Fenchel's duality theorem

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

Let ƒ be a proper convex function on Rⁿ and let g be a proper concave function on Rⁿ. Then, if regularity conditions are satisfied,

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

where ƒ ^* is the convex conjugate of ƒ (also referred to as the Fenchel–Legendre transform) and g _* is the concave conjugate of g. That is,

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

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