Supporting functional explained
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
\phi\left(x0\right)=\sigma=\supx\phi(x)>infx\phi(x)
then
defines a supporting hyperplane to
C at
.
[2] Notes and References
- Book: Foundations of mathematical optimization: convex analysis without linearity. 323. Diethard. Pallaschke. Stefan. Rolewicz. Springer. 1997. 978-0-7923-4424-7.
- Book: Borwein . Jonathan . Jonathan Borwein . Lewis . Adrian . Convex Analysis and Nonlinear Optimization: Theory and Examples. 2 . 2006 . Springer . 978-0-387-29570-1 . 240.
