v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 193(314) we get a partial dual to (393); id est, assuming x∈aff cone X{ }x ∈ K ⇔ γ ∗T x ≥ 0 for all γ ∗ ∈ Γ ∗ i , i=1... N ⊂ ∂K ∗ ∩ aff K (408)⇔ X † x ≽ 0 (409)that leads to a partial halfspace-description,K = { x∈aff coneX | X † x ≽ 0 } (410)For γ ∗ =X †T e i , any x =Xa , and for all i we have e T i X † Xa = e T i a ≥ 0only when a ≽ 0. Hence x∈ K .When X is full-rank, then the unique biorthogonal expansion of x ∈ Kbecomes (403)x = XX † x =N∑i=1Γ i Γ ∗Ti x (411)whose coordinates Γ ∗Ti x must be nonnegative because K is assumed pointedclosed and convex. Whenever X is full-rank, so is its pseudoinverse X † .(E) In the present case, the columns of X †T are linearly independentand generators of the dual cone K ∗ ∩ aff K ; hence, the columns constituteits extreme directions. (2.10.2) That section of the dual cone is itself apolyhedral cone (by (287) or the cone intersection theorem,2.7.2.1.1) havingthe same number of extreme directions as K .2.13.8.2 x ∈ aff KThe extreme directions of K and K ∗ ∩ aff K have a distinct relationship;because X † X = I , then for i,j = 1... N , Γ T i Γ ∗ i = 1, while for i ≠ j ,Γ T i Γ ∗ j = 0. Yet neither set of extreme directions, {Γ i } nor {Γ ∗ i } , isnecessarily orthogonal. This is a biorthogonality condition, precisely,[365,2.2.4] [202] implying each set of extreme directions is linearlyindependent. (B.1.1.1)The biorthogonal expansion therefore applies more broadly; meaning,for any x ∈ aff K , vector x can be uniquely expressed x =Xb whereb∈R N because aff K contains the origin. Thus, for any such x∈ R(X)(conferE.1.1), biorthogonal expansion (411) becomes x =XX † Xb =Xb .