v2010.10.26 - Convex Optimization

146 CHAPTER 2. CONVEX GEOMETRY{x l ∈ R n , l=1... n + 1} be affinely independent (we want vector x n+1 notparallel to ∂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}(285)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.2.10.2.0.2 Exercise. Enumerating conically independent directions.Describe a nonpointed polyhedral cone in three dimensions having more than8 conically independent generators. (confer Table 2.10.0.0.1) 2.10.3 Utility of conic independencePerhaps the most useful application of conic independence is determinationof the intersection of closed convex cones from their halfspace-descriptions,or representation of the sum of closed convex cones from theirvertex-descriptions.⋂Ki∑KiA halfspace-description for the intersection of any number of closedconvex cones K i can be acquired by pruning normals; specifically,only the conically independent normals from the aggregate of all thehalfspace-descriptions need be retained.Generators for the sum of any number of closed convex cones K i canbe determined by retaining only the conically independent generatorsfrom the aggregate of all the vertex-descriptions.Such conically independent sets are not necessarily unique or minimal.2.11 When extreme means exposedFor any convex full-dimensional polyhedral set in R n , distinction betweenthe terms extreme and exposed vanishes [330,2.4] [113,2.2] for faces of