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 \( {\displaystyle \phi \not =0} \) and \( \phi (x)\leq \phi (x_{0}) \) 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 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
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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Hellenica World - Scientific Library
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