v2009.01.01 - Convex Optimization

2.7. CONES 95
A pointed closed convex cone K induces partial order on R n or R m×n , [20,1]
[281, p.7] essentially defined by vector or matrix inequality;
x ≼
K
z ⇔ z − x ∈ K (164)
x ≺
K
z ⇔ z − x ∈ rel int K (165)
Neither x or z is necessarily a member of K for these relations to hold. Only
when K is the nonnegative orthant do these inequalities reduce to ordinary
entrywise comparison. (2.13.4.2.3) Inclusive of that special case, we ascribe
nomenclature generalized inequality to comparison with respect to a pointed
closed convex cone.
Visceral mechanics of actually comparing points when cone K is
not an orthant are well illustrated in the example of Figure 55 which
relies on the equivalent membership-interpretation in definition (164) or
(165). Comparable points and the minimum element of some vector- or
matrix-valued partially ordered set are thus well defined, so nonincreasing
sequences with respect to cone K can therefore converge in this sense: Point
x ∈ C is the (unique) minimum element of set C with respect to cone K iff
for each and every z ∈ C we have x ≼ z ; equivalently, iff C ⊆ x + K . 2.29
Further properties of partial ordering with respect to pointed closed
convex cone K are:
homogeneity (x≼y , λ≥0 ⇒ λx≼λz),
additivity (x≼z , u≼v ⇒ x+u ≼ z+v),
(x≺y , λ>0 ⇒ λx≺λz)
(x≺z , u≼v ⇒ x+u ≺ z+v)
A closely related concept, minimal element, is useful for partially ordered
sets having no minimum element: Point x ∈ C is a minimal element of set C
with respect to K if and only if (x − K) ∩ C = x . (Figure 34) No uniqueness
is implied here, although implicit is the assumption: dim K ≥ dim aff C .
2.29 Borwein & Lewis [48,3.3, exer.21] ignore possibility of equality to x + K in this
condition, and require a second condition: . . . and C ⊂ y + K for some y in R n implies
x ∈ y + K .