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 ${\displaystyle C\subset X}$ be a convex set, then the continuous linear functional ${\displaystyle \phi :X\to \mathbb {R} }$ is a supporting functional of C at the point ${\displaystyle x_{0}}$ if ${\displaystyle \phi \not =0}$ and ${\displaystyle \phi (x)\leq \phi (x_{0})}$ for every ${\displaystyle x\in C}$.[1]

Relation to support function

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

Relation to supporting hyperplane

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

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

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