v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
472 CHAPTER 6. CONE OF DISTANCE MATRICES 6.7 Vectorization & projection interpretation InE.7.2.0.2 we learn: −V DV can be interpreted as orthogonal projection [5,2] of vectorized −D ∈ S N h on the subspace of geometrically centered symmetric matrices S N c = {G∈ S N | G1 = 0} (1874) = {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )} = {V Y V | Y ∈ S N } ⊂ S N (1875) ≡ {V N AVN T | A ∈ SN−1 } (895) because elementary auxiliary matrix V is an orthogonal projector (B.4.1). Yet there is another useful projection interpretation: Revising the fundamental matrix criterion for membership to the EDM cone (793), 6.10 〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0 D ∈ S N h } ⇔ D ∈ EDM N (1140) this is equivalent, of course, to the Schoenberg criterion −VN TDV } N ≽ 0 ⇔ D ∈ EDM N (817) D ∈ S N h because N(11 T ) = R(V N ). When D ∈ EDM N , correspondence (1140) means −z T Dz is proportional to a nonnegative coefficient of orthogonal projection (E.6.4.2, Figure 123) of −D in isometrically isomorphic R N(N+1)/2 on the range of each and every vectorized (2.2.2.1) symmetric dyad (B.1) in the nullspace of 11 T ; id est, on each and every member of T ∆ = { svec(zz T ) | z ∈ N(11 T )= R(V N ) } ⊂ svec ∂ S N + = { svec(V N υυ T V T N ) | υ ∈ RN−1} (1141) whose dimension is dim T = N(N − 1)/2 (1142) 6.10 N(11 T )= N(1 T ) and R(zz T )= R(z)
6.7. VECTORIZATION & PROJECTION INTERPRETATION 473 The set of all symmetric dyads {zz T |z∈R N } constitute the extreme directions of the positive semidefinite cone (2.8.1,2.9) S N + , hence lie on its boundary. Yet only those dyads in R(V N ) are included in the test (1140), 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} , for example, only one extreme direction of the PSD cone is involved: zz T = [ 1 −1 −1 1 ] (1143) Any nonnegative scaling of vectorized zz T belongs to the set T illustrated in Figure 122 and Figure 123. 6.7.1 Face of PSD cone S N + containing V In any case, set T (1141) constitutes the vectorized extreme directions of an N(N −1)/2-dimensional face of the PSD cone S N + containing auxiliary matrix V ; a face isomorphic with S N−1 + = S rank V + (2.9.2.3). To show this, we must first find the smallest face that contains auxiliary matrix V and then determine its extreme directions. From (203), 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 (1144) where the equivalence ≡ is from5.6.1 while isomorphic equality ≃ with transformed EDM cone is from (925). Projector V belongs to F ( S N + ∋V ) because V N V † N V †T N V N T = V . (B.4.3) Each and every rank-one matrix belonging to this face is therefore of the form: V N υυ T V T N | υ ∈ R N−1 (1145) 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 origin constitutes the extreme points of F , and auxiliary matrix V is some convex combination of those extreme points and directions by the extremes theorem (2.8.1.1.1).
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6.7. VECTORIZATION & PROJECTION INTERPRETATION 473<br />
The set of all symmetric dyads {zz T |z∈R N } constitute the extreme<br />
directions of the positive semidefinite cone (2.8.1,2.9) S N + , hence lie on<br />
its boundary. Yet only those dyads in R(V N ) are included in the test (1140),<br />
thus only a subset T of all vectorized extreme directions of S N + is observed.<br />
In the particularly simple case D ∈ EDM 2 = {D ∈ S 2 h | d 12 ≥ 0} , for<br />
example, only one extreme direction of the PSD cone is involved:<br />
zz T =<br />
[<br />
1 −1<br />
−1 1<br />
]<br />
(1143)<br />
Any nonnegative scaling of vectorized zz T belongs to the set T illustrated<br />
in Figure 122 and Figure 123.<br />
6.7.1 Face of PSD cone S N + containing V<br />
In any case, set T (1141) constitutes the vectorized extreme directions of<br />
an N(N −1)/2-dimensional face of the PSD cone S N + containing auxiliary<br />
matrix V ; a face isomorphic with S N−1<br />
+ = S rank V<br />
+ (2.9.2.3).<br />
To show this, we must first find the smallest face that contains auxiliary<br />
matrix V and then determine its extreme directions. From (203),<br />
F ( S N + ∋V ) = {W ∈ S N + | N(W) ⊇ N(V )} = {W ∈ S N + | N(W) ⊇ 1}<br />
= {V Y V ≽ 0 | Y ∈ S N } ≡ {V N BV T N | B ∈ SN−1 + }<br />
≃ S rank V<br />
+ = −V T N EDMN V N (1144)<br />
where the equivalence ≡ is from5.6.1 while isomorphic equality ≃ with<br />
transformed EDM cone is from (925). Projector V belongs to F ( S N + ∋V )<br />
because V N V † N V †T<br />
N V N T = V . (B.4.3) Each and every rank-one matrix<br />
belonging to this face is therefore of the form:<br />
V N υυ T V T N | υ ∈ R N−1 (1145)<br />
Because F ( S N + ∋V ) is isomorphic with a positive semidefinite cone S N−1<br />
+ ,<br />
then T constitutes the vectorized extreme directions of F , the origin<br />
constitutes the extreme points of F , and auxiliary matrix V is some convex<br />
combination of those extreme points and directions by the extremes theorem<br />
(2.8.1.1.1).