v2007.09.13 - Convex Optimization

86 CHAPTER 2. CONVEX GEOMETRYFamiliar examples of convex cones include an unbounded ice-cream coneunited with its interior (a.k.a: second-order cone, quadratic cone, circularcone (2.9.2.5.1), Lorentz cone (confer Figure 34) [46, exmps.2.3 & 2.25]),{[ ]}xK l = ∈ R n × R | ‖x‖tl ≤ t , l=2 (147)and any polyhedral cone (2.12.1.0.1); e.g., any orthant generated byCartesian half-axes (2.1.3). Esoteric examples of convex cones includethe point at the origin, any line through the origin, any ray having theorigin as base such as the nonnegative real line R + in subspace R , anyhalfspace partially bounded by a hyperplane through the origin, the positivesemidefinite cone S M + (160), the cone of Euclidean distance matrices EDM N(707) (6), any subspace, and Euclidean vector space R n .2.7.2.1 cone invariance(confer Figures: 15, 22, 23, 24, 25, 26, 27, 29, 31, 38, 41, 44, 46, 47,48, 49, 50, 51, 52, 94, 107, 129) More Euclidean bodies are cones,it seems, than are not. This class of convex body, the convex cone, isinvariant to nonnegative scaling, vector summation, affine and inverse affinetransformation, Cartesian product, and intersection, [228, p.22] but is notinvariant to projection; e.g., Figure 33.2.7.2.1.1 Theorem. Cone intersection. [228,2,19]The intersection of an arbitrary collection of convex cones is a convex cone.Intersection of an arbitrary collection of closed convex cones is a closedconvex cone. [188,2.3] Intersection of a finite number of polyhedral cones(2.12.1.0.1, Figure 38 p.123) is polyhedral.⋄The property pointedness is associated with a convex cone.2.7.2.1.2 Definition. Pointed convex cone. (confer2.12.2.2)A convex cone K is pointed iff it contains no line. Equivalently, K is notpointed iff there exists any nonzero direction Γ ∈ K such that −Γ ∈ K .[46,2.4.1] If the origin is an extreme point of K or, equivalently, ifK ∩ −K = {0} (148)then K is pointed, and vice versa. [245,2.10]△