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 $$x_0$$.