v2007.11.26 - Convex Optimization

124 CHAPTER 2. CONVEX GEOMETRY{extreme directions} ⇒ {c.i.}Conversely, when a conically independent set of directions from pointedclosed convex cone K is known a priori to comprise generators, then alldirections from that set must be extreme directions of the cone;{extreme directions} ⇔ {c.i. generators of pointed closed convex K}Barker & Carlson [18,1] call the extreme directions a minimal generatingset for a pointed closed convex cone. A minimal set of generators is thereforea conically independent set of generators, and vice versa, 2.39 for a pointedclosed convex cone.Any collection of n or fewer extreme directions from pointed closedconvex cone K ⊂ R n must be linearly independent;{≤ n extreme directions in R n } ⇒ {l.i.}Conversely, because l.i. ⇒ c.i.,{extreme directions} ⇐ {l.i. generators of pointed closed convex K}2.10.2.0.1 Example. Vertex-description of halfspace H about origin.From n + 1 points in R n we can make a vertex-description of a convexcone that is a halfspace H , where {x l ∈ R n , l=1... n} constitutes aminimal set of generators for a hyperplane ∂H through the origin. Anexample is illustrated in Figure 41. By demanding the augmented set{x l ∈ R n , l=1... n + 1} be affinely independent (we want x n+1 not parallelto ∂H), thenH = ⋃ (ζ x n+1 + ∂H)ζ ≥0= {ζ x n+1 + cone{x l ∈ R n , l=1... n} | ζ ≥0}= cone{x l ∈ R n , l=1... n + 1}(245)a union of parallel hyperplanes. Cardinality is one step beyond dimension ofthe ambient space, but {x l ∀l} is a minimal set of generators for this convexcone H which has no extreme elements.2.39 This converse does not hold for nonpointed closed convex cones as Table 2.10.0.0.1implies; e.g., ponder four conically independent generators for a plane (case n=2).