Closed convex function

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

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  • 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 hf (called the affine minorants of f).

References

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  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.