v2007.09.13 - Convex Optimization

2.12. CONVEX POLYHEDRA 127It follows from definition (245) by exposure that each face of a convexpolyhedron is a convex polyhedron.The projection of any polyhedron on a subspace remains a polyhedron.More generally, the image of a polyhedron under any linear transformationis a polyhedron. [20,I.9]When b and d in (245) 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 hyperplanes andhalfspaces constituting a convex cone is possibly but not necessarily infinite.When the number is finite, the convex cone is termed polyhedral. That isthe primary distinguishing feature between the set of all convex cones andpolyhedra; all polyhedra, including polyhedral cones, are finitely generated[228,19]. We distinguish polyhedral cones in the set of all convex cones forthis reason, although all convex cones of dimension 2 or less are polyhedral.2.12.1.0.1 Definition. Polyhedral cone, halfspace-description. 2.41(confer (252)) 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(246)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, possibly has emptyinterior, and only a finite number of generators (2.8.1.2), and vice versa.(Minkowski/Weyl) [245,2.8] [228, thm.19.1]2.41 Rockafellar [228,19] proposes affine sets be handled via complementary pairs of affineinequalities; e.g., Cy ≽d and Cy ≼d .