v2006.03.09 - Convex Optimization

328 CHAPTER 5. EDM CONE5.6.0.0.2 Example. ⎡ Extreme ⎤ rays versus rays on the boundary.0 1 4The EDM D = ⎣ 1 0 1 ⎦ is an extreme direction of EDM 3 where[ ]4 1 01u = in (760). Because −VN 2TDV N has eigenvalues {0, 5}, the ray whosedirection is D also lies on the relative [ boundary ] of EDM 3 .0 1In exception, EDM D = κ , for any particular κ > 0, is an1 0extreme direction of EDM 2 but −VN TDV N has only one eigenvalue: {κ} .Because EDM 2 is a ray whose relative boundary (2.6.1.3.1) is the origin,this conventional boundary does not include D which belongs to the relativeinterior in this dimension. (2.7.0.0.1)5.6.1 Gram-form correspondence to S N−1+With respect to D(G)=δ(G)1 T + 1δ(G) T − 2G (472) the linear Gram-formEDM operator, results in4.6.1 provide [1,2.6]EDM N = D ( V(EDM N ) ) ≡ D ( )V N S N−1+ VNT(765)V N S N−1+ VN T ≡ V ( D ( ))V N S N−1+ VN T = V(EDM N ) = ∆ −V EDM N V 1 = 2 SN c ∩ S N +(766)a one-to-one correspondence between EDM N and S N−1+ .5.6.2 EDM cone by elliptopeDefining the elliptope parametrized by scalar t>0E N t∆= S N + ∩ {Φ∈ S N | δ(Φ)=t1} (644)then following Alfakih [7] we haveEDM N = cone{11 T − E N 1 } = {t(11 T − E N 1 ) | t ≥ 0} (767)Identification E N = E1Nparametrized elliptope.equates the standard elliptope (4.9.1.0.1) to our