Supporting functional

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

Let X be a locally convex topological space, and C ⊂ X {\displaystyle C\subset X} be a convex set, then the continuous linear functional ϕ : X → R {\displaystyle \phi :X\to \mathbb {R} } is a supporting functional of C at the point x 0 {\displaystyle x_{0}} if ϕ ≠ 0 {\displaystyle \phi \not =0} and ϕ ( x ) ≤ ϕ ( x 0 ) {\displaystyle \phi (x)\leq \phi (x_{0})} for every x ∈ C {\displaystyle x\in C} .^[1]

If h C : X ∗ → R {\displaystyle h_{C}:X^{*}\to \mathbb {R} } (where X ∗ {\displaystyle X^{*}} is the dual space of X {\displaystyle X} ) is a support function of the set C, then if h C ( x ∗ ) = x ∗ ( x 0 ) {\displaystyle h_{C}\left(x^{*}\right)=x^{*}\left(x_{0}\right)} , it follows that h C {\displaystyle h_{C}} defines a supporting functional ϕ : X → R {\displaystyle \phi :X\to \mathbb {R} } of C at the point x 0 {\displaystyle x_{0}} such that ϕ ( x ) = x ∗ ( x ) {\displaystyle \phi (x)=x^{*}(x)} for any x ∈ X {\displaystyle x\in X} .

If ϕ {\displaystyle \phi } is a supporting functional of the convex set C at the point x 0 ∈ C {\displaystyle x_{0}\in C} such that

ϕ ( x 0 ) = σ = sup x ∈ C ϕ ( x ) > inf x ∈ C ϕ ( x ) {\displaystyle \phi \left(x_{0}\right)=\sigma =\sup _{x\in C}\phi (x)>\inf _{x\in C}\phi (x)}

then H = ϕ − 1 ( σ ) {\displaystyle H=\phi ^{-1}(\sigma )} defines a supporting hyperplane to C at x 0 {\displaystyle x_{0}} .^[2]