v2010.10.26 - Convex Optimization

2.7. CONES 103cone that is a halfline, emanating from the origin in R n , has relative boundary0 while its relative interior is the halfline itself excluding 0. Pointed areany Lorentz cone, cone of Euclidean distance matrices EDM N in symmetrichollow subspace S N h , and positive semidefinite cone S M + in ambient S M .2.7.2.1.3 Theorem. Pointed cones. [55,3.3.15, exer.20]A closed convex cone K ⊂ R n is pointed if and only if there exists a normal αsuch that the setC {x ∈ K | 〈x,α〉 = 1} (180)is closed, bounded, and K = cone C . Equivalently, K is pointed if and onlyif there exists a vector β normal to a hyperplane strictly supporting K ;id est, for some positive scalar ǫ〈x, β〉 ≥ ǫ‖x‖ ∀x∈ K (181)If closed convex cone K is not pointed, then it has no extreme point. 2.29Yet a pointed closed convex cone has only one extreme point [42,3.3]: theexposed point residing at the origin; its vertex. Pointedness is invariantto Cartesian product by (179). And from the cone intersection theorem itfollows that an intersection of convex cones is pointed if at least one of thecones is; implying, each and every nonempty exposed face of a pointed closedconvex cone is a pointed closed convex cone.⋄2.7.2.2 Pointed closed convex cone induces partial orderRelation ≼ represents partial order on some set if that relation possesses 2.30reflexivity (x≼x)antisymmetry (x≼z , z ≼x ⇒ x=z)transitivity (x≼y , y ≼z ⇒ x≼z),(x≼y , y ≺z ⇒ x≺z)2.29 nor does it have extreme directions (2.8.1).2.30 A set is totally ordered if it further obeys a comparability property of the relation: foreach and every x and y from the set, x ≼ y or y ≼ x; e.g., one-dimensional real vectorspace R is the smallest unbounded totally ordered and connected set.