v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
94 CHAPTER 2. CONVEX GEOMETRY C 1 C 2 (a) x + K y x (b) y − K Figure 34: (a) Point x is the minimum element of set C 1 with respect to cone K because cone translated to x∈ C 1 contains set. (b) Point y is a minimal element of set C 2 with respect to cone K because negative cone translated to y ∈ C 2 contains only y . (Cones drawn truncated in R 2 .)
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 .
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94 CHAPTER 2. CONVEX GEOMETRY<br />
C 1<br />
C 2<br />
(a)<br />
x + K<br />
y<br />
x<br />
(b)<br />
y − K<br />
Figure 34: (a) Point x is the minimum element of set C 1 with respect to<br />
cone K because cone translated to x∈ C 1 contains set. (b) Point y is a<br />
minimal element of set C 2 with respect to cone K because negative cone<br />
translated to y ∈ C 2 contains only y . (Cones drawn truncated in R 2 .)
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