [math]\displaystyle{ \alpha \in \mathbb{R} }[/math], the sublevel set [math]\displaystyle{ \{ x \in \mbox{dom} f \vert f(x) \leq \alpha \} }[/math] is a closed set.
Equivalently, if the epigraph defined by [math]\displaystyle{ \mbox{epi} f = \{ (x,t) \in \mathbb{R}^{n+1} \vert x \in \mbox{dom} f,\; f(x) \leq t\} }[/math] is closed, then the function [math]\displaystyle{ f }[/math] is closed.
This definition is valid for any function, but most used for convex functions. A proper convex function is closed if and only if it is lower semi-continuous.[1] For a convex function that is not proper, there is disagreement as to the definition of the closure of the function.[citation needed]
Properties
- If [math]\displaystyle{ f: \mathbb{R}^n \rightarrow \mathbb{R} }[/math] is a continuous function and [math]\displaystyle{ \mbox{dom} f }[/math] is closed, then [math]\displaystyle{ f }[/math] is closed.
- If [math]\displaystyle{ f: \mathbb R^n \rightarrow \mathbb R }[/math] is a continuous function and [math]\displaystyle{ \mbox{dom} f }[/math] is open, then [math]\displaystyle{ f }[/math] is closed if and only if it converges to [math]\displaystyle{ \infty }[/math] along every sequence converging to a boundary point of [math]\displaystyle{ \mbox{dom} f }[/math].[2]
- A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that h ≤ f (called the affine minorants of f).
References
- ↑ Convex Optimization Theory. Athena Scientific. 2009. pp. 10, 11. ISBN 978-1886529311.
- ↑ Boyd, Stephen; Vandenberghe, Lieven (2004). Convex optimization. New York: Cambridge. pp. 639–640. ISBN 978-0521833783. https://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf.
- Rockafellar, R. Tyrrell (1997). Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.
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