In
convex analysis, the Fenchel–Moreau theorem (named after
Werner Fenchel and
Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a
theorem which gives
necessary and sufficient conditions for a function to be equal to its
biconjugate
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformatio ...
. This is in contrast to the general property that for any function
.
This can be seen as a generalization of the
bipolar theorem.
It is used in
duality theory to prove
strong duality Strong duality is a condition in mathematical optimization in which the primal optimal objective and the dual optimal objective are equal. This is as opposed to weak duality (the primal problem has optimal value smaller than or equal to the dual ...
(via the
perturbation function In mathematical optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact that any such function defines a perturbation of the initial problem. In many cases this takes the form ...
).
Statement
Let
be a
Hausdorff locally convex space, for any
extended real valued function
it follows that
if and only if one of the following is true
#
is a
proper,
lower semi-continuous, and
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
,
#
, or
#
.
References
{{DEFAULTSORT:Fenchel-Moreau theorem
Convex analysis
Theorems in analysis
Theorems involving convexity