Optimality for smooth
function figure 1

The figure (
Optimality for
smooth function figure 1
) illustrates the condition
.
The painted triangle is the constraint set
.
The ellipses are the level curves of a function
with the internal ellipse is the level curve with the smallest value. The
slightly transparent triangle is the set
.
The arrow is the vector
.
The
is orthogonal to the level curve that passes through
and points to the direction of increase of
.
The
points in direction of decrease. Because the
lies within the
the point
minimizes
over
.
The alternative situation is presented on the picture
(
Optimality for smooth
function figure 2
). Here,
lies outside of the
.
In addition the
must be orthogonal to the level curve. Therefore, the level curve must cross
into
thus preventing
from being the minimum.
Optimality for smooth
function figure 2

