Closed convex function

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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.[citation needed]

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 hf (called the affine minorants of f).

References[edit]

  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.