v2006.03.09 - Convex Optimization

332 CHAPTER 5. EDM CONEThe set of all symmetric dyads {zz T |z∈R N } constitute the extremedirections of the positive semidefinite cone (2.8.1,2.9) S N + , hence lie onits boundary. Yet only those dyads in R(V N ) are included in the test (777),thus only a subset T of all vectorized extreme directions of S N + is observed.In the particularly simple case D ∈ EDM 2 = {D ∈ S 2 h | d 12 ≥ 0} , forexample, only one extreme direction of the PSD cone is involved:zz T =[1 −1−1 1](780)Any nonnegative scaling of vectorized zz T belongs to the set T illustratedin Figure 82 and Figure 83.5.7.1 Face of PSD cone S N + containing VIn any case, set T (778) constitutes the vectorized extreme directions ofan N(N −1)/2-dimensional face of the PSD cone S N + containing auxiliarymatrix V ; a face isomorphic with S N−1+ = S rank V+ (2.9.2.2).To show this, we must first find the smallest face that contains auxiliarymatrix V and then determine its extreme directions. From (181),F ( S N + ∋V ) = {W ∈ S N + | N(W) ⊇ N(V )} = {W ∈ S N + | N(W) ⊇ 1}= {V Y V ≽ 0 | Y ∈ S N } ≡ {V N BV T N | B ∈ SN−1 + }≃ S rank V+ = −V T N EDMN V N (781)where the equivalence ≡ is from4.6.1 while isomorphic equality ≃ withtransformed EDM cone is from (569). Projector V belongs to F ( S N + ∋V )because V N V † N V †TN V N T = V . (B.4.3) Each and every rank-one matrixbelonging to this face is therefore of the form:V N υυ T V T N | υ ∈ R N−1 (782)Because F ( S N + ∋V ) is isomorphic with a positive semidefinite cone S N−1+ ,then T constitutes the vectorized extreme directions of F , the originconstitutes the extreme points of F , and auxiliary matrix V is some convexcombination of those extreme points and directions by the extremes theorem(2.8.1.1.1).