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Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation

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Abstract

One of the challenging problems in collaborative position localization arises when the distance measurements contain non-line-of-sight (NLOS) biases. Convex optimization has played a major role in modelling such problems and numerical algorithm developments. One of the successful examples is the semi-definite programming (SDP), which translates Euclidean distances into the constraints of positive semidefinite matrices, leading to a large number of constraints in the case of NLOS biases. In this paper, we propose a new convex optimization model that is built upon the concept of Euclidean distance matrix (EDM). The resulting EDM optimization has an advantage that its Lagrangian dual problem is well structured and hence is conducive to algorithm developments. We apply a recently proposed 3-block alternating direction method of multipliers to the dual problem and tested the algorithm on some real as well as simulated data of large scale. In particular, the EDM model significantly outperforms the existing SDP model and several others.

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Notes

  1. Let \(x_1,\ldots , x_n\) be independent, identically distributed real random variables. The corresponding empirical distribution function \(F_n(t)\) is defined as \(F_n(t)=\frac{1}{n}\sum _{i=1}^{n}{} \mathbf{1}_{x_i\le t}\), where \(\mathbf{1}_{x_i\le t}\) is the indicator of event \(x_i\le t\).

  2. The data can be downloaded from http://web.eecs.umich.edu/~hero/localize/.

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Acknowledgments

We would like to thank the two referees for their constructive comments that have helped to improve the quality of the paper. This work is supported in part by Engineering and Physical Science Research Council (UK) Project EP/K007645/1.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China

    Chao Ding

  2. School of Mathematics, The University of Southampton, Highfield, Southampton, SO17 1BJ, UK

    Hou-Duo Qi

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  1. Chao Ding
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Correspondence to Chao Ding.

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Ding, C., Qi, HD. Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation. Comput Optim Appl 66, 187–218 (2017). https://doi.org/10.1007/s10589-016-9858-5

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