v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1952.13.9.2 Simplicial caseWhen a convex cone is simplicial (2.12.3), Cone Table 1 simplifies becausethen aff coneX = R n : For square X and assuming simplicial K such thatrank(X ∈ R n×N ) = N dim aff K = n (419)we haveCone Table S K K ∗vertex-description X X †Thalfspace-description X † X TFor example, vertex-description (418) simplifies toK ∗ = {X †T b | b ≽ 0} ⊂ R n (420)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 (297) of ordinary dual cone K ∗ , in ambient vectorspace R , that its determination instead in subspace S ⊆ R is identicalto its intersection with S ; id est, assuming closed convex cone K ⊆ S andK ∗ ⊆ R(K ∗ were ambient S) ≡ (K ∗ in ambient R) ∩ S (421)because{y ∈ S | 〈y , x〉 ≥ 0 for all x ∈ K} = {y ∈ R | 〈y , x〉 ≥ 0 for all x ∈ K} ∩ S(422)From this, a constrained membership relation for the ordinary dual coneK ∗ ⊆ R , assuming x,y ∈ S and closed convex cone K ⊆ Sy ∈ K ∗ ∩ S ⇔ 〈y , x〉 ≥ 0 for all x ∈ K (423)By closure in subspace S we have conjugation (2.13.1.1):x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ ∩ S (424)