v2007.09.13 - Convex Optimization

2.12. CONVEX POLYHEDRA 129Coefficient indices in (249) may or may not be overlapping, but allthe coefficients are assumed constrained. From (67), (75), and (83), wesummarize how the coefficient conditions may be applied;affine sets −→ a T 1:k 1 = 1polyhedral cones −→ a m:N ≽ 0}←− convex hull (m ≤ k) (250)It is always possible to describe a convex hull in the region of overlappingindices because, for 1 ≤ m ≤ k ≤ N{a m:k | a T m:k1 = 1, a m:k ≽ 0} ⊆ {a m:k | a T 1:k1 = 1, a m:N ≽ 0} (251)Members of a generating list are not necessarily vertices of thecorresponding polyhedron; certainly true for (75) and (249), some subsetof list members reside in the polyhedron’s relative interior. Conversely, whenboundedness (75) applies, the convex hull of the vertices is a polyhedronidentical to the convex hull of the generating list.2.12.2.1 Vertex-description of polyhedral coneGiven closed convex cone K in a subspace of R n having any set of generatorsfor it arranged in a matrix X ∈ R n×N as in (240), then that cone is describedsetting m=1 and k=0 in vertex-description (249):K = cone(X) = {Xa | a ≽ 0} ⊆ R n (252)a conic hull, like (83), of N generators.This vertex description is extensible to an infinite number of generators;which follows from the extremes theorem (2.8.1.1.1) and Example 2.8.1.2.1.2.12.2.2 Pointedness[245,2.10] Assuming all generators constituting the columns of X ∈ R n×Nare nonzero, polyhedral cone K is pointed (2.7.2.1.2) if and only ifthere is no nonzero a ≽ 0 that solves Xa=0; id est, iff N(X) ∩ R N + = 0.(If rankX = n , then the dual cone K ∗ is pointed. (268))A polyhedral proper cone in R n must have at least n linearly independentgenerators, or be the intersection of at least n halfspaces whose partialboundaries have normals that are linearly independent. Otherwise, the cone