Editing Convex conjugate (section)

== 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 [[Polyhedron|polyhedral]] [[Epigraph (mathematics)|epigraph]]) is again a polyhedral convex function.

=== Order reversing===

Declare that <math>f \le g</math> if and only if <math>f(x) \le g(x)</math> for all <math>x.</math> Then convex-conjugation is [[Order theory|order-reversing]], which by definition means that if <math>f \le g</math> then <math>f^* \ge g^*.</math> 

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

:<math>\left(\inf_\alpha f_\alpha\right)^*(x^*) = \sup_\alpha f_\alpha^*(x^*),</math>

and from the [[max–min inequality]] that 

:<math>\left(\sup_\alpha f_\alpha\right)^*(x^*) \le \inf_\alpha f_\alpha^*(x^*).</math>

=== Biconjugate ===
The convex conjugate of a function is always [[lower semi-continuous]]. The '''biconjugate''' <math>f^{**}</math> (the convex conjugate of the convex conjugate) is also the [[closed convex hull]], i.e. the largest [[lower semi-continuous]] convex function with <math>f^{**} \le f.</math> 
For [[Proper convex function|proper functions]] <math>f,</math> 

:<math>f = f^{**}</math> [[if and only if]] <math>f</math> 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 <math>x \in X</math>  and {{nowrap|<math>p \in X^{*}</math>:}}

:<math>\left\langle p,x \right\rangle \le f(x) + f^*(p).</math>

Furthermore, the equality holds only when <math>p \in \partial f(x)</math>.
The proof follows from the definition of convex conjugate: <math>f^*(p) = \sup_{\tilde x} \left\{ \langle p,\tilde x \rangle - f(\tilde x) \right\} \ge \langle p,x \rangle - f(x).</math>

=== Convexity ===
For two functions <math>f_0</math> and <math>f_1</math> and a number <math>0 \le \lambda \le 1</math> the convexity relation 

:<math>\left((1-\lambda) f_0 + \lambda f_1\right)^{*} \le (1-\lambda) f_0^{*} + \lambda f_1^{*}</math>

holds. The <math>{*}</math> operation is a convex mapping itself.

=== Infimal convolution ===
The '''infimal convolution''' (or epi-sum) of two functions <math>f</math> and <math>g</math> is defined as

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

Let <math>f_1, \ldots, f_{m}</math> be [[Proper convex function|proper]], convex and [[Semi-continuity|lower semicontinuous]] functions on <math>\mathbb{R}^{n}.</math> Then the infimal convolution is convex and lower semicontinuous (but not necessarily proper),<ref>{{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]}}</ref> and satisfies

:<math>\left( f_1 \operatorname{\Box} \cdots \operatorname{\Box} f_m \right)^{*} = f_1^{*} + \cdots + f_m^{*}.</math>

The infimal convolution of two functions has a geometric interpretation:  The (strict) [[epigraph (mathematics)|epigraph]] of the infimal convolution of two functions is the [[Minkowski sum]] of the (strict) epigraphs of those functions.<ref>{{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 }}</ref>

=== Maximizing argument ===
If the function <math>f</math> is differentiable, then its derivative is the maximizing argument in the computation of the convex conjugate:
:<math>f^\prime(x) = x^*(x):= \arg\sup_{x^{*}} {\langle x, x^{*}\rangle} -f^{*}\left( x^{*} \right)</math> and
:<math>f^{{*}\prime}\left( x^{*} \right) = x\left( x^{*} \right):= \arg\sup_x {\langle x, x^{*}\rangle} - f(x);</math>

hence
:<math>x = \nabla f^{{*}}\left( \nabla f(x) \right),</math>
:<math>x^{*} = \nabla f\left( \nabla f^{{*}}\left( x^{*} \right)\right),</math>

and moreover
:<math>f^{\prime\prime}(x) \cdot f^{{*}\prime\prime}\left( x^{*}(x) \right) = 1,</math>
:<math>f^{{*}\prime\prime}\left( x^{*} \right) \cdot f^{\prime\prime}\left( x(x^{*}) \right) = 1.</math>

=== Scaling properties ===
If for some <math>\gamma>0,</math> <math>g(x) = \alpha + \beta x + \gamma \cdot f\left( \lambda x + \delta \right)</math>, then 
:<math>g^{*}\left( x^{*} \right)= - \alpha - \delta\frac{x^{*}-\beta} \lambda + \gamma \cdot f^{*}\left(\frac {x^{*}-\beta}{\lambda \gamma}\right).</math>

=== Behavior under linear transformations ===
Let <math>A : X \to Y</math> be a [[bounded linear operator]]. For any convex function <math>f</math> on <math>X,</math> 

:<math>\left(A f\right)^{*} = f^{*} A^{*}</math>

where

:<math>(A f)(y) = \inf\{ f(x) : x \in X , A x = y \}</math>

is the preimage of <math>f</math> with respect to <math>A</math> and <math>A^{*}</math> is the [[adjoint operator]] of <math>A.</math><ref>Ioffe, A.D. and Tichomirov, V.M. (1979), ''Theorie der Extremalaufgaben''. [[Deutscher Verlag der Wissenschaften]]. Satz 3.4.3</ref>

A closed convex function <math>f</math> is symmetric with respect to a given set <math>G</math> of [[orthogonal matrix|orthogonal linear transformation]]s,

:<math>f(A x) = f(x)</math> for all <math>x</math> and all <math>A \in G</math>

if and only if its convex conjugate <math>f^{*}</math> is symmetric with respect to <math>G.</math>