Closed convex function explained

f: R ^{n} → R is said to be closed if for each \alpha \in R , the sublevel set\{ x \in dom f \vert f(x) \leq \alpha \} is a closed set.

Equivalently, if the epigraph defined by

epi f = \{ (x,t) \in R ^{n+1} \vert x \in 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.^[1]

Properties

• If

f: R ^{n} → R is a continuous function and dom f is closed, then f is closed.

• If

f: R ^{n} → R is a continuous function and dom f is open, then f is closed if and only if it converges to inf ty along every sequence converging to a boundary point of 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

• Book: Rockafellar, R. Tyrrell. Rockafellar, R. Tyrrell. Convex Analysis. Princeton University Press. Princeton, NJ. 1997. 1970. 978-0-691-01586-6.

Notes and References

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