Fenchel's duality theorem

In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel.

Let ${\displaystyle f}$ be a proper convex function on ${\displaystyle \mathbb {R} ^{n}}$ and let ${\displaystyle g}$ be a proper concave function on ${\displaystyle \mathbb {R} ^{n}}$. Then, if regularity conditions are satisfied,

${\displaystyle \inf _{x}(f(x)-g(x))=\sup _{p}(g_{*}(p)-f^{*}(p)).}$

where ${\displaystyle f^{*}}$ is the convex conjugate of ${\displaystyle f}$ (also referred to as the Fenchel–Legendre transform) and ${\displaystyle g_{*}}$ is the concave conjugate of ${\displaystyle g}$. That is,

${\displaystyle f^{*}\left(x^{*}\right):=\sup \left\{\left.\left\langle x^{*},x\right\rangle -f\left(x\right)\right|x\in \mathbb {R} ^{n}\right\}}$

${\displaystyle g_{*}\left(x^{*}\right):=\inf \left\{\left.\left\langle x^{*},x\right\rangle -g\left(x\right)\right|x\in \mathbb {R} ^{n}\right\}}$