Supporting functional

Last updated March 22, 2019

In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

Convex analysis branch of mathematics devoted to the study of properties of convex functions and convex sets

Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.

In mathematics, computer science and operations research, mathematical optimization or mathematical programming is the selection of a best element from some set of available alternatives.

In geometry, a supporting hyperplane of a set $in Euclidean space is a hyperplane that has both of the following two properties:$

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 $\phi \not =0$ and $\phi (x)\leq \phi (x_{0})$ for every $x\in C$ .^[1]

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Relation to support function

If $h_{C}:X^{*}\to \mathbb {R}$ (where $X^{*}$ is the dual space of $X$ ) 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 $x\in X$ .

In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

In mathematics, the support functionh_A of a non-empty closed convex set A in $describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on . Any non-empty closed convex set A is uniquely determined by h A . Furthermore, the support function, as a function of the set A is compatible with many natural geometric operations, like scaling, translation, rotation and Minkowski addition. Due to these properties, the support function is one of the most central basic concepts in convex geometry.$

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]

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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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[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.