Supporting functional

Mathematical definition

Let X be a locally convex topological space, and C \subset X be a convex set, then the continuous linear functional \phi: X \to \mathbb{R} is a supporting functional of C at the point x_0 if \phi \not=0 and \phi(x) \leq \phi(x_0) for every x \in C.

Relation to support function

If h_C: X^* \to \mathbb{R} (where X^* is the dual space of X) is a support function 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{R} of C at the point x_0 such that \phi(x) = x^*(x) for any x \in X.

Relation to supporting hyperplane

If \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_{x \in C} \phi(x) \inf_{x \in C} \phi(x) then H = \phi^{-1}(\sigma) defines a supporting hyperplane to C at x_0.

References

References

1. (1997). "Foundations of mathematical optimization: convex analysis without linearity". Springer.
2. (2006). "Convex Analysis and Nonlinear Optimization: Theory and Examples". Springer.