Closed convex function

In mathematics, a function ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ is said to be closed if for each ${\displaystyle \alpha \in \mathbb {R} }$, the sublevel set ${\displaystyle \{x\in {\mbox{dom}}f\vert f(x)\leq \alpha \}}$ is a closed set.

Equivalently, if the epigraph defined by ${\displaystyle {\mbox{epi}}f=\{(x,t)\in \mathbb {R} ^{n+1}\vert x\in {\mbox{dom}}f,\;f(x)\leq t\}}$ is closed, then the function ${\displaystyle f}$ 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.[1] For a convex function which is not proper there is disagreement as to the definition of the closure of the function.

Properties

• If ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ is a continuous function and ${\displaystyle {\mbox{dom}}f}$ is closed, then ${\displaystyle f}$ is closed.
• If ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ is a continuous function and ${\displaystyle {\mbox{dom}}f}$ is open, then ${\displaystyle f}$ is closed if and only if it converges to ${\displaystyle \infty }$ along every sequence converging to a boundary point of ${\displaystyle {\mbox{dom}}f}$[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

1. Convex Optimization Theory. Athena Scientific. 2009. pp. 10, 11. ISBN 978-1886529311.
2. Boyd, Stephen; Vandenberghe, Lieven (2004). Convex optimization (PDF). New York: Cambridge. pp. 639–640. ISBN 978-0521833783.

• Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.