v2007.09.13 - Convex Optimization

6.6. CORRESPONDENCE TO PSD CONE S N−1+ 4116.6.0.0.2 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 (1020). Because −VN 2TDV N has eigenvalues {0, 5}, the raywhose direction 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)6.6.1 Gram-form correspondence to S N−1+With respect to D(G)=δ(G)1 T + 1δ(G) T − 2G (717) the linear Gram-formEDM operator, results in5.6.1 provide [1,2.6]EDM N = D ( V(EDM N ) ) ≡ D ( )V N S N−1+ VNT(1025)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 +(1026)a one-to-one correspondence between EDM N and S N−1+ .6.6.2 EDM cone by elliptope(confer5.10.1) Defining the elliptope parametrized by scalar t>0then following Alfakih [7] we haveE N t = S N + ∩ {Φ∈ S N | δ(Φ)=t1} (900)EDM N = cone{11 T − E N 1 } = {t(11 T − E N 1 ) | t ≥ 0} (1027)Identification E N = E N 1 equates the standard elliptope (5.9.1.0.1, Figure 86)to our parametrized elliptope.