Convex functional

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A functional, defined on a convex subset of a linear vector space, the supergraph of which is a convex set. A functional which does not assume the value on a convex set is convex on if and only if the inequality

is satisfied. If the inequality sign is reversed, the functional is called concave. Operations which convert a convex functional into a convex functional include addition , multiplication by a positive number, taking the upper bound

and the infimal convolution

A convex functional bounded from above in a neighbourhood of some point is continuous at that point. If a convex functional is finite at some point , it has a (finite or infinite) derivative in any direction at that point. Closed convex functionals (i.e. functionals with convex and closed supergraphs) in locally convex linear topological spaces may be described in a dual way: Such a functional is the least upper bound of the affine functions which it dominates. This duality makes it possible to relate to each convex functional a dual object, the conjugate functional

Properties of convex functionals, operations on such functionals, and the relationship between a convex functional and its conjugate, are studied in convex analysis.


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How to Cite This Entry:
Convex functional. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article