v2009.01.01 - Convex Optimization

2.10. CONIC INDEPENDENCE (C.I.) 129
2.10.1 Preservation of conic independence
Independence in the linear (2.1.2.1), affine (2.4.2.4), and conic senses can
be preserved under linear transformation. Suppose a matrix X ∈ R n×N (253)
holds a conically independent set columnar. Consider the transformation
T(X) : R n×N → R n×N ∆ = XY (255)
where the given matrix Y = ∆ [y 1 y 2 · · · y N ]∈ R N×N is represented by linear
operator T . Conic independence of {Xy i ∈ R n , i=1... N} demands, by
definition (252),
Xy i ζ i + · · · + Xy j ζ j − Xy l ζ l = 0, i≠ · · · ≠j ≠l = 1... N (256)
have no nontrivial solution ζ ∈ R N + . That is ensured by conic independence
of {y i ∈ R N } and by R(Y )∩ N(X) = 0 ; seen by factoring out X .
2.10.1.1 linear maps of cones
[20,7] If K is a convex cone in Euclidean space R and T is any linear
mapping from R to Euclidean space M , then T(K) is a convex cone in M
and x ≼ y with respect to K implies T(x) ≼ T(y) with respect to T(K).
If K is closed or has nonempty interior in R , then so is T(K) in M .
If T is a linear bijection, then x ≼ y ⇔ T(x) ≼ T(y). Further, if F is
a face of K , then T(F) is a face of T(K).
2.10.2 Pointed closed convex K & conic independence
The following bullets can be derived from definitions (168) and (252) in
conjunction with the extremes theorem (2.8.1.1.1):
The set of all extreme directions from a pointed closed convex cone K ⊂ R n
is not necessarily a linearly independent set, yet it must be a conically
independent set; (compare Figure 20 on page 65 with Figure 44(a))
{extreme directions} ⇒ {c.i.}
Conversely, when a conically independent set of directions from pointed
closed convex cone K is known to comprise generators, then all directions
from that set must be extreme directions of the cone;
{extreme directions} ⇔ {c.i. generators of pointed closed convex K}