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 $\phi (x)\leq \phi (x_{0})$ for every $x \in C$.^[1]

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 $x_{0}$.^[2]

References

1. Pallaschke, Diethard; Rolewicz, Stefan (1997). Foundations of mathematical optimization: convex analysis without linearity. Springer. p. 323. ISBN 978-0-7923-4424-7.
2. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. p. 240. ISBN 978-0-387-29570-1.