Closed convex function
In mathematics, a function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is said to be closed if for each α ∈ R {\displaystyle \alpha \in \mathbb {R} } , the sublevel set { x ∈ dom f | f ( x ) ≤ α } {\displaystyle \{x\in {\mbox{dom}}f\vert f(x)\leq \alpha \}} is a closed set. Equivalently, if the epigraph defined by epi f = { ( x , t ) ∈ R n + 1 | x ∈ dom f , f ( x ) ≤ t } {\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 f {\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.
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.