The Phase Problem: A Problem in Constrained Global Optimization

Abstract

It is now almost 200 years since Gauss, a teenager at the time, formulated his famous principle of least-squares and used it to determine, for the first time, the orbit of one of the asteroids, a problem which had defeated astronomers for years. When applied to the crystallographic phase problem, least-squares leads directly to the formulation of the minimal principle, which effectively replaces the phase problem by one of constrained global minimization. Shake-and-Bake, the computer software package which implements this formulation of the phase problem, provides a completely automatic solution of this problem. Shake-and-Bake requires that diffraction intensities to a resolution of 1.2Å, at least, be available. Structures having as many as 600 independent non-hydrogen atoms have been routinely solved in this way; the ultimate potential of the method is still not known.

When single-wavelength anomalous scattering (SAS) diffraction data are available, the phase problem may again be formulated as a problem in global optimization. Although the objective function has a myriad of local maxima, its global maximum is readily accessible and easily identified by virtue of its isolation. The ability to determine the global maximum of the objective function represents the latest and most successful attempt to go directly from the known probabilistic estimates of the three-phase structure invariants to the values of the individual phases. The relationship between the maxima of the objective function and the solutions of the newly formulated system of SAS tangent equations plays a key role in this development.