The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the given Euclidean space. Solution algorithms are generally constructed using local or global nonlinear optimization techniques. We present a new modelling technique for this problem where, instead of deciding vertex positions, formulations decide the length of the segments representing the edges in each cycle in the graph, projected in every dimension. We propose an exact formulation and a relaxation based on a Eulerian cycle. We then compare computational results from protein conformation instances obtained with stochastic global optimization techniques on the new cycle-based formulation and on the existing edge-based formulation. While edge-based formulations take less time to reach termination, cycle-based formulations are generally better on solution quality measures.

翻译：距离几何问题要求在一个给定维度K的欧氏空间中，找出给定简单边赋权图的一种实现，使得边被实现为长度等于（或尽可能接近）边权重的直线段。该问题通常被建模为一个包含决策变量的数学规划形式，这些决策变量用于确定顶点在给定欧氏空间中的位置。求解算法一般基于局部或全局非线性优化技术构建。我们提出了一种针对该问题的新建模技术，其中决策变量不是顶点位置，而是每个环中边在每一维度上的投影段长度。我们给出了一个精确模型以及一个基于欧拉环的松弛模型。随后，我们比较了在基于环的新模型和现有基于边的模型上，采用随机全局优化技术处理蛋白质构象实例时得到的计算结果。虽然基于边的模型达到终止所需时间更短，但基于环的模型通常在解的质量指标上表现更优。