v2010.10.26 - Convex Optimization

110 CHAPTER 2. CONVEX GEOMETRY2.8.1.1.1 Theorem. (Klee) Extremes. [330,3.6] [307,18, p.166](confer2.3.2,2.12.2.0.1) Any closed convex set containing no lines can beexpressed as the convex hull of its extreme points and extreme rays. ⋄It follows that any element of a convex set containing no lines maybe expressed as a linear combination of its extreme elements; e.g.,Example 2.9.2.7.1.2.8.1.2 GeneratorsIn the narrowest sense, generators for a convex set comprise any collectionof points and directions whose convex hull constructs the set.When the extremes theorem applies, the extreme points and directionsare called generators of a convex set. An arbitrary collection of generatorsfor a convex set includes its extreme elements as a subset; the set of extremeelements 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, coniccombination of generators during construction is implicit as shown inExample 2.8.1.2.1 and Example 2.10.2.0.1. So, a vertex at the origin (if itexists) becomes benign.We can, of course, generate affine sets by taking the affine hull of anycollection of points and directions. We broaden, thereby, the meaning ofgenerator 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 avertex-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 thextreme direction by Γ i ∈ R n , then their convex hull is (86)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 } (188)⊂ R na closed convex set that is simply a conic hull like (103).