closed convex function

In mathematics, a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is said to be closed if for each $\alpha \in \mathbb{R}$ , the sublevel set

$\{ x \in \mbox{dom} f \vert f(x) \leq \alpha \}$

Equivalently, if the epigraph defined by

$\mbox{epi} f = \{ (x,t) \in \mathbb{R}^{n+1} \vert x \in \mbox{dom} f,\; f(x) \leq t\}$

is closed, then the function $f$ 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.{{Cite book|title = Convex Optimization Theory|publisher = Athena Scientific|year = 2009|isbn = 978-1886529311|pages=10, 11 }}

Properties

If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a continuous function and $\mbox{dom} f$ is closed, then $f$ is closed.
If $f: \mathbb R^n \rightarrow \mathbb R$ is a continuous function and $\mbox{dom} f$ is open, then $f$ is closed if and only if it converges to $\infty$ along every sequence converging to a boundary point of $\mbox{dom} f$ .{{cite book|last1=Boyd|first1=Stephen|first2=Lieven|last2=Vandenberghe|title=Convex optimization|date=2004|publisher=Cambridge|location=New York|isbn=978-0521833783|pages=639–640|url=https://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf}}
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).

{{cite book|author=Rockafellar, R. Tyrrell|author-link=Rockafellar, R. Tyrrell|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|orig-year=1970|isbn=978-0-691-01586-6}}