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A least-squares approach for discretizable distance geometry problems with inexact distances

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Abstract

A branch-and-prune (BP) algorithm is presented for the discretizable distance geometry problem in \(\mathbb {R}^K\) with inexact distances. The algorithm consists in a sequential buildup procedure where possible positions for each new point to be localized are computed by using distances to at least K previously placed reference points and solving a system of quadratic equations. Such a system is solved in a least-squares sense, by finding the best positive semidefinite rank K approximation for an induced Gram matrix. When only K references are available, a second candidate position is obtained by reflecting the least-squares solution through the hyperplane defined by the reference points. This leads to a search tree which is explored by BP, where infeasible branches are pruned on the basis of Schoenberg’s theorem. In order to study the influence of the noise level, numerical results on some instances with distances perturbed by a small additive noise are presented.

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Notes

  1. 1.

    As detailed in [19], the squared volume of the simplex defined by \(\{v_{i-K}, \dots , v_{i-1} \}\) is directly proportional to the absolute value of \(CM(\{v_{i-K}, \dots , v_{i-1} \})\).

  2. 2.

    This sort of matrix is also known in the literature as nonnegative hollow matrix [1].

  3. 3.

    In fact, if the problem is only to determine whether D is an EDM, without obtaining a realization, it is enough to obtain the smallest eigenvalue of G, e.g., by the inverse shifted power method.

  4. 4.

    Recall the Frobenius norm of a matrix A is given by \(\Vert A \Vert _F^2 = \sum _{i} \sum _{j} a_{ij}^2 = \text {trace}(A^T A)\).

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Acknowledgements

I would like to thank Prof. Nathan Krislock for helping with SNLSDPClique package. I am grateful to the anonymous reviewers for their comments and suggestions that helped to improve this work. I also thank the Brazilian research agency CNPq for financial support.

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Correspondence to Douglas S. Gonçalves.

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Gonçalves, D.S. A least-squares approach for discretizable distance geometry problems with inexact distances. Optim Lett 14, 423–437 (2020). https://doi.org/10.1007/s11590-017-1225-7

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Keywords

  • Distance geometry
  • EDM
  • Least-squares
  • Gram matrix