v2010.10.26 - Convex Optimization

150 CHAPTER 2. CONVEX GEOMETRYof polyhedra in terms of that same finite-length list X (76):2.12.2.0.1 Definition. Convex polyhedra, vertex-description.(confer2.8.1.1.1) Denote the truncated a-vector[ ]aia i:l = .a l(288)By discriminating a suitable finite-length generating list (or set) arrangedcolumnar in X ∈ R n×N , then any particular polyhedron may be describedP = { Xa | a T 1:k1 = 1, a m:N ≽ 0, {1... k} ∪ {m ... N} = {1... N} } (289)where 0≤k ≤N and 1≤m≤N+1. Setting k=0 removes the affineequality condition. Setting m=N+1 removes the inequality. △Coefficient indices in (289) may or may not be overlapping, but allthe coefficients are assumed constrained. From (78), (86), and (103), 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) (290)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} (291)Members of a generating list are not necessarily vertices of thecorresponding polyhedron; certainly true for (86) and (289), some subsetof list members reside in the polyhedron’s relative interior. Conversely, whenboundedness (86) applies, convex hull of the vertices is a polyhedron identicalto 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 (280), then that cone is describedsetting m=1 and k=0 in vertex-description (289):a conic hull of N generators.K = coneX = {Xa | a ≽ 0} ⊆ R n (103)