v2006.03.09 - Convex Optimization

100 CHAPTER 2. CONVEX GEOMETRYThe visceral mechanics of actually comparing vectors when the cone is notan orthant is well illustrated in the example of Figure 46 which relies on theequivalent membership-interpretation in (145) or (146). Comparable pointsand the minimum element of some vector- or matrix-valued set are thus welldefined (and decreasing sequences with respect to K therefore converge):Vector x ∈ C is the (unique) minimum element of set C with respect tocone K if and only if for each and every z ∈ C we have x ≼ z ; equivalently,iff C ⊆ x + K . 2.21 A closely related concept minimal element, is useful forsets having no minimum element: Vector x ∈ C is a minimal element of setC with respect to K if and only if (x − K) ∩ C = x .More properties of partial ordering with respect to K are catalogedin [38,2.4] with respect to proper cones: 2.22 such as reflexivity (x≼x),antisymmetry (x≼z , z ≼x ⇒ x=z), transitivity (x≼y , y ≼z ⇒ x≼z),additivity (x≼z , u≼v ⇒ x+u ≼ z+v), strict properties, and preservationunder nonnegative scaling or limiting operations.2.7.2.2.1 Definition. Proper cone: [38,2.4.1] A cone that ispointedclosedconvexhas nonempty interior△A proper cone remains proper under injective linear transformation.[51,5.1]2.21 Borwein & Lewis [36,3.3, exer.21] ignore possibility of equality to x + K in thiscondition, and require a second condition: . . . and C ⊂ y + K for some y in R n impliesx ∈ y + K .2.22 We distinguish pointed closed convex cones here because the generalized inequalityand membership corollary (2.13.2.0.1) remains intact. [123,A.4.2.7]