Supporting functional

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

Mathematical definition

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 and for every .[1]

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 .[2]

References

  1. ^ Pallaschke, Diethard; Rolewicz, Stefan (1997). Foundations of mathematical optimization: convex analysis without linearity. Springer. p. 323. ISBN 978-0-7923-4424-7.
  2. ^ Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. p. 240. ISBN 978-0-387-29570-1.

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