v2006.03.09 - Convex Optimization

228 CHAPTER 4. EUCLIDEAN DISTANCE MATRIXD ∈ EDM N⇔{−V DV ∈ SN+D ∈ S N h(484)Of particular utility when D ∈ EDM N is the fact, (B.4.2 no.20) (456)tr ( −V DV 1 2)=12N∑i,jd ij = 12N vec(X)T (∑i,jΦ ij ⊗ I= tr(V GV ) , G ≽ 0)vec X∑= trG = N ‖x l ‖ 2 = ‖X‖ 2 F , X1 = 0l=1(485)where ∑ Φ ij ∈ S N + (458), therefore convex in vecX . We will find this traceuseful as a heuristic to minimize affine dimension of an unknown list arrangedcolumnar in X , (7.2.2) but it tends to facilitate reconstruction of a listconfiguration having least energy; id est, it compacts a reconstructed list byminimizing total norm-square of the vertices.By substituting G=−V DV 1 (482) into D(G) (472), assuming X1=02(confer4.6.1)D = δ ( −V DV 2) 1 1 T + 1δ ( −V DV 2) 1 T ( )− 2 −V DV12(486)These relationships will allow combination of distance and Gramconstraints in any optimization problem we may pose:Constraining all main-diagonal entries of a Gram matrix to 1, forexample, is equivalent to the constraint that all points lie on ahypersphere (4.9.1.0.2) of radius 1 centered at the origin. Thisis equivalent to the EDM constraint: D1 = 2N1. [50, p.116] Anyfurther constraint on that Gram matrix then applies only to interpointangle Ψ .More generally, interpoint angle Ψ can be constrained by fixing all theindividual point lengths δ(G) 1/2 ; thenΨ = − 1 2 δ2 (G) −1/2 V DV δ 2 (G) −1/2 (487)