v2009.01.01 - Convex Optimization

126 CHAPTER 2. CONVEX GEOMETRY
When the intersection A ∩ S N + is known a priori to consist only of a
single point, then Barvinok’s proposition provides the greatest upper bound
on its rank not exceeding N . The intersection can be a single nonzero point
only if the number of linearly independent hyperplanes m constituting A
satisfies 2.42 N(N + 1)/2 − 1 ≤ m ≤ N(N + 1)/2 (251)
2.10 Conic independence (c.i.)
In contrast to extreme direction, the property conically independent direction
is more generally applicable, inclusive of all closed convex cones (not
only pointed closed convex cones). Similar to the definition for linear
independence, arbitrary given directions {Γ i ∈ R n , i=1... N} are conically
independent if and only if, for all ζ ∈ R N +
Γ i ζ i + · · · + Γ j ζ j − Γ l ζ l = 0, i≠ · · · ≠j ≠l = 1... N (252)
has only the trivial solution ζ =0; in words, iff no direction from the
given set can be expressed as a conic combination of those remaining.
(Figure 43, for example. A Matlab implementation of test (252) is given on
Wıκımization.) It is evident that linear independence (l.i.) of N directions
implies their conic independence;
l.i. ⇒ c.i.
Arranging any set of generators for a particular convex cone in a matrix
columnar,
X ∆ = [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (253)
then the relationship l.i. ⇒ c.i. suggests: the number of l.i. generators in
the columns of X cannot exceed the number of c.i. generators. Denoting by
k the number of conically independent generators contained in X , we have
the most fundamental rank inequality for convex cones
dim aff K = dim aff[0 X ] = rankX ≤ k ≤ N (254)
Whereas N directions in n dimensions can no longer be linearly independent
once N exceeds n , conic independence remains possible:
2.42 For N >1, N(N+1)/2 −1 independent hyperplanes in R N(N+1)/2 can make a line