v2010.10.26 - Convex Optimization

2.12. CONVEX POLYHEDRA 149convex polyhedraboundedpolyhedrapolyhedralconesconvex conesFigure 52: Polyhedral cones are finitely generated, unbounded, and convex.2.12.1.0.2 Exercise. Unbounded convex polyhedra.Illustrate an unbounded polyhedron that is not a cone or its translation. From 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 flavor (proper) Lorentz cone (317), anysubspace, and R n . More polyhedral cones are illustrated in Figure 50 andFigure 24.2.12.2 Vertices of convex polyhedraBy definition, a vertex (2.6.1.0.1) always lies on the relative boundary of aconvex polyhedron. [223, def.115/6 p.358] In Figure 20, each vertex of thepolyhedron is located at an intersection of three or more facets, and everyedge belongs to precisely two facets [26,VI.1 p.252]. In Figure 24, the onlyvertex of that polyhedral cone lies at the origin.The set of all polyhedral cones is clearly a subset of convex polyhedraand a subset of convex cones (Figure 52). Not all convex polyhedra arebounded; evidently, neither can they all be described by the convex hull of abounded set of points as defined in (86). Hence a universal vertex-description