v2009.01.01 - Convex Optimization

134 CHAPTER 2. CONVEX GEOMETRY
From the definition it follows that any single hyperplane through the
origin, or any halfspace partially bounded by a hyperplane through the origin
is a polyhedral cone. The most familiar example of polyhedral cone is any
quadrant (or orthant,2.1.3) generated by Cartesian half-axes. Esoteric
examples of polyhedral cone include the point at the origin, any line through
the origin, any ray having the origin as base such as the nonnegative real
line R + in subspace R , polyhedral flavors of the (proper) Lorentz cone
(confer (160))
{[ ]
}
x
K l = ∈ R n × R | ‖x‖
t
l ≤ t , l=1 or ∞ (260)
any subspace, and R n .
Figure 20.
More examples are illustrated in Figure 44 and
2.12.2 Vertices of convex polyhedra
By definition, a vertex (2.6.1.0.1) always lies on the relative boundary of a
convex polyhedron. [194, def.115/6, p.358] In Figure 16, each vertex of the
polyhedron is located at the intersection of three or more facets, and every
edge belongs to precisely two facets [25,VI.1, p.252]. In Figure 20, the only
vertex of that polyhedral cone lies at the origin.
The set of all polyhedral cones is clearly a subset of convex polyhedra and
a subset of convex cones. Not all convex polyhedra are bounded; evidently,
neither can they all be described by the convex hull of a bounded set of
points as defined in (78). Hence we propose a universal vertex-description of
polyhedra in terms of that same finite-length list X (68):
2.12.2.0.1 Definition. Convex polyhedra, vertex-description.
(confer2.8.1.1.1) Denote the truncated a-vector
[ ]
ai
a i:l = .
a l
(261)
By discriminating a suitable finite-length generating list (or set) arranged
columnar in X ∈ R n×N , then any particular polyhedron may be described
P = { Xa | a T 1:k1 = 1, a m:N ≽ 0, {1... k} ∪ {m ... N} = {1... N} } (262)
where 0≤k ≤N and 1≤m≤N+1. Setting k=0 removes the affine
equality condition. Setting m=N+1 removes the inequality. △