v2009.01.01 - Convex Optimization

376 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
Figure 103: A depiction of molecular conformation. [99]
This semidefinite program realizes the wireless location problem illustrated in
Figure 101. Location X ⋆ (:, 1) is taken as solution, although measurement
noise will often cause rankG ⋆ to exceed 2. Randomized search for a rank-2
optimal solution is not so easy here as in Example 5.4.2.2.7. We introduce a
method in4.4 for enforcing the stronger rank-constraint (845). To formulate
this same problem in three dimensions, point list X is simply redimensioned
in the semidefinite program.
5.4.2.2.9 Example. (Biswas, Nigam, Ye) Molecular Conformation.
The subatomic measurement technique called nuclear magnetic resonance
spectroscopy (NMR) is employed to ascertain physical conformation of
molecules; e.g., Figure 3, Figure 103. From this technique, distance, angle,
and dihedral angle measurements can be obtained. Dihedral angles arise
consequent to a phenomenon where atom subsets are physically constrained
to Euclidean planes.
In the rigid covalent geometry approximation, the bond lengths
and angles are treated as completely fixed, so that a given spatial
structure can be described very compactly indeed by a list of
torsion angles alone... These are the dihedral angles between
the planes spanned by the two consecutive triples in a chain of
four covalently bonded atoms. [77,1.1]