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

after-content-x4

${displaystyle langle cdot ,cdot rangle :X^{*}times Xto mathbb {R} }$

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

${displaystyle f^{*}:X^{*}to mathbb {R} cup {-infty ,+infty }}$

whose value at

${displaystyle x^{*}in X^{*}}$

is defined to be the supremum:

${displaystyle f^{*}left(x^{*}right):=sup left{leftlangle x^{*},xrightrangle -f(x)~colon ~xin Xright},}$

or, equivalently, in terms of the infimum:

${displaystyle f^{*}left(x^{*}right):=-inf left{f(x)-leftlangle x^{*},xrightrangle ~colon ~xin Xright}.}$

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
  ${displaystyle f(x)=leftlangle a,xrightrangle -b}$

  is

  $${displaystyle f^{*}left(x^{*}right)={begin{cases}b,&x^{*}=a\+infty ,&x^{*}neq a.end{cases}}}$$

  

• The convex conjugate of a power function
  ${displaystyle f(x)={frac {1}{p}}|x|^{p},1$

  is

  $${displaystyle f^{*}left(x^{*}right)={frac {1}{q}}|x^{*}|^{q},1$$

  

• The convex conjugate of the absolute value function
  ${displaystyle f(x)=left|xright|}$

  is

  $${displaystyle f^{*}left(x^{*}right)={begin{cases}0,&left|x^{*}right|leq 1\infty ,&left|x^{*}right|>1.end{cases}}}$$

  
  ${displaystyle f(x)=e^{x}}$

  is

  $${displaystyle f^{*}left(x^{*}right)={begin{cases}x^{*}ln x^{*}-x^{*},&x^{*}>0\0,&x^{*}=0\infty ,&x^{*}<0.end{cases}}}$$

  

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]

See this article for example.

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,

${displaystyle f^{text{inc}}=f}$

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

${displaystyle left(inf _{alpha }f_{alpha }right)^{*}(x^{*})=sup _{alpha }f_{alpha }^{*}(x^{*}),}$

and from the max–min inequality that

${displaystyle left(sup _{alpha }f_{alpha }right)^{*}(x^{*})leq inf _{alpha }f_{alpha }^{*}(x^{*}).}$

Biconjugate[edit]

The convex conjugate of a function is always lower semi-continuous. The biconjugate

${displaystyle f^{**}}$

(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

${displaystyle f,}$

${displaystyle f=f^{**}}$

if and only if

${displaystyle f}$

is convex and lower semi-continuous, by the Fenchel–Moreau theorem.

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

${displaystyle xin X}$

and

${displaystyle pin X^{*}}$

:

${displaystyle leftlangle p,xrightrangle leq f(x)+f^{*}(p).}$

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

${displaystyle left((1-lambda )f_{0}+lambda f_{1}right)^{*}leq (1-lambda )f_{0}^{*}+lambda f_{1}^{*}}$

holds. The

${displaystyle {*}}$

operation is a convex mapping itself.

Infimal convolution[edit]

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

${displaystyle f}$

and

${displaystyle g}$

is defined as

${displaystyle left(foperatorname {Box } gright)(x)=inf left{f(x-y)+g(y)mid yin mathbb {R} ^{n}right}.}$

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

${displaystyle left(f_{1}operatorname {Box } cdots operatorname {Box } f_{m}right)^{*}=f_{1}^{*}+cdots +f_{m}^{*}.}$

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:

${displaystyle f^{prime }(x)=x^{*}(x):=arg sup _{x^{*}}{langle x,x^{*}rangle }-f^{*}left(x^{*}right)}$

and

${displaystyle f^{{*}prime }left(x^{*}right)=xleft(x^{*}right):=arg sup _{x}{langle x,x^{*}rangle }-f(x);}$

whence

${displaystyle x=nabla f^{*}left(nabla f(x)right),}$

${displaystyle x^{*}=nabla fleft(nabla f^{*}left(x^{*}right)right),}$

and moreover

${displaystyle f^{prime prime }(x)cdot f^{{*}prime prime }left(x^{*}(x)right)=1,}$

${displaystyle f^{{*}prime prime }left(x^{*}right)cdot f^{prime prime }left(x(x^{*})right)=1.}$

Scaling properties[edit]

If for some

${displaystyle gamma >0,}$

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

, then

${displaystyle g^{*}left(x^{*}right)=-alpha -delta {frac {x^{*}-beta }{lambda }}+gamma cdot f^{*}left({frac {x^{*}-beta }{lambda gamma }}right).}$

Behavior under linear transformations[edit]

Let

${displaystyle A:Xto Y}$

be a bounded linear operator. For any convex function

${displaystyle f}$

on

${displaystyle X,}$

${displaystyle left(Afright)^{*}=f^{*}A^{*}}$

where

${displaystyle (Af)(y)=inf{f(x):xin X,Ax=y}}$

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,

${displaystyle f(Ax)=f(x)}$

for all

${displaystyle x}$

and all

${displaystyle Ain G}$

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]

Further reading[edit]