Supporting functional

In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

Mathematical definition

Let X be a locally convex topological space, and $C\subset X$ be a convex set, then the continuous linear functional $\phi :X\to \mathbb {R}$ is a supporting functional of C at the point $x_{0}$ if $\phi \not =0$ and $\phi (x)\leq \phi (x_{0})$ for every $x\in C$.

Relation to support function

If $h_{C}:X^{*}\to \mathbb {R}$ (where $X^{*}$ is the dual space of $X$) is a support function of the set C, then if $h_{C}\left(x^{*}\right)=x^{*}\left(x_{0}\right)$, it follows that $h_{C}$ defines a supporting functional $\phi :X\to \mathbb {R}$ of C at the point $x_{0}$ such that $\phi (x)=x^{*}(x)$ for any $x\in X$.

Relation to supporting hyperplane

If $\phi$ is a supporting functional of the convex set C at the point $x_{0}\in C$ such that

$\phi \left(x_{0}\right)=\sigma =\sup _{x\in C}\phi (x)>\inf _{x\in C}\phi (x)$

then $H=\phi ^{-1}(\sigma )$ defines a supporting hyperplane to C at $x_{0}$.