Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in K-dimensional space such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called … Read more
The interval Distance Geometry Problem (iDGP) consists in finding a realization in R^K of a simple undirected graph G=(V,E) with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. Our aim is to determine … Read more
The Distance Geometry Problem in three dimensions consists in finding an embedding in R^3 of a given nonnegatively weighted simple undirected graph such that edge weights are equal to the corresponding Euclidean distances in the embedding. This is a continuous search problem that can be discretized under some assumptions on the minimum degree of the … Read more
A Euclidean distance matrix is one in which the $(i,j)$ entry specifies the squared distance between particle $i$ and particle $j$. Given a partially-specified symmetric matrix $A$ with zero diagonal, the Euclidean distance matrix completion problem (EDMCP) is to determine the unspecified entries to make $A$ a Euclidean distance matrix. We survey three different approaches … Read more
Distance geometry problems arise from the need to position entities in the Euclidean $K$-space given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph(graph drawing). In the context of molecular distance geometry, the distances are usually known because of chemical properties … Read more
In this paper we consider global optimization algorithms based on multiple local searches for the Molecular Distance Geometry Problem (MDGP). Three distinct approaches (Multistart, Monotonic Basin Hopping, Population Basin Hopping) are presented and for each of them a computational analysis is performed. The results are also compared with those of two other approaches in the … Read more
We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of this polynomial, we propose a new structured SOS relaxation, … Read more