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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