v2009.01.01 - Convex Optimization

2.12. CONVEX POLYHEDRA 135
Coefficient indices in (262) may or may not be overlapping, but all
the coefficients are assumed constrained. From (70), (78), and (94), we
summarize how the coefficient conditions may be applied;
affine sets −→ a T 1:k 1 = 1
polyhedral cones −→ a m:N ≽ 0
}
←− convex hull (m ≤ k) (263)
It is always possible to describe a convex hull in the region of overlapping
indices because, for 1 ≤ m ≤ k ≤ N
{a m:k | a T m:k1 = 1, a m:k ≽ 0} ⊆ {a m:k | a T 1:k1 = 1, a m:N ≽ 0} (264)
Members of a generating list are not necessarily vertices of the
corresponding polyhedron; certainly true for (78) and (262), some subset
of list members reside in the polyhedron’s relative interior. Conversely, when
boundedness (78) applies, the convex hull of the vertices is a polyhedron
identical to the convex hull of the generating list.
2.12.2.1 Vertex-description of polyhedral cone
Given closed convex cone K in a subspace of R n having any set of generators
for it arranged in a matrix X ∈ R n×N as in (253), then that cone is described
setting m=1 and k=0 in vertex-description (262):
a conic hull of N generators. 2.47
K = coneX = {Xa | a ≽ 0} ⊆ R n (94)
2.12.2.2 Pointedness
[286,2.10] Assuming all generators constituting the columns of X ∈ R n×N
are nonzero, polyhedral cone K is pointed (2.7.2.1.2) if and only if there is
no nonzero a ≽ 0 that solves Xa=0; id est, iff N(X) ∩ R N + = 0 . 2.48
2.47 This vertex-description is extensible to an infinite number of generators; which follows,
for a pointed cone, from the extremes theorem (2.8.1.1.1) and Example 2.8.1.2.1.
2.48 If rankX = n , then the dual cone K ∗ (2.13.1) is pointed. (280)