v2010.10.26 - Convex Optimization

24 CHAPTER 1. OVERVIEWFigure 3: [188] [116] Distance data collected via nuclear magnetic resonance(NMR) helped render this 3-dimensional depiction of a protein molecule.At the beginning of the 1980s, Kurt Wüthrich [Nobel laureate] developedan idea about how NMR could be extended to cover biological moleculessuch as proteins. He invented a systematic method of pairing each NMRsignal with the right hydrogen nucleus (proton) in the macromolecule. Themethod is called sequential assignment and is today a cornerstone of all NMRstructural investigations. He also showed how it was subsequently possible todetermine pairwise distances between a large number of hydrogen nuclei anduse this information with a mathematical method based on distance-geometryto calculate a three-dimensional structure for the molecule. −[281] [378] [183]Geometric problems involving distance between points can sometimes bereduced to convex optimization problems. Mathematics of this combinedstudy of geometry and optimization is rich and deep. Its applicationhas already proven invaluable discerning organic molecular conformationby measuring interatomic distance along covalent bonds; e.g., Figure 3.[88] [351] [139] [46] Many disciplines have already benefitted and simplifiedconsequent to this theory; e.g., distance-based pattern recognition (Figure 4),localization in wireless sensor networks by measurement of intersensordistance along channels of communication, [47] [389] [45] wireless locationof a radio-signal source such as a cell phone by multiple measurements ofsignal strength, the global positioning system (GPS), and multidimensionalscaling which is a numerical representation of qualitative data by finding alow-dimensional scale.