v2009.01.01 - Convex Optimization

6.8. DUAL EDM CONE 479
Proof. First, we observe membership of H −P S N
+
(V H V ) to K 2 because
(
)
P S N
+
(V H V ) − H = P S N
+
(V H V ) − V H V + (V H V − H) (1159)
The term P S N
+
(V H V ) − V H V necessarily belongs to the (dual) positive
semidefinite cone by Theorem E.9.2.0.1. V 2 = V , hence
(
)
−V H −P S N
+
(V H V ) V ≽ 0 (1160)
by Corollary A.3.1.0.5.
Next, we require
Expanding,
〈P K2 H −H , P K2 H 〉 = 0 (1161)
〈−P S N
+
(V H V ) , H −P S N
+
(V H V )〉 = 0 (1162)
〈P S N
+
(V H V ) , (P S N
+
(V H V ) − V H V ) + (V H V − H)〉 = 0 (1163)
〈P S N
+
(V H V ) , (V H V − H)〉 = 0 (1164)
Product V H V belongs to the geometric center subspace; (E.7.2.0.2)
V H V ∈ S N c = {Y ∈ S N | N(Y )⊇1} (1165)
Diagonalize V H V ∆ =QΛQ T (A.5) whose nullspace is spanned by
the eigenvectors corresponding to 0 eigenvalues by Theorem A.7.3.0.1.
Projection of V H V on the PSD cone (7.1) simply zeroes negative
eigenvalues in diagonal matrix Λ . Then
from which it follows:
N(P S N
+
(V H V )) ⊇ N(V H V ) (⊇ N(V )) (1166)
P S N
+
(V H V ) ∈ S N c (1167)
so P S N
+
(V H V ) ⊥ (V H V −H) because V H V −H ∈ S N⊥
c .
Finally, we must have P K2 H −H =−P S N
+
(V H V )∈ K ∗ 2 .
From6.7.1
we know dual cone K ∗ 2 =−F ( S N + ∋V ) is the negative of the positive
semidefinite cone’s smallest face that contains auxiliary matrix V . Thus
P S N
+
(V H V )∈ F ( S N + ∋V ) ⇔ N(P S N
+
(V H V ))⊇ N(V ) (2.9.2.3) which was
already established in (1166).