Supriya Ghosh

Supporting functional

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

Contents

Mathematical definition

Let X be a locally convex topological space, and C X be a convex set, then the continuous linear functional ϕ : X R is a supporting functional of C at the point x 0 if ϕ ( x ) ϕ ( x 0 ) for every x C .

Relation to support function

If h C : X R (where X is the dual space of X ) is a support function of the set C, then if h C ( x ) = x ( x 0 ) , it follows that h C defines a supporting functional ϕ : X R of C at the point x 0 such that ϕ ( x ) = x ( x ) for any x X .

Relation to supporting hyperplane

If ϕ is a supporting functional of the convex set C at the point x 0 C such that

ϕ ( x 0 ) = σ = sup x C ϕ ( x ) > inf x C ϕ ( x )

then H = ϕ 1 ( σ ) defines a supporting hyperplane to C at x 0 .

References

Supporting functional Wikipedia


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