A distance geometry procedure using the Levenberg-Marquardt algorithm and with applications in biology but not only

International audienceWe revisit a simple, yet capable to provide good solutions, procedure for solving the Distance Geometry Problem (DGP). This procedure combines two main components: the first one identifying an initial approximated solution via semidefinite programming, which is thereafter projected to the target dimension via PCA; and another component where this initial solution is refined by locally minimizing the Smooth STRESS function. In this work, we propose the use of the projected Levenberg-Marquart algorithm for this second step. In spite of the simplicity, as well as of its heuristic character, our experiments show that this procedure is able to exhibit good performances in terms of quality of the solutions for most of the instances we have selected for our experiments. Moreover, it seems to be promising not only for the DGP application arising in structural biology, which we considered in our computational experiments, but also in other ongoing studies related to the DGP and its applications: we finally provide a general discussion on how extending the presented ideas to other applications

An Approach to Dynamical Distance Geometry

International audienceWe introduce the dynamical distance geometry problem (dynDGP), where vertices of a given simple weighted undirected graph are to be embedded at different times t. Solutions to the dynDGP can be seen as motions of a given set of objects. In this work, we focus our attention on a class of instances where motion inter-frame distances are not available, and reduce the problem of embedding every motion frame as a static distance geometry problem. Some preliminary computational experiments are presented