Diagonally Dominant Programming in Distance Geometry

Distance geometry is a branch of geometry which puts the concept of distance at its core. The fundamental problem of distance geometry asks to find a realization of a finite, but partially specified, metric space in a Euclidean space of given dimension. An associated problem asks the same question in a Euclidean space of any dimension. Both problems have many applications to science and engineering, and many methods have been proposed to solve them. Unless some structure is known about the structure of the instance, it is notoriously difficult to solve these problems computationally, and most methods will either not scale up to useful sizes, or will be unlikely to identify good solutions. We propose a new heuristic algorithm based on a semidefinite programming formulation, a diagonally-dominant inner approximation of Ahmadi and Hall’s, a randomized-type rank reduction method of Barvinok’s, and a call to a local nonlinear programming solver.