v2010.10.26 - Convex Optimization

516 CHAPTER 6. CONE OF DISTANCE MATRICES6.5.2.0.1 Expository. Normal cone, tangent cone, elliptope.Define T E (11 T ) to be the tangent cone to the elliptope E at point 11 T ;id est,T E (11 T ) {t(E − 11 T ) | t≥0} (1223)The normal cone K ⊥ E (11T ) to the elliptope at 11 T is a closed convex conedefined (E.10.3.2.1, Figure 175)K ⊥ E (11 T ) {B | 〈B , Φ − 11 T 〉 ≤ 0, Φ∈ E } (1224)The polar cone of any set K is the closed convex cone (confer (297))K ◦ {B | 〈B , A〉≤0, for all A∈ K} (1225)The normal cone is well known to be the polar of the tangent cone,and vice versa; [199,A.5.2.4]K ⊥ E (11 T ) = T E (11 T ) ◦ (1226)K ⊥ E (11 T ) ◦ = T E (11 T ) (1227)From Deza & Laurent [113, p.535] we have the EDM coneEDM = −T E (11 T ) (1228)The polar EDM cone is also expressible in terms of the elliptope. From (1226)we haveEDM ◦ = −K ⊥ E (11 T ) (1229)⋆In5.10.1 we proposed the expression for EDM DD = t11 T − E ∈ EDM N (1098)where t∈ R + and E belongs to the parametrized elliptope E N t . We furtherpropose, for any particular t>0EDM N = cone{t11 T − E N t } (1230)Proof. Pending.6.5.2.0.2 Exercise. EDM cone from elliptope.Relationship of the translated negated elliptope with the EDM cone isillustrated in Figure 141. Prove whether it holds thatEDM N = limt→∞t11 T − E N t (1231)