Vaia Anastasia, Sahinidis Nikolaos V
Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue, Urbana, IL 61801, USA.
Acta Crystallogr A. 2003 Sep;59(Pt 5):452-8. doi: 10.1107/S0108767303012972. Epub 2003 Aug 29.
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.
本文所解决的问题是通过X射线衍射测量确定中心对称晶体的三维结构。采用某种量仅通过晶体结构达到最小化的“最小原理”来解决相位问题。最小原理的数学表述是一个非凸非线性优化问题。迄今为止，已使用局部优化技术和先进的计算机架构来解决这个可能存在大量局部最优解的问题。在本文中，针对中心对称结构的情况，将最小原理模型根据缺失相位重新表述为一个整数规划问题。这种表述可通过成熟的组合优化技术求解，这些技术保证能在有限步骤内提供全局最优解，而无需明确列举所有可能的相位组合。文中给出了在所提出方法应用于一些中等复杂程度结构上的计算经验，结果表明该方法产生了一种快速且可靠的方法，可解决中心对称结构情况下的晶体学相位问题。
