View source for Supporting functional

In [[Convex analysis|convex analysis]] and [[Mathematical optimization|mathematical optimization]], the '''supporting functional''' is a generalization of the [[Supporting hyperplane|supporting hyperplane]] of a set.
== Mathematical definition ==
Let ''X'' be a locally convex [[Topological space|topological space]], and <math>C \subset X</math> be a [[Convex set|convex set]], then the continuous linear functional <math>\phi: X \to \mathbb{R}</math> is a supporting functional of ''C'' at the point <math>x_0</math> if <math>\phi \not=0</math> and <math>\phi(x) \leq \phi(x_0)</math> for every <math>x \in C</math>.<ref>{{cite book|title=Foundations of mathematical optimization: convex analysis without linearity|page=323|first1=Diethard|last1=Pallaschke|first2=Stefan|last2=Rolewicz|publisher=Springer|year=1997|isbn=978-0-7923-4424-7}}</ref>

== Relation to support function ==
If <math>h_C: X^* \to \mathbb{R}</math> (where <math>X^*</math> is the [[Dual space|dual space]] of <math>X</math>) is a [[Support function|support function]] of the set ''C'', then if <math>h_C\left(x^*\right) = x^*\left(x_0\right)</math>, it follows that <math>h_C</math> defines a supporting functional <math>\phi: X \to \mathbb{R}</math> of ''C'' at the point <math>x_0</math> such that <math>\phi(x) = x^*(x)</math> for any <math>x \in X</math>.

== Relation to supporting hyperplane ==
If <math>\phi</math> is a supporting functional of the convex set ''C'' at the point <math>x_0 \in C</math> such that
:<math>\phi\left(x_0\right) = \sigma = \sup_{x \in C} \phi(x) > \inf_{x \in C} \phi(x)</math>
then <math>H = \phi^{-1}(\sigma)</math> defines a supporting hyperplane to ''C'' at <math>x_0</math>.<ref>{{cite book |last1=Borwein |first1=Jonathan |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1 |page = 240}}</ref>

== References ==
{{Reflist}}

[[Category:Functional analysis]]
[[Category:Duality theories]]
[[Category:Types of functions]]

{{Sourceattribution|Supporting functional}}