v2007.11.26 - Convex Optimization

126 CHAPTER 2. CONVEX GEOMETRY2.11 When extreme means exposedFor any convex polyhedral set in R n having nonempty interior, distinctionbetween the terms extreme and exposed vanishes [249,2.4] [77,2.2] forfaces of all dimensions except n ; their meanings become equivalent as wesaw in Figure 14 (discussed in2.6.1.2). In other words, each and every faceof any polyhedral set (except the set itself) can be exposed by a hyperplane,and vice versa; e.g., Figure 17.Lewis [180,6] [158,2.3.4] claims nonempty extreme proper subsets andthe exposed subsets coincide for S n + ; id est, each and every face of the positivesemidefinite cone, whose dimension is less than the dimension of the cone,is exposed. A more general discussion of cones having this property can befound in [259]; e.g., the Lorentz cone (148) [17,II.A].2.12 Convex polyhedraEvery polyhedron, such as the convex hull (76) of a bounded list X , canbe expressed as the solution set of a finite system of linear equalities andinequalities, and vice versa. [77,2.2]2.12.0.0.1 Definition. Convex polyhedra, halfspace-description.[46,2.2.4] A convex polyhedron is the intersection of a finite number ofhalfspaces and hyperplanes;P = {y | Ay ≽ b, Cy = d} ⊆ R n (246)where coefficients A and C generally denote matrices. Each row of C is avector normal to a hyperplane, while each row of A is a vector inward-normalto a hyperplane partially bounding a halfspace.△By the halfspaces theorem in2.4.1.1.1, a polyhedron thus described is aclosed convex set having possibly empty interior; e.g., Figure 14. Convexpolyhedra 2.40 are finite-dimensional comprising all affine sets (2.3.1),polyhedral cones, line segments, rays, halfspaces, convex polygons, solids[164, def.104/6, p.343], polychora, polytopes, 2.41 etcetera.2.40 We consider only convex polyhedra throughout, but acknowledge the existence ofconcave polyhedra. [284, Kepler-Poinsot Solid]2.41 Some authors distinguish bounded polyhedra via the designation polytope. [77,2.2]