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 $\varphi :X\to ℝ$ is a supporting functional of C at the point $x_0$ if $\varphi ≠0$ and $\varphi (x)\le \varphi (x_0)$ for every $x\in C$.

Relation to support function

If $h_C:X^{\ast }\to ℝ$ (where $X^{\ast }$ is the dual space of $X$) is a support function of the set C, then if $h_C\left(x^{\ast }\right)=x^{\ast }\left(x_0\right)$, it follows that $h_C$ defines a supporting functional $\varphi :X\to ℝ$ of C at the point $x_0$ such that $\varphi (x)=x^{\ast }(x)$ for any $x\in X$.

Relation to supporting hyperplane

If $\varphi$ is a supporting functional of the convex set C at the point $x_0\in C$ such that

$\varphi \left(x_0\right)=\sigma =\underset{x\in C}{sup}\varphi (x)>\underset{x\in C}{inf}\varphi (x)$

then $H={\varphi }^{-1}(\sigma )$ defines a supporting hyperplane to C at $x_0$.