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 or Fenchel transformation (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality.
Let be a real topological vector space, and let be the dual space to . Denote the dual pairing by
For a function
taking values on the extended real number line, the convex conjugate
is defined in terms of the supremum by
or, equivalently, in terms of the infimum by
This definition can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes. 
The convex conjugate of an affine function
The convex conjugate of a power function
The convex conjugate of the absolute value function
The convex conjugate of the exponential function is
Convex conjugate and Legendre transform of the exponential function agree except that the domain of the convex conjugate is strictly larger as the Legendre transform is only defined for positive real numbers.
Connection with expected shortfall (average value at risk)
Let F denote a cumulative distribution function of a random variable X. Then (integrating by parts),
has the convex conjugate
A particular interpretation has the transform
as this is a nondecreasing rearrangement of the initial function f; in particular, for ƒ nondecreasing.
The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.
Convex-conjugation is order-reversing: if then . Here
For a family of functions it follows from the fact that supremums may be interchanged that
and from the max–min inequality that
The convex conjugate of a function is always lower semi-continuous. The biconjugate (the convex conjugate of the convex conjugate) is also the closed convex hull, i.e. the largest lower semi-continuous convex function with . For proper functions f,
- if and only if f is convex and lower semi-continuous by Fenchel–Moreau theorem.
For any function f and its convex conjugate f *, Fenchel's inequality (also known as the Fenchel–Young inequality) holds for every x ∈ X and p ∈ X * :
The proof follows immediately from the definition of convex conjugate: .
For two functions and and a number the convexity relation
holds. The operation is a convex mapping itself.
The infimal convolution (or epi-sum) of two functions f and g is defined as
Let f1, …, fm be proper, convex and lower semicontinuous functions on Rn. Then the infimal convolution is convex and lower semicontinuous (but not necessarily proper), and satisfies
The infimal convolution of two functions has a geometric interpretation: The (strict) epigraph of the infimal convolution of two functions is the Minkowski sum of the (strict) epigraphs of those functions.
If the function is differentiable, then its derivative is the maximizing argument in the computation of the convex conjugate:
If, for some , , then
In case of an additional parameter (α, say) moreover
where is chosen to be the maximizing argument.
Behavior under linear transformations
Let A be a bounded linear operator from X to Y. For any convex function f on X, one has
is the preimage of f w.r.t. A and A* is the adjoint operator of A.
A closed convex function f is symmetric with respect to a given set G of orthogonal linear transformations,
if and only if its convex conjugate f* is symmetric with respect to G.
Table of selected convex conjugates
The following table provides Legendre transforms for many common functions as well as a few useful properties.
|(where )||(where )|
|(where )||(where )|
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