Definition
(Directional derivative). For a function
the directional derivative is defined
by
Let
be a convex function
.
We use the notation
,
,
.
Fix
.
A hyperplane
that passes through the point
and has the normal vector
is given by the
relationship
Equivalently,
The
lies above
iff
or


(Subgradient)

The following statements are verified with similar techniques.
