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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 be a convex set, then the continuous linear functional is a supporting functional of C at the point if for every .
Relation to support function
If (where is the dual space of ) is a support function of the set C, then if , it follows that defines a supporting functional of C at the point such that for any .
Relation to supporting hyperplane
If is a supporting functional of the convex set C at the point such that
then defines a supporting hyperplane to C at .