research papers 452 Vaia and Sahinidis  Phase problem for centrosymmetric structures Acta Cryst. (2003). A59, 452±458 Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Received 8 April 2003 Accepted 11 June 2003 # 2003 International Union of Crystallography Printed in Great Britain ± all rights reserved An integer programming approach to the phase problem for centrosymmetric structures Anastasia Vaia and Nikolaos V. Sahinidis* Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana- Champaign, 600 South Mathews Avenue, Urbana, IL 61801, USA. Correspondence e-mail: nikos@uiuc.edu 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 ®nite 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. 1. Introduction Since the mid-1900's, analysis of X-ray diffraction data of crystals has been used extensively for the determination of molecular structure and properties. While the method is employed almost on a routine basis worldwide, it is often a major challenge to identify the three-dimensional structure that best ®ts the diffraction data. A key obstacle, in particular, is the identi®cation of the phases of the diffracted rays from measurements of intensities alone. Methods developed for the phase problem have included the tangent formula (Karle & Hauptman, 1956), the maximum entropy (Bricogne, 1984), the minimal principle (Debaerde- maeker & Woolfson, 1983), and variants of the above (Germain & Woolfson, 1968; Germain et al., 1971; Olthof & Schenk, 1982; Gull et al., 1987; Hauptman, 1988; Sheldrick, 1990; Altomare et al. , 1993; Miller et al., 1993; Gilmore, 1996; Sheldrick, 1997; Chang et al., 1997; Giacovazzo, 1998; Hauptman et al., 1999). Most of the methods for the phase problem make use of a merit function to score potential structures based on how well they match the experimental data. The complexity of the resulting phase-estimation problem is signi®cant because of the existence of multiple local optima in the underlying opti- mization formulations. To this date, crystallographers have resorted to combinations of local optimization and stochastic global optimization techniques to solve these models. For example, the Shake-and-Bake approach (Miller et al., 1993) is based on alternating phase re®nement in reciprocal space with a peak-picking technique in real space and terminates once a prespeci®ed number of iterations has been reached. For centrosymmetric structures, it has long been observed that the phases can only take values of 0 or . While this, effectively, makes the phase problem a discrete optimization problem, no current solution strategy exploits the mathema- tical properties of the problem to effectively resolve the phase problem. Yet a very large number of crystal structures are centrosymmetric. For instance, nearly 76% of the over a quarter of a million crystal structures in the Cambridge Structural Database are centrosymmetric (Allen, 2002). In this paper, we address the problem of using X-ray measurements to determine structures with a center of symmetry. Our starting point is the minimal principle model. In the general case, this model requires the solution of a highly nonlinear nonconvex optimization problem with trigono- metric terms in its objective function. When the structure is centrosymmetric, we show that the underlying optimization problem can be reformulated in a way that avoids the trig- onometric terms. Through the introduction of a suitable set of binary variables, the objective function is rendered linear and the model is reduced to a mathematically equivalent integer linear optimization problem. Off-the-shelf optimization soft- ware of the branch-and-bound type can be utilized to solve the integer model. These algorithms require no starting point and