v2010.10.26 - Convex Optimization

148 CHAPTER 2. CONVEX GEOMETRYtransformation remains a convex polyhedron; [26,I.9] [307, thm.19.3] theforemost consequence being, invariance of polyhedral set closedness.When b and d in (286) are 0, the resultant is a polyhedral cone. Theset of all polyhedral cones is a subset of convex cones:2.12.1 Polyhedral coneFrom our study of cones, we see: the number of intersecting hyperplanesand halfspaces constituting a convex cone is possibly but not necessarilyinfinite. When the number is finite, the convex cone is termed polyhedral.That is the primary distinguishing feature between the set of all convexcones and polyhedra; all polyhedra, including polyhedral cones, are finitelygenerated [307,19]. (Figure 52) We distinguish polyhedral cones in the setof all convex cones for this reason, although all convex cones of dimension 2or less are polyhedral.2.12.1.0.1 Definition. Polyhedral cone, halfspace-description. 2.55(confer (103)) A polyhedral cone is the intersection of a finite number ofhalfspaces and hyperplanes about the origin;K = {y | Ay ≽ 0, Cy = 0} ⊆ R n (a)= {y | Ay ≽ 0, Cy ≽ 0, Cy ≼ 0} (b)⎧ ⎡ ⎤ ⎫⎨ A ⎬=⎩ y | ⎣ C ⎦y ≽ 0(c)⎭−C(287)where coefficients A and C generally denote matrices of finite dimension.Each row of C is a vector normal to a hyperplane containing the origin,while each row of A is a vector inward-normal to a hyperplane containingthe origin and partially bounding a halfspace.△A polyhedral cone thus defined is closed, convex (2.4.1.1), has onlya finite number of generators (2.8.1.2), and can be not full-dimensional.(Minkowski) Conversely, any finitely generated convex cone is polyhedral.(Weyl) [330,2.8] [307, thm.19.1]2.55 Rockafellar [307,19] proposes affine sets be handled via complementary pairs of affineinequalities; e.g., Cy ≽d and Cy ≼d which, when taken together, can present severedifficulties to some interior-point methods of numerical solution.