v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
424 CHAPTER 6. EDM CONEFrom the results inE.7.2.0.2, we know matrix product V H V is theorthogonal projection of H ∈ S N on the geometric center subspace S N c . Thusthe projection productP K2 H = H − P S N+P S N cH (1065)6.8.1.1.1 Lemma. Projection on PSD cone ∩ geometric center subspace.P S N+ ∩ S N c= P S N+P S N c(1066)⋄Proof. For each and every H ∈ S N , projection of P S N cH on the positivesemidefinite cone remains in the geometric center subspaceS N c = {G∈ S N | G1 = 0} (1761)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1762)(795)That is because: eigenvectors of P S N cH corresponding to 0 eigenvaluesspan its nullspace N(P S N cH). (A.7.3.0.1) To project P S N cH on the positivesemidefinite cone, its negative eigenvalues are zeroed. (7.1.2) The nullspaceis thereby expanded while eigenvectors originally spanning N(P S N cH)remain intact. Because the geometric center subspace is invariant toprojection on the PSD cone, then the rule for projection on a convex setin a subspace governs (E.9.5, projectors do not commute) and statement(1066) follows directly. From the lemma it followsThen from (1788){P S N+P S N cH | H ∈ S N } = {P S N+ ∩ S N c H | H ∈ SN } (1067)− ( S N c ∩ S N +) ∗= {H − PS N+P S N cH | H ∈ S N } (1068)From (272) we get closure of a vector sumK 2 = − ( )S N c ∩ S N ∗+ = SN⊥c − S N + (1069)therefore the new equalityEDM N = K 1 ∩ K 2 = S N h ∩ ( )S N⊥c − S N +(1070)
6.8. DUAL EDM CONE 425whose veracity is intuitively evident, in hindsight, [61, p.109] from themost fundamental EDM definition (705). Formula (1070) is not a matrixcriterion for membership to the EDM cone, and it is not an equivalencebetween EDM operators. Rather, it is a recipe for constructing the EDMcone whole from large Euclidean bodies: the positive semidefinite cone,orthogonal complement of the geometric center subspace, and symmetrichollow subspace. A realization of this construction in low dimension isillustrated in Figure 104 and Figure 105.The dual EDM cone follows directly from (1070) by standard propertiesof cones (2.13.1.1):EDM N∗ = K ∗ 1 + K ∗ 2 = S N⊥h − S N c ∩ S N + (1071)which bears strong resemblance to (1050).6.8.1.2 nonnegative orthantThat EDM N is a proper subset of the nonnegative orthant is not obviousfrom (1070). We wish to verifyEDM N = S N h ∩ ( )S N⊥c − S N + ⊂ RN×N+ (1072)While there are many ways to prove this, it is sufficient to show that allentries of the extreme directions of EDM N must be nonnegative; id est, forany particular nonzero vector z = [z i , i=1... N]∈ N(1 T ) (6.5.3.1),δ(zz T )1 T + 1δ(zz T ) T − 2zz T ≥ 0 (1073)where the inequality denotes entrywise comparison. The inequality holdsbecause the i,j th entry of an extreme direction is squared: (z i − z j ) 2 .We observe that the dyad 2zz T ∈ S N + belongs to the positive semidefinitecone, the doubletδ(zz T )1 T + 1δ(zz T ) T ∈ S N⊥c (1074)to the orthogonal complement (1763) of the geometric center subspace, whiletheir difference is a member of the symmetric hollow subspace S N h .Here is an algebraic method provided by Trosset to prove nonnegativity:Suppose we are given A∈ S N⊥c and B = [B ij ]∈ S N + and A −B ∈ S N h .Then we have, for some vector u , A = u1 T + 1u T = [A ij ] = [u i + u j ] andδ(B)= δ(A)= 2u . Positive semidefiniteness of B requires nonnegativityA −B ≥ 0 because(e i −e j ) T B(e i −e j ) = (B ii −B ij )−(B ji −B jj ) = 2(u i +u j )−2B ij ≥ 0 (1075)
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424 CHAPTER 6. EDM CONEFrom the results inE.7.2.0.2, we know matrix product V H V is theorthogonal projection of H ∈ S N on the geometric center subspace S N c . Thusthe projection productP K2 H = H − P S N+P S N cH (1065)6.8.1.1.1 Lemma. Projection on PSD cone ∩ geometric center subspace.P S N+ ∩ S N c= P S N+P S N c(1066)⋄Proof. For each and every H ∈ S N , projection of P S N cH on the positivesemidefinite cone remains in the geometric center subspaceS N c = {G∈ S N | G1 = 0} (1761)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1762)(795)That is because: eigenvectors of P S N cH corresponding to 0 eigenvaluesspan its nullspace N(P S N cH). (A.7.3.0.1) To project P S N cH on the positivesemidefinite cone, its negative eigenvalues are zeroed. (7.1.2) The nullspaceis thereby expanded while eigenvectors originally spanning N(P S N cH)remain intact. Because the geometric center subspace is invariant toprojection on the PSD cone, then the rule for projection on a convex setin a subspace governs (E.9.5, projectors do not commute) and statement(1066) follows directly. From the lemma it followsThen from (1788){P S N+P S N cH | H ∈ S N } = {P S N+ ∩ S N c H | H ∈ SN } (1067)− ( S N c ∩ S N +) ∗= {H − PS N+P S N cH | H ∈ S N } (1068)From (272) we get closure of a vector sumK 2 = − ( )S N c ∩ S N ∗+ = SN⊥c − S N + (1069)therefore the new equalityEDM N = K 1 ∩ K 2 = S N h ∩ ( )S N⊥c − S N +(1070)