v2009.01.01 - Convex Optimization

2.8. CONE BOUNDARY 99
2.8.1.1 extreme distinction, uniqueness
An extreme direction is unique, but its vector representation Γ ε is not
because any positive scaling of it produces another vector in the same
(extreme) direction. Hence an extreme direction is unique to within a positive
scaling. When we say extreme directions are distinct, we are referring to
distinctness of rays containing them. Nonzero vectors of various length in
the same extreme direction are therefore interpreted to be identical extreme
directions. 2.34
The extreme directions of the polyhedral cone in Figure 20 (p.65), for
example, correspond to its three edges. For any pointed polyhedral cone,
there is a one-to-one correspondence of one-dimensional faces with extreme
directions.
The extreme directions of the positive semidefinite cone (2.9) comprise
the infinite set of all symmetric rank-one matrices. [20,6] [170,III] It
is sometimes prudent to instead consider the less infinite but complete
normalized set, for M >0 (confer (206))
{zz T ∈ S M | ‖z‖= 1} (169)
The positive semidefinite cone in one dimension M =1, S + the nonnegative
real line, has one extreme direction belonging to its relative interior; an
idiosyncrasy of dimension 1.
Pointed closed convex cone K = {0} has no extreme direction because
extreme directions are nonzero by definition.
If closed convex cone K is not pointed, then it has no extreme directions
and no vertex. [20,1]
Conversely, pointed closed convex cone K is equivalent to the convex hull
of its vertex and all its extreme directions. [266,18, p.167] That is the
practical utility of extreme direction; to facilitate construction of polyhedral
sets, apparent from the extremes theorem:
2.34 Like vectors, an extreme direction can be identified by the Cartesian point at the
vector’s head with respect to the origin.