An integer programming approach to the phase problem for centrosymmetric structures

The problem addressed in this paper is the determination of three-dimensional structures of centrosymmetric crystals from X-ray diffraction measurements. The 'minimal principle' that a certain quantity is minimized only by the crystal structure is employed to solve the phase problem. The mathematical formulation of the minimal principle is a nonconvex nonlinear optimization problem. To date, local optimization techniques and advanced computer architectures have been used to solve this problem, which may have a very large number of local optima. In this paper, the minimal principle model is reformulated for the case of centrosymmetric structures into an integer programming problem in terms of the missing phases. This formulation is solvable by well established combinatorial optimization techniques that are guaranteed to provide the global optimum in a finite number of steps without explicit enumeration of all possible combinations of phases. Computational experience with the proposed method on a number of structures of moderate complexity is provided and demonstrates that the approach yields a fast and reliable method that resolves the crystallographic phase problem for the case of centrosymmetric structures.