v2007.09.13 - Convex Optimization

94 CHAPTER 2. CONVEX GEOMETRYWhen 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 the convex hull is (75)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 } (157)⊂ R na closed convex set that is simply a conic hull like (83).2.8.2 Exposed direction2.8.2.0.1 Definition. Exposed point & direction of pointed convex cone.[228,18] (confer2.6.1.0.1)When a convex cone has a vertex, an exposed point, it resides at theorigin; there can be only one.In the closure of a pointed convex cone, an exposed direction is thedirection of a one-dimensional exposed face that is a ray emanatingfrom the origin.{exposed directions} ⊆ {extreme directions}△