v2007.09.13 - Convex Optimization

320 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXFigure 84: Example of molecular conformation. [80]in Figure 82. Location X ⋆ (:, 1) is taken as solution, although measurementnoise will often cause rankG ⋆ to exceed 2. Randomized search for a rank-2optimal solution is not so easy here as in Example 5.4.2.2.6. We introduce amethod in4.4 for enforcing the stronger rank-constraint (752). To formulatethis same problem in three dimensions, point list X is simply redimensionedin the semidefinite program.5.4.2.2.8 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 84. From this technique, distance, angle,and dihedral angle data can be obtained. Dihedral angles arise consequent toa phenomenon where atom subsets are physically constrained to Euclideanplanes.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. [60,1.1]Crippen & Havel recommend working exclusively with distance databecause they consider angle data to be mathematically cumbersome. Thepresent example shows instead how inclusion of dihedral angle data into aproblem statement can be made elegant and convex.