v2010.10.26 - Convex Optimization

428 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXFigure 127: A depiction of molecular conformation. [116]noise will often cause rankG ⋆ to exceed 2. Randomized search for a rank-2optimal solution is not so easy here as in Example 5.4.2.3.8. We introduce amethod in4.4 for enforcing the stronger rank-constraint (940). To formulatethis same problem in three dimensions, point list X is simply redimensionedin the semidefinite program.5.4.2.3.10 Example. (Biswas, Nigam, Ye) Molecular Conformation.The subatomic measurement technique called nuclear magnetic resonancespectroscopy (NMR) is employed to ascertain physical conformation ofmolecules; e.g., Figure 3, Figure 127. From this technique, distance, angle,and dihedral angle measurements can be obtained. Dihedral angles ariseconsequent to a phenomenon where atom subsets are physically constrainedto Euclidean planes.In the rigid covalent geometry approximation, the bond lengthsand angles are treated as completely fixed, so that a given spatialstructure can be described very compactly indeed by a list oftorsion angles alone... These are the dihedral angles betweenthe planes spanned by the two consecutive triples in a chain offour covalently bonded atoms.−G. M. Crippen & T. F. Havel (1988) [88,1.1]