v2007.09.13 - Convex Optimization

27Figure 7: These bees construct a honeycomb by solving a convex optimizationproblem. (5.4.2.2.3) The most dense packing of identical spheres about acentral sphere in two dimensions is 6. Packed sphere centers describe aregular lattice.The EDM is studied in chapter 5, Euclidean distance matrix, itsproperties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical properties of the Euclidean metric;thereby, observing existence of an infinity of properties of the Euclideanmetric beyond the triangle inequality. We proceed by deriving the fifthEuclidean metric property and then explain why furthering this endeavor isinefficient because the ensuing criteria (while describing polyhedra in angleor area, volume, content, and so on ad infinitum) grow linearly in complexityand number with problem size.Reconstruction methods are explained and applied to a map of theUnited States; e.g., Figure 6. We also generate a distorted but recognizableisotonic map using only comparative distance information (only ordinaldistance data). We demonstrate an elegant method for including dihedral(or torsion) angle constraints into a molecular conformation problem. Moregeometrical problems solvable via EDMs are presented with the best methodsfor posing them, EDM problems are posed as convex optimizations, andwe show how to recover exact relative position given incomplete noiselessinterpoint distance information.The set of all Euclidean distance matrices forms a pointed closed convexcone called the EDM cone, EDM N . We offer a new proof of Schoenberg’sseminal characterization of EDMs:{−VTD ∈ EDM NN DV N ≽ 0⇔(724)D ∈ S N hOur proof relies on fundamental geometry; assuming, any EDM mustcorrespond to a list of points contained in some polyhedron (possibly at