In mathematics, a function is said to be closed if for each , the sublevel set is a closed set.
Equivalently, if the epigraph defined by is closed, then the function 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]
Properties
[edit]- If is a continuous function and is closed, then is closed.
- If is a continuous function and is open, then is closed if and only if it converges to along every sequence converging to a boundary point of .[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
[edit]- ^ Convex Optimization Theory. Athena Scientific. 2009. pp. 10, 11. ISBN 978-1886529311.
- ^ 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.