v2007.11.26 - Convex Optimization

128 CHAPTER 2. CONVEX GEOMETRYFrom the definition it follows that any single hyperplane through theorigin, or any halfspace partially bounded by a hyperplane through the originis a polyhedral cone. The most familiar example of polyhedral cone is anyquadrant (or orthant,2.1.3) generated by Cartesian half-axes. Esotericexamples of polyhedral cone include the point at the origin, any line throughthe origin, any ray having the origin as base such as the nonnegative realline R + in subspace R , polyhedral flavors of the (proper) Lorentz cone(confer (148)){[ ]}xK l = ∈ R n × R | ‖x‖tl ≤ t , l=1 or ∞ (248)any subspace, and R n . More examples are illustrated in Figure 40 andFigure 17.2.12.2 Vertices of convex polyhedraBy definition, a vertex (2.6.1.0.1) always lies on the relative boundary of aconvex polyhedron. [164, def.115/6, p.358] In Figure 14, each vertex of thepolyhedron is located at the intersection of three or more facets, and everyedge belongs to precisely two facets [20,VI.1, p.252]. In Figure 17, the onlyvertex of that polyhedral cone lies at the origin.The set of all polyhedral cones is clearly a subset of convex polyhedra anda subset of convex cones. Not all convex polyhedra are bounded, evidently,neither can they all be described by the convex hull of a bounded set of pointsas we defined it in (76). Hence we propose a universal vertex-description ofpolyhedra in terms of that same finite-length list X (66):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(249)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} } (250)where 0 ≤ k ≤ N and 1 ≤ m ≤ N + 1 . Setting k = 0 removes the affineequality condition. Setting m=N + 1 removes the inequality. △