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] For a convex function which is not proper there is disagreement as to the definition of the closure of the function.

Properties

References

  1. Convex Optimization Theory. Athena Scientific. 2009. pp. 10, 11. ISBN 978-1886529311.


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