v2007.09.13 - Convex Optimization

2.7. CONES 89nomenclature generalized inequality to comparison with respect to a pointedclosed convex cone.The visceral mechanics of actually comparing points when the cone Kis not an orthant is well illustrated in the example of Figure 49 whichrelies on the equivalent membership-interpretation in definition (151) or(152). Comparable points and the minimum element of some vector- ormatrix-valued partially ordered set are thus well defined, so decreasingsequences with respect to cone K can therefore converge in this sense: Pointx ∈ C is the (unique) minimum element of set C with respect to cone K ifffor each and every z ∈ C we have x ≼ z ; equivalently, iff C ⊆ x + K . 2.23Further properties of partial ordering with respect to pointed closedconvex cone K are:reflexivity (x≼x)antisymmetry (x≼z , z ≼x ⇒ x=z)transitivity (x≼y , y ≼z ⇒ x≼z),homogeneity (x≼y , λ≥0 ⇒ λx≼λz),additivity (x≼z , u≼v ⇒ x+u ≼ z+v),(x≼y , y ≺z ⇒ x≺z)(x≺y , λ>0 ⇒ λx≺λz)(x≺z , u≼v ⇒ x+u ≺ z+v)A closely related concept, minimal element, is useful for partially orderedsets having no minimum element: Point x ∈ C is a minimal element of set Cwith respect to K if and only if (x − K) ∩ C = x . (Figure 28) No uniquenessis implied here, although implicit is the assumption: dim K ≥ dim aff C .2.7.2.2.1 Definition. Proper cone: [46,2.4.1] a cone that ispointedclosedconvexhas nonempty interior (is full-dimensional).△2.23 Borwein & Lewis [41,3.3, 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 .