v2010.10.26 - Convex Optimization

5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS 461In fact, any positive semidefinite matrix whose entries belong to {±1} isa rank-one correlation matrix; and vice versa: 5.465.9.1.0.2 Theorem. Elliptope vertices rank-one. [125,2.1.1]For Y ∈ S n , y ∈ R n , and all i,j∈{1... n} (confer2.3.1.0.2)Y ≽ 0, Y ij ∈ {±1} ⇔ Y = yy T , y i ∈ {±1} (1078)⋄The elliptope for dimension n = 2 is a line segment in isometricallyisomorphic R n(n+1)/2 (Figure 131). Obviously, cone(E n )≠ S n + . The elliptopefor dimension n = 3 is realized in Figure 130.5.9.1.0.3 Lemma. Hypersphere. (confer bullet p.408) [17,4]Matrix Ψ = [Ψ ij ]∈ S N belongs to the elliptope in S N iff there exist N points pon the boundary of a hypersphere in R rankΨ having radius 1 such that‖p i − p j ‖ 2 = 2(1 − Ψ ij ) , i,j=1... N (1079)⋄There is a similar theorem for Euclidean distance matrices:We derive matrix criteria for D to be an EDM, validating (910) usingsimple geometry; distance to the polyhedron formed by the convex hull of alist of points (76) in Euclidean space R n .5.9.1.0.4 EDM assertion.D is a Euclidean distance matrix if and only if D ∈ S N h and distances-squarefrom the origin{‖p(y)‖ 2 = −y T V T NDV N y | y ∈ S − β} (1080)correspond to points p in some bounded convex polyhedronP − α = {p(y) | y ∈ S − β} (1081)having N or fewer vertices embedded in an r-dimensional subspace A − α ofR n , where α ∈ A = aff P and where domain of linear surjection p(y) is theunit simplex S ⊂ R N−1+ shifted such that its vertex at the origin is translatedto −β in R N−1 . When β = 0, then α = x 1 .⋄5.46 As there are few equivalent conditions for rank constraints, this device is ratherimportant for relaxing integer, combinatorial, or Boolean problems.