# Convex conjugate – Wikipedia

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 transformation**, or **Fenchel conjugate** (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality.

## Definition[edit]

Let

${displaystyle X}$be a real topological vector space and let

${displaystyle X^{*}}$be the dual space to

${displaystyle X}$. Denote by

the canonical dual pairing, which is defined by

${displaystyle left(x^{*},xright)mapsto x^{*}(x).}$

For a function

${displaystyle f:Xto mathbb {R} cup {-infty ,+infty }}$ taking values on the extended real number line, its ** convex conjugate** is the function

whose value at

${displaystyle x^{*}in X^{*}}$is defined to be the supremum:

or, equivalently, in terms of the infimum:

This definition can be interpreted as an encoding of the convex hull of the function’s epigraph in terms of its supporting hyperplanes.^{[1]}

## Examples[edit]

For more examples, see § Table of selected convex conjugates.

- 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 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)[edit]

Let *F* denote a cumulative distribution function of a random variable *X*. Then (integrating by parts),

has the convex conjugate

### Ordering[edit]

A particular interpretation has the transform

as this is a nondecreasing rearrangement of the initial function *f*; in particular,

for *f* nondecreasing.

## Properties[edit]

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.

### Order reversing[edit]

Declare that

${displaystyle fleq g}$if and only if

${displaystyle f(x)leq g(x)}$for all

${displaystyle x.}$Then convex-conjugation is order-reversing, which by definition means that if

${displaystyle fleq g}$then

${displaystyle f^{*}geq g^{*}.}$

For a family of functions

${displaystyle left(f_{alpha }right)_{alpha }}$it follows from the fact that supremums may be interchanged that

and from the max–min inequality that

### Biconjugate[edit]

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

${displaystyle f^{**}leq f.}$

For proper functions

### Fenchel’s inequality[edit]

For any function f and its convex conjugate *f* *, **Fenchel’s inequality** (also known as the **Fenchel–Young inequality**) holds for every

and

${displaystyle pin X^{*}}$:

The proof follows from the definition of convex conjugate:

${displaystyle f^{*}(p)=sup _{tilde {x}}left{langle p,{tilde {x}}rangle -f({tilde {x}})right}geq langle p,xrangle -f(x).}$

### Convexity[edit]

For two functions

${displaystyle f_{0}}$and

${displaystyle f_{1}}$and a number

${displaystyle 0leq lambda leq 1}$the convexity relation

holds. The

${displaystyle {*}}$operation is a convex mapping itself.

### Infimal convolution[edit]

The **infimal convolution** (or epi-sum) of two functions

and

${displaystyle g}$is defined as

Let

${displaystyle f_{1},ldots ,f_{m}}$be proper, convex and lower semicontinuous functions on

${displaystyle mathbb {R} ^{n}.}$ Then the infimal convolution is convex and lower semicontinuous (but not necessarily proper),^{[2]} 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.^{[3]}

### Maximizing argument[edit]

If the function

${displaystyle f}$is differentiable, then its derivative is the maximizing argument in the computation of the convex conjugate:

whence

and moreover

### Scaling properties[edit]

If for some

${displaystyle gamma >0,}$${displaystyle g(x)=alpha +beta x+gamma cdot fleft(lambda x+delta right)}$

, then

### Behavior under linear transformations[edit]

Let

${displaystyle A:Xto Y}$be a bounded linear operator. For any convex function

${displaystyle f}$on

${displaystyle X,}$

where

is the preimage of

${displaystyle f}$with respect to

${displaystyle A}$and

${displaystyle A^{*}}$is the adjoint operator of

${displaystyle A.}$^{[4]}

A closed convex function

${displaystyle f}$is symmetric with respect to a given set

${displaystyle G}$of orthogonal linear transformations,

if and only if its convex conjugate

${displaystyle f^{*}}$is symmetric with respect to

${displaystyle G.}$

## Table of selected convex conjugates[edit]

The following table provides Legendre transforms for many common functions as well as a few useful properties.^{[5]}

## See also[edit]

## References[edit]

**^**“Legendre Transform”. Retrieved April 14, 2019.**^**Phelps, Robert (1993).*Convex Functions, Monotone Operators and Differentiability*(2 ed.). Springer. p. 42. ISBN 0-387-56715-1.**^**Bauschke, Heinz H.; Goebel, Rafal; Lucet, Yves; Wang, Xianfu (2008). “The Proximal Average: Basic Theory”.*SIAM Journal on Optimization*.**19**(2): 766. CiteSeerX 10.1.1.546.4270. doi:10.1137/070687542.**^**Ioffe, A.D. and Tichomirov, V.M. (1979),*Theorie der Extremalaufgaben*. Deutscher Verlag der Wissenschaften. Satz 3.4.3**^**Borwein, Jonathan; Lewis, Adrian (2006).*Convex Analysis and Nonlinear Optimization: Theory and Examples*(2 ed.). Springer. pp. 50–51. ISBN 978-0-387-29570-1.

## Further reading[edit]

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