v2007.09.13 - Convex Optimization

2.7. CONES 87Thus the simplest and only bounded [277, p.75] convex coneK = {0} ⊆ R n is pointed by convention, but has empty interior. Its relativeboundary is the empty set (141) while its relative interior is the pointitself (11). The pointed convex cone that is a halfline emanating from theorigin in R n has the origin as relative boundary while its relative interior isthe halfline itself, excluding the origin.2.7.2.1.3 Theorem. Pointed cones. [41,3.3.15, exer.20]A closed convex cone K ⊂ R n is pointed if and only if there exists a normalα such that the setC ∆ = {x ∈ K | 〈x,α〉=1} (149)is closed, bounded, and K = cone C . Equivalently, if and only if there existsa vector β and positive scalar ǫ such that〈x, β〉 ≥ ǫ‖x‖ ∀x∈ K (150)is K pointed.⋄If closed convex cone K is not pointed, then it has no extreme point. Yeta pointed closed convex cone has only one extreme point; it resides at theorigin. [30,3.3]From the cone intersection theorem it follows that an intersection ofconvex cones is pointed if at least one of the cones is.2.7.2.2 Pointed closed convex cone and partial orderA pointed closed convex cone K induces partial order [279] on R n or R m×n ,[18,1] [240, p.7] respectively defined by vector or matrix inequality;x ≼Kx ≺Kz ⇔ z − x ∈ K (151)z ⇔ z − x ∈ rel int K (152)Neither x or z is necessarily a member of K for these relations to hold. Onlywhen K is the nonnegative orthant do these inequalities reduce to ordinaryentrywise comparison. (2.13.4.2.3) Inclusive of that special case, we ascribe