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 [math]\displaystyle{ C \subset X }[/math] be a convex set, then the continuous linear functional [math]\displaystyle{ \phi: X \to \mathbb{R} }[/math] is a supporting functional of C at the point [math]\displaystyle{ x_0 }[/math] if [math]\displaystyle{ \phi \not=0 }[/math] and [math]\displaystyle{ \phi(x) \leq \phi(x_0) }[/math] for every [math]\displaystyle{ x \in C }[/math].[1]
Relation to support function
If [math]\displaystyle{ h_C: X^* \to \mathbb{R} }[/math] (where [math]\displaystyle{ X^* }[/math] is the dual space of [math]\displaystyle{ X }[/math]) is a support function of the set C, then if [math]\displaystyle{ h_C\left(x^*\right) = x^*\left(x_0\right) }[/math], it follows that [math]\displaystyle{ h_C }[/math] defines a supporting functional [math]\displaystyle{ \phi: X \to \mathbb{R} }[/math] of C at the point [math]\displaystyle{ x_0 }[/math] such that [math]\displaystyle{ \phi(x) = x^*(x) }[/math] for any [math]\displaystyle{ x \in X }[/math].
Relation to supporting hyperplane
If [math]\displaystyle{ \phi }[/math] is a supporting functional of the convex set C at the point [math]\displaystyle{ x_0 \in C }[/math] such that
- [math]\displaystyle{ \phi\left(x_0\right) = \sigma = \sup_{x \in C} \phi(x) \gt \inf_{x \in C} \phi(x) }[/math]
then [math]\displaystyle{ H = \phi^{-1}(\sigma) }[/math] defines a supporting hyperplane to C at [math]\displaystyle{ x_0 }[/math].[2]
References
- ↑ Pallaschke, Diethard; Rolewicz, Stefan (1997). Foundations of mathematical optimization: convex analysis without linearity. Springer. p. 323. ISBN 978-0-7923-4424-7.
- ↑ Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. p. 240. ISBN 978-0-387-29570-1.
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