v2007.09.17 - Convex Optimization

168 CHAPTER 2. CONVEX GEOMETRYFor example, vertex-description (359) simplifies toK ∗ = {X †T b | b ≽ 0} ⊂ R n (361)Now, because dim R(X)= dim R(X †T ) , (E) the dual cone K ∗ is simplicialwhenever K is.2.13.9.3 Cone membership relations in a subspaceIt is obvious by definition (258) of the ordinary dual cone K ∗ in ambient vectorspace R that its determination instead in subspace M⊆ R is identical toits intersection with M ; id est, assuming closed convex cone K ⊆ M andK ∗ ⊆ R(K ∗ were ambient M) ≡ (K ∗ in ambient R) ∩ M (362)because{y ∈ M | 〈y , x〉 ≥ 0 for all x ∈ K}={y ∈ R | 〈y , x〉 ≥ 0 for all x ∈ K}∩ M(363)From this, a constrained membership relation for the ordinary dual coneK ∗ ⊆ R , assuming x,y ∈ M and closed convex cone K ⊆ My ∈ K ∗ ∩ M ⇔ 〈y , x〉 ≥ 0 for all x ∈ K (364)By closure in subspace M we have conjugation (2.13.1.1):x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ ∩ M (365)This means membership determination in subspace M requires knowledgeof the dual cone only in M . For sake of completeness, for proper cone Kwith respect to subspace M (confer (282))x ∈ int K ⇔ 〈y , x〉 > 0 for all y ∈ K ∗ ∩ M, y ≠ 0 (366)x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ int K ∗ ∩ M (367)(By conjugation, we also have the dual relations.) Yet when M equals aff Kfor K a closed convex conex ∈ rel int K ⇔ 〈y , x〉 > 0 for all y ∈ K ∗ ∩ aff K , y ≠ 0 (368)x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ rel int(K ∗ ∩ aff K) (369)