v2010.10.26 - Convex Optimization

408 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXwhere ∑ Φ ij ∈ S N + (889), 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 (912) into D(G) (903), assuming X1=02D = δ ( −V DV 2) 1 1 T + 1δ ( −V DV 2) 1 T ( )− 2 −V DV12(999)Details of this bijection can be found in5.6.1.1.5.4.2.3 hypersphereThese foregoing relationships allow combination of distance and Gramconstraints in any optimization problem we might pose:Interpoint angle Ψ can be constrained by fixing all the individual pointlengths δ(G) 1/2 ; thenΨ = − 1 2 δ2 (G) −1/2 V DV δ 2 (G) −1/2 (916)(confer5.9.1.0.3, (1098) (1241)) Constraining all main diagonal entriesg ii of a Gram matrix to 1, for example, is equivalent to the constraintthat all points lie on a hypersphere of radius 1 centered at the origin.D = 2(g 11 11 T − G) ∈ EDM N (917)Requiring 0 geometric center then becomes equivalent to the constraint:D1 = 2N1. [89, p.116] Any further constraint on that Gram matrixapplies only to interpoint angle matrix Ψ = G .Because any point list may be constrained to lie on a hypersphere boundarywhose affine dimension exceeds that of the list, a Gram matrix may alwaysbe constrained to have equal positive values along its main diagonal.(Laura Klanfer 1933 [312,3]) This observation renewed interest in theelliptope (5.9.1.0.1).