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Convex conjugate

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In mathematics, convex conjugation is a generalization of the Legendre transformation. It is also known as Legendre-Fenchel transformation or Fenchel transformation.

1 Definition
2 Examples
3 Properties
4 References
5 External links

Definition

Let <math>X</math> be a real normed vector space, and let <math>X^{*}</math> be the dual space to <math>X</math>. Denote the dual pairing by

<math>\langle \cdot , \cdot \rangle : X^{*} \times X \to \mathbb{R}</math>.

For a function

<math>f : X \to \mathbb{R} \cup \{ + \infty \}</math>

taking values on the extended real number line the convex conjugate

<math>f^\star : X^{*} \to \mathbb{R} \cup \{ + \infty \}</math>

is defined by

<math>f^{\star} \left( x^{*} \right) := \sup \left \{ \left. \left\langle x^{*} , x \right\rangle - f \left( x \right) \right| x \in X \right\},</math>

or, equivalently, by

<math>f^{\star} \left( x^{*} \right) := - \inf \left \{ \left. f \left( x \right) - \left\langle x^{*} , x \right\rangle \right| x \in X \right\}.</math>

Examples

The convex conjugate of an affine function

<math>

f(x) = \left\langle a,x \right\rangle - b,\, a \in \mathbb{R}^n, b \in \mathbb{R} </math> is

<math>

f^\star\left(x^{*} \right) = \begin{cases} b, & x^{*} = a

            \\ \infty, & x^{*}  \ne a
 \end{cases}

</math> The convex conjugate of the absolute value function

<math>f(x) = \left| x \right|</math>

<math>

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

            \\ \infty, & \left|x^{*} \right|  >  1
 \end{cases}

</math> The convex conjugate of the exponential function is

<math>

\exp^\star\left(x^{*} \right) = \begin{cases} x^{*} \ln x^{*} - x^{*} , & x^{*} > 0

            \\ 0                            , & x^{*}  = 0
            \\ \infty                       , & x^{*}  < 0
 \end{cases}

</math> 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.

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. Convex-conjugation is order-reversing: if <math>f \le g</math> then <math>f^* \ge g^*</math>. Here <math> (f \le g ) :\iff (\forall x, f(x) \le g(x))</math>.

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 smaller than f. For proper functions f, f = f^** if and only if f is convex and lower semi-continuous.

Fenchel's inequality

For any proper convex function f and its convex conjugate f^* Fenchel's inequality (also known as the Fenchel-Young inequality) holds:

<math>

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

Behavior under linear transformations

Let A be a linear transformation from Rⁿ to R^m. For any convex function f on Rⁿ, one has

<math> \left(A f\right)^\star = f^\star A^\star </math>

where A^* is the adjoint operator of A defined by

<math> \left \langle Ax, y^\star \right \rangle = \left \langle x, A^\star y^\star \right \rangle </math>

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

<math>f\left(A x\right) = f(x), \; \forall x, \; \forall A \in G </math>

if and only if its convex conjugate f^* is symmetric with respect to G.

Infimal convolution

The infimal convolution of two functions f and g is defined as

<math> \left(f \star_\inf g\right)(x) = \inf \left \{ f(x-y) + g(y) \, | \, y \in \mathbb{R}^n \right \} </math>

Let f₁, …, f_m be proper convex functions on Rⁿ. Then

<math> \left( f_1 \star_\inf \cdots \star_\inf f_m \right)^\star = f_1^\star + \cdots + f_m^\star </math>

References

Arnol'd, Vladimir Igorevich (1989). Mathematical Methods of Classical Mechanics (second edition). Springer. ISBN 0-387-96890-3.
Rockafellar, Ralph Tyrell (1970). Convex Analysis. Princeton University Press. ISBN 0-691-01586-4.