We study the problem of satisfying the maximum number of distance geometry constraints with minimum experimental error. This models the determination of the shape of proteins from atomic distance data which are obtained from nuclear magnetic resonance experiments and exhibit experimental and systematic errors. Experimental errors are represented by interval constraints on Euclidean distances. Systematic errors occur from a misassignment of distances to wrong atomic pairs: we represent such errors by maximizing the number of satisfiable distance constraints. We present many mathematical programming formulations, as well as a “matheuristic” algorithm based on reformulations, relaxations, restrictions and refinement. We show that this algorithm works on protein graphs with hundreds of atoms and thousands of distances.

我们研究了以最小的实验误差满足最大距离几何约束的问题。这将根据从原子核磁共振实验获得的原子距离数据确定蛋白质的形状，并表现出实验和系统误差。实验误差由对欧几里得距离的间隔约束表示。系统错误是由于错误地将距离分配给错误的原子对而发生的：我们通过最大化可满足的距离约束的数量来表示此类错误。我们提出了许多数学程序设计公式，以及基于公式化，松弛，限制和细化的“数学”算法。我们证明了该算法适用于具有数百个原子和数千个距离的蛋白质图。