v2007.11.26 - Convex Optimization

436 CHAPTER 6. CONE OF DISTANCE MATRICESThe ordinary dual EDM cone cannot, therefore, be pointed. (2.13.1.1)When N = 1, the EDM cone is the point at the origin in R . Auxiliarymatrix V N is empty [ ∅ ] , and dual cone EDM ∗ is the real line.When N = 2, the EDM cone is a nonnegative real line in isometricallyisomorphic R 3 ; there S 2 h is a real line containing the EDM cone. Dual coneEDM 2∗ is the particular halfspace in R 3 whose boundary has inward-normalEDM 2 . Diagonal matrices {δ(u)} in (1080) are represented by a hyperplanethrough the origin {d | [ 0 1 0 ]d = 0} while the term cone{V N υυ T V T N }is represented by the halfline T in Figure 108 belonging to the positivesemidefinite cone boundary. The dual EDM cone is formed by translatingthe hyperplane along the negative semidefinite halfline −T ; the union ofeach and every translation. (confer2.10.2.0.1)When cardinality N exceeds 2, the dual EDM cone can no longer bepolyhedral simply because the EDM cone cannot. (2.13.1.1)6.8.1.1 EDM cone and its dual in ambient S NConsider the two convex conessoK ∆ 1 = S N hK ∆ 2 =⋂{A ∈ S N | 〈yy T , −A〉 ≥ 0 }y∈N(1 T )= { A ∈ S N | −z T V AV z ≥ 0 ∀zz T (≽ 0) } (1082)= { A ∈ S N | −V AV ≽ 0 }K 1 ∩ K 2 = EDM N (1083)The dual cone K ∗ 1 = S N⊥h ⊆ S N (63) is the subspace of diagonal matrices.From (1080) via (273),K ∗ 2 = − cone { V N υυ T V T N | υ ∈ R N−1} ⊂ S N (1084)Gaffke & Mathar [99,5.3] observe that projection on K 1 and K 2 havesimple closed forms: Projection on subspace K 1 is easily performed bysymmetrization and zeroing the main diagonal or vice versa, while projectionof H ∈ S N on K 2 isP K2 H = H − P S N+(V H V ) (1085)