v2009.01.01 - Convex Optimization

100 CHAPTER 2. CONVEX GEOMETRY
2.8.1.1.1 Theorem. (Klee) Extremes. [286,3.6] [266,18, p.166]
(confer2.3.2,2.12.2.0.1) Any closed convex set containing no lines can be
expressed as the convex hull of its extreme points and extreme rays. ⋄
It follows that any element of a convex set containing no lines may
be expressed as a linear combination of its extreme elements; e.g.,
Example 2.9.2.4.1.
2.8.1.2 Generators
In the narrowest sense, generators for a convex set comprise any collection
of points and directions whose convex hull constructs the set.
When the extremes theorem applies, the extreme points and directions
are called generators of a convex set. An arbitrary collection of generators
for a convex set includes its extreme elements as a subset; the set of extreme
elements of a convex set is a minimal set of generators for that convex set.
Any polyhedral set has a minimal set of generators whose cardinality is finite.
When the convex set under scrutiny is a closed convex cone, conic
combination of generators during construction is implicit as shown in
Example 2.8.1.2.1 and Example 2.10.2.0.1. So, a vertex at the origin (if it
exists) becomes benign.
We can, of course, generate affine sets by taking the affine hull of any
collection of points and directions. We broaden, thereby, the meaning of
generator to be inclusive of all kinds of hulls.
Any hull of generators is loosely called a vertex-description. (2.3.4)
Hulls encompass subspaces, so any basis constitutes generators for a
vertex-description; span basis R(A).
2.8.1.2.1 Example. Application of extremes theorem.
Given an extreme point at the origin and N extreme rays, denoting the i th
extreme direction by Γ i ∈ R n , then the convex hull is (78)
P = { [0 Γ 1 Γ 2 · · · Γ N ] aζ | a T 1 = 1, a ≽ 0, ζ ≥ 0 }
{
= [Γ1 Γ 2 · · · Γ N ] aζ | a T 1 ≤ 1, a ≽ 0, ζ ≥ 0 }
{
= [Γ1 Γ 2 · · · Γ N ] b | b ≽ 0 } (170)
⊂ R n
a closed convex set that is simply a conic hull like (94).