Closed convex function
Terms in Maths
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
- 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
- Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.
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Convex analysis and variational analysis
- Convex combination
- Convex function
- Convex set
- Convex hull
- (Orthogonally, Pseudo-) Convex set
- Effective domain
- Epigraph
- Hypograph
- John ellipsoid
- Lens
- Radial set/Algebraic interior
- Zonotope
- Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
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