Convex conjugate

{{Short description|Generalization of the Legendre transformation}}

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). The convex conjugate is widely used for constructing the dual problem in optimization theory, thus generalizing Lagrangian duality.

Definition

Let $X$ be a real topological vector space and let $X^{*}$ be the dual space to $X$ . Denote by

: $\langle \cdot , \cdot \rangle : X^{*} \times X \to \mathbb{R}$

the canonical dual pairing, which is defined by $\left\langle x^*, x \right\rangle \mapsto x^* (x).$

For a function $f : X \to \mathbb{R} \cup \{ - \infty, + \infty \}$ taking values on the extended real number line, its {{em|convex conjugate}} is the function

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

whose value at $x^* \in X^{*}$ is defined to be the supremum:

: $f^{*} \left( x^{*} \right) := \sup \left\{ \left\langle x^{*}, x \right\rangle - f (x) ~\colon~ x \in X \right\},$

or, equivalently, in terms of the infimum:

: $f^{*} \left( x^{*} \right) := - \inf \left\{ f (x) - \left\langle x^{*}, x \right\rangle ~\colon~ x \in X \right\}.$

This definition can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes.{{cite web|url=https://physics.stackexchange.com/a/9360/821 |title=Legendre Transform |accessdate=April 14, 2019}}

Examples

For more examples, see {{Section link||Table of selected convex conjugates}}.

The convex conjugate of an affine function $f(x) = \left\langle a, x \right\rangle - b$ is $f^{*}\left(x^{*} \right)$

= \begin{cases} b, & x^{*} = a \\ +\infty, & x^{*} \ne a. \end{cases}

The convex conjugate of a power function $f(x) = \frac{1}{p}|x|^p, 1 < p < \infty$ is

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

The convex conjugate of the absolute value function $f(x) = \left| x \right|$ is

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

The convex conjugate of the exponential function $f(x)= e^x$ is

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)=

See [https://link.springer.com/article/10.1007/s10107-014-0801-1 this article for example.]

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

$f(x):= \int_{-\infty}^x F(u) \, du = \operatorname{E}\left[\max(0,x-X)\right] = x-\operatorname{E} \left[\min(x,X)\right]$

has the convex conjugate

$f^{*}(p)= \int_0^p F^{-1}(q) \, dq = (p-1)F^{-1}(p)+\operatorname{E}\left[\min(F^{-1}(p),X)\right]
= p F^{-1}(p)-\operatorname{E}\left[\max(0,F^{-1}(p)-X)\right].$

= Ordering =

A particular interpretation has the transform

$f^\text{inc}(x):= \arg \sup_t t\cdot x-\int_0^1 \max\{t-f(u),0\} \, du,$

as this is a nondecreasing rearrangement of the initial function f; in particular, $f^\text{inc}= f$ for f nondecreasing.

Properties

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=

Declare that $f \le g$ if and only if $f(x) \le g(x)$ for all $x.$ Then convex-conjugation is order-reversing, which by definition means that if $f \le g$ then $f^* \ge g^*.$

For a family of functions $\left(f_\alpha\right)_\alpha$ it follows from the fact that supremums may be interchanged that

: $\left(\inf_\alpha f_\alpha\right)^*(x^*) = \sup_\alpha f_\alpha^*(x^*),$

and from the max–min inequality that

: $\left(\sup_\alpha f_\alpha\right)^*(x^*) \le \inf_\alpha f_\alpha^*(x^*).$

= Biconjugate =

The convex conjugate of a function is always lower semi-continuous. The biconjugate $f^{**}$ (the convex conjugate of the convex conjugate) is also the closed convex hull, i.e. the largest lower semi-continuous convex function with $f^{**} \le f.$

For proper functions $f,$

: $f = f^{**}$ if and only if $f$ is convex and lower semi-continuous, by the Fenchel–Moreau theorem.

= Fenchel's inequality =

For any function {{mvar|f}} and its convex conjugate {{math|f *}}, Fenchel's inequality (also known as the Fenchel–Young inequality) holds for every $x \in X$ and {{nowrap| $p \in X^{*}$ :}}

: $\left\langle p,x \right\rangle \le f(x) + f^*(p).$

Furthermore, the equality holds only when $p \in \partial f(x)$ .

The proof follows from the definition of convex conjugate: $f^*(p) = \sup_{\tilde x} \left\{ \langle p,\tilde x \rangle - f(\tilde x) \right\} \ge \langle p,x \rangle - f(x).$

= Convexity =

For two functions $f_0$ and $f_1$ and a number $0 \le \lambda \le 1$ the convexity relation

: $\left((1-\lambda) f_0 + \lambda f_1\right)^{*} \le (1-\lambda) f_0^{*} + \lambda f_1^{*}$

holds. The ${*}$ operation is a convex mapping itself.

= Infimal convolution =

The infimal convolution (or epi-sum) of two functions $f$ and $g$ is defined as

: $\left( f \operatorname{\Box} g \right)(x) = \inf \left\{ f(x-y) + g(y) \mid y \in \mathbb{R}^n \right\}.$

Let $f_1, \ldots, f_{m}$ be proper, convex and lower semicontinuous functions on $\mathbb{R}^{n}.$ Then the infimal convolution is convex and lower semicontinuous (but not necessarily proper),{{cite book |last=Phelps |first=Robert |authorlink=Robert R. Phelps |title=Convex Functions, Monotone Operators and Differentiability|url=https://archive.org/details/convexfunctionsm00phel |url-access=limited | edition=2 |year=1993|publisher=Springer |isbn= 0-387-56715-1|page= [https://archive.org/details/convexfunctionsm00phel/page/n50 42]}} and satisfies

: $\left( f_1 \operatorname{\Box} \cdots \operatorname{\Box} f_m \right)^{*} = f_1^{*} + \cdots + f_m^{*}.$

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.{{cite journal |doi=10.1137/070687542 |title=The Proximal Average: Basic Theory |year=2008 |last1=Bauschke |first1=Heinz H. |last2=Goebel |first2=Rafal |last3=Lucet |first3=Yves |last4=Wang |first4=Xianfu |journal=SIAM Journal on Optimization |volume=19 |issue=2 |pages=766|citeseerx=10.1.1.546.4270 }}

= Maximizing argument =

If the function $f$ is differentiable, then its derivative is the maximizing argument in the computation of the convex conjugate:

: $f^\prime(x) = x^*(x):= \arg\sup_{x^{*}} {\langle x, x^{*}\rangle} -f^{*}\left( x^{*} \right)$ and

: $f^{{*}\prime}\left( x^{*} \right) = x\left( x^{*} \right):= \arg\sup_x {\langle x, x^{*}\rangle} - f(x);$

hence

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

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

and moreover

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

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

= Scaling properties =

If for some $\gamma>0,$ $g(x) = \alpha + \beta x + \gamma \cdot f\left( \lambda x + \delta \right)$ , then

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

= Behavior under linear transformations =

Let $A : X \to Y$ be a bounded linear operator. For any convex function $f$ on $X,$

: $\left(A f\right)^{*} = f^{*} A^{*}$

where

: $(A f)(y) = \inf\{ f(x) : x \in X , A x = y \}$

is the preimage of $f$ with respect to $A$ and $A^{*}$ is the adjoint operator of $A.$ Ioffe, A.D. and Tichomirov, V.M. (1979), Theorie der Extremalaufgaben. Deutscher Verlag der Wissenschaften. Satz 3.4.3

A closed convex function $f$ is symmetric with respect to a given set $G$ of orthogonal linear transformations,

: $f(A x) = f(x)$ for all $x$ and all $A \in G$

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.{{cite book |last1=Borwein |first1=Jonathan |authorlink1=Jonathan Borwein|last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples|url=https://archive.org/details/convexanalysisno00borw_812 |url-access=limited | edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1|pages=[https://archive.org/details/convexanalysisno00borw_812/page/n62 50]–51}}

$g(x)$	$\operatorname{dom}(g)$	$g^(x^)$	$\operatorname{dom}(g^*)$
class="wikitable"
$f(ax)$ (where $a \neq 0$ )	$X$	$f^\left(\frac{x^}{a}\right)$	$X^*$
$f(x + b)$	$X$	$f^(x^) - \langle b,x^* \rangle$	$X^*$
$a f(x)$ (where $a > 0$ )	$X$	$a f^\left(\frac{x^}{a}\right)$	$X^*$
$\alpha+ \beta x+ \gamma \cdot f(\lambda x+\delta)$	$X$	$-\alpha- \delta\frac{x^-\beta}\lambda+ \gamma \cdot f^ \left(\frac {x^*-\beta}{\gamma \lambda}\right)\quad (\gamma>0)$	$X^*$
$\frac{\|x\|^p}{p}$ (where $p > 1$ )	$\mathbb{R}$	$\frac{\|x^*\|^q}{q}$ (where $\frac{1}{p} + \frac{1}{q} = 1$ )	$\mathbb{R}$
$\frac{-x^p}{p}$ (where $0 < p < 1$ )	$\mathbb{R}_+$	$\frac{-(-x^*)^q}q$ (where $\frac 1 p + \frac 1 q = 1$ )	$\mathbb{R}_{--}$
$\sqrt{1 + x^2}$	$\mathbb{R}$	$-\sqrt{1 - (x^*)^2}$	$[-1,1]$
$-\log(x)$	$\mathbb{R}_{++}$	$-(1 + \log(-x^*))$	$\mathbb{R}_{--}$
$e^x$	$\mathbb{R}$	$\begin{cases}x^* \log(x^) - x^ & \text{if }x^* > 0\\ 0 & \text{if }x^* = 0\end{cases}$	$\mathbb{R}_{+}$
$\log\left(1 + e^x\right)$	$\mathbb{R}$	$\begin{cases}x^* \log(x^) + (1 - x^) \log(1 - x^) & \text{if }0 < x^ < 1\\ 0 & \text{if }x^* = 0,1\end{cases}$	$[0,1]$
$-\log\left(1 - e^x\right)$	$\mathbb{R}_{--}$	$\begin{cases}x^* \log(x^) - (1 + x^) \log(1 + x^) & \text{if }x^ > 0\\ 0 & \text{if }x^* = 0\end{cases}$	$\mathbb{R}_+$

References

{{cite book

| authorlink=Vladimir Igorevich Arnol'd

| last=Arnol'd

| first=Vladimir Igorevich

| title=Mathematical Methods of Classical Mechanics

| edition=Second

| publisher=Springer

| year=1989

| isbn=0-387-96890-3

| mr=997295

| url-access=registration

| url=https://archive.org/details/mathematicalmeth0000arno

}}

{{Rockafellar Wets Variational Analysis 2009 Springer}}
{{cite book

| last = Rockafellar

| first = R. Tyrell

| authorlink = R. Tyrrell Rockafellar

| title = Convex Analysis

| publisher = Princeton University Press

| year = 1970

| location = Princeton

| isbn=0-691-01586-4

| mr = 0274683

}}