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 minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of s ...

and

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

, the supporting functional is a generalization of the

supporting hyperplane In geometry, a supporting hyperplane of a 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 half-spaces bounded by the hyperplane, * S has at lea ...

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 ve ...

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

, and

C \subset X

be a

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

, 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 const ...

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 n ...

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