Convex Conjugate - Examples

Examples

The convex conjugate of an affine function

$f(x) = \left\langle a,x \right\rangle - b,\, a \in \mathbb{R}^n, b \in \mathbb{R}$

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

The convex conjugate of a power function

$f(x) = \frac{1}{p}|x|^p,\,1<p<\infty$

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

where

The convex conjugate of the absolute value function

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

The convex conjugate of the exponential function is

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

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.

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