v2010.10.26 - Convex Optimization

6.8. DUAL EDM CONE 531Proof. 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) (1254)The term P S N+(V H V ) − V H V necessarily belongs to the (dual) positivesemidefinite cone by Theorem E.9.2.0.1. V 2 = V , hence()−V H −P S N+(V H V ) V ≽ 0 (1255)by Corollary A.3.1.0.5.Next, we requireExpanding,〈P K2 H −H , P K2 H 〉 = 0 (1256)〈−P S N+(V H V ) , H −P S N+(V H V )〉 = 0 (1257)〈P S N+(V H V ) , (P S N+(V H V ) − V H V ) + (V H V − H)〉 = 0 (1258)〈P S N+(V H V ) , (V H V − H)〉 = 0 (1259)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} (1260)Diagonalize V H V QΛQ T (A.5) whose nullspace is spanned bythe 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 negativeeigenvalues in diagonal matrix Λ . ThenN(P S N+(V H V )) ⊇ N(V H V ) (⊇ N(V )) (1261)from which it follows:P S N+(V H V ) ∈ S N c (1262)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 .From6.6.1we know dual cone K ∗ 2 =−F ( S N + ∋V ) is the negative of the positivesemidefinite cone’s smallest face that contains auxiliary matrix V . ThusP S N+(V H V )∈ F ( S N + ∋V ) ⇔ N(P S N+(V H V ))⊇ N(V ) (2.9.2.3) which wasalready established in (1261).