v2007.09.17 - Convex Optimization

6.6. CORRESPONDENCE TO PSD CONE S N−1+ 4156.6.2.0.1 Expository. Define T E (11 T ) to be the tangent cone to theelliptope E at point 11 T ; id est,T E (11 T ) ∆ = {t(E − 11 T ) | t≥0} (1032)The normal cone K ⊥ E (11T ) to the elliptope at 11 T is a closed convex conedefined (E.10.3.2.1, Figure 130)K ⊥ E (11 T ) ∆ = {B | 〈B , Φ − 11 T 〉 ≤ 0, Φ∈ E } (1033)The polar cone of any set K is the closed convex cone (confer (258))K ◦ ∆ = {B | 〈B , A〉≤0, for all A∈ K} (1034)The normal cone is well known to be the polar of the tangent cone,and vice versa; [148,A.5.2.4]K ⊥ E (11 T ) = T E (11 T ) ◦ (1035)K ⊥ E (11 T ) ◦ = T E (11 T ) (1036)From Deza & Laurent [77, p.535] we have the EDM coneEDM = −T E (11 T ) (1037)The polar EDM cone is also expressible in terms of the elliptope. From(1035) we haveEDM ◦ = −K ⊥ E (11 T ) (1038)⋆In5.10.1 we proposed the expression for EDM DD = t11 T − E ∈ EDM N (905)where t∈ R + and E belongs to the parametrized elliptope E N t . We furtherpropose, for any particular t>0Proof. Pending.EDM N = cone{t11 T − E N t } (1039)Relationship of the translated negated elliptope with the EDM cone isillustrated in Figure 102. We speculateEDM N = limt→∞t11 T − E N t (1040)