v2010.10.26 - Convex Optimization

2.7. CONES 105A pointed closed convex cone K induces partial order on R n or R m×n , [21,1][324, p.7] essentially defined by vector or matrix inequality;x ≼Kz ⇔ z − x ∈ K (182)x ≺Kz ⇔ z − x ∈ rel int K (183)Neither x or z is necessarily a member of K for these relations to hold.Only when K is a nonnegative orthant R n + do these inequalities reduceto ordinary entrywise comparison (2.13.4.2.3) while partial order lingers.Inclusive of that special case, we ascribe nomenclature generalized inequalityto comparison with respect to a pointed closed convex cone.We say two points x and y are comparable when x ≼ y or y ≼ xwith respect to pointed closed convex cone K . Visceral mechanics ofactually comparing points, when cone K is not an orthant, are wellillustrated in the example of Figure 63 which relies on the equivalentmembership-interpretation in definition (182) or (183).Comparable points and the minimum element of some vector- ormatrix-valued partially ordered set are thus well defined, so nonincreasingsequences 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 . 2.31A closely related concept, minimal element, is useful for partially orderedsets having no minimum element: Point x ∈ C is a minimal elementof set C with respect to pointed closed convex cone K if and only if(x − K) ∩ C = x . (Figure 40) No uniqueness is implied here, althoughimplicit is the assumption: dim K ≥ dim aff C . In words, a point that isa minimal element is smaller (with respect to K) than any other point in theset to which it is comparable.Further properties of partial order with respect to pointed closed convexcone K are not defining:homogeneity (x≼y , λ≥0 ⇒ λx≼λz),(x≺y , λ>0 ⇒ λx≺λz)additivity (x≼z , u≼v ⇒ x+u ≼ z+v),(x≺z , u≼v ⇒ x+u ≺ z+v)2.31 Borwein & Lewis [55,3.3 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 .