v2006.03.09 - Convex Optimization

338 CHAPTER 5. EDM CONEProof. 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) (796)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 (797)by Corollary A.3.1.0.5.Next, we requireExpanding,〈P K2 H −H , P K2 H 〉 = 0 (798)〈−P S N+(V H V ) , H −P S N+(V H V )〉 = 0 (799)〈P S N+(V H V ) , (P S N+(V H V ) − V H V ) + (V H V − H)〉 = 0 (800)〈P S N+(V H V ) , (V H V − H)〉 = 0 (801)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} (802)Diagonalize V H V ∆ =QΛQ T (A.5) whose nullspace is spanned bythe eigenvectors corresponding to 0 eigenvalues by Theorem A.7.2.0.1.Projection of V H V on the PSD cone (7.1) simply zeros negative eigenvaluesin diagonal matrix Λ . Thenfrom which it follows:N(P S N+(V H V )) ⊇ N(V H V ) (⊇ N(V )) (803)P S N+(V H V ) ∈ S N c (804)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 .From5.7.1we 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 ) (2.9.2.2) which wasalready established in (803).