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

and

mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

, the supporting functional is a generalization of the

supporting hyperplane In geometry, a supporting hyperplane of a Set (mathematics), set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties: * S is entirely contained in one of the two closed set, closed Half-space (geometry), h ...

of a set.

Mathematical definition

Let ''X'' be a

locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

, and

C \subset X

be a

convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

, then the

continuous linear functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear ...

\phi: X \to \mathbb

is a supporting functional of ''C'' at the point

x_0

\phi \not=0

and

\phi(x) \leq \phi(x_0)

for every

x \in C

Relation to support function

h_C: X^* \to \mathbb

(where

X^*

is the

dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...

X

) is a

support function In mathematics, the support function ''h'A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any ...

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

of ''C'' at the point

x_0

such that

\phi(x) = x^*(x)

for any

x \in X

Relation to supporting hyperplane

\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_ \phi(x) > \inf_ \phi(x)

then

H = \phi^(\sigma)

defines a supporting hyperplane to ''C'' at

x_0

References

{{Reflist Functional analysis Duality theories Types of functions