# Edge-based semidefinite programming relaxation of sensor network localization with lower bound constraints

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## Abstract

In this paper, we strengthen the edge-based semidefinite programming relaxation (ESDP) recently proposed by Wang, Zheng, Boyd, and Ye (SIAM J. Optim. 19:655–673, 2008) by adding lower bound constraints. We show that, when distances are exact, zero individual trace is necessary and sufficient for a sensor to be correctly positioned by an interior solution. To extend this characterization of accurately positioned sensors to the noisy case, we propose a noise-aware version of ESDP_{lb} (*ρ*-ESDP_{lb}) and show that, for small noise, a small individual trace is equivalent to the sensor being accurately positioned by a certain analytic center solution. We then propose a postprocessing heuristic based on *ρ*-ESDP_{lb} and a distributed algorithm to solve it. Our computational results show that, when applied to a solution obtained by solving *ρ*-ESDP proposed of Pong and Tseng (Math. Program. doi: 10.1007/s10107-009-0338-x), this heuristics usually improves the RMSD by at least 10%. Furthermore, it provides a certificate for identifying accurately positioned sensors in the refined solution, which is not common for existing refinement heuristics.

### Keywords

Sensor network localization Semidefinite programming relaxation Error bound Log-barrier Coordinate gradient descent## Notes

### Acknowledgements

The author would like to thank the anonymous referees for their many comments that help improve the manuscript. The author is indebted to Paul Tseng for suggesting this topic, his suggestion to use *ρ*-ESDP_{lb} as a postprocessing refinement heuristic, providing a possible explanation for the improvement in localization error by using strategy **F** and many other fruitful discussions. This paper was originally prepared as part of the PhD dissertation of the author, under supervision of Paul Tseng. The author would also like to thank Maryam Fazel, Anthony Man-Cho So and Rekha Thomas, for reading and commenting on an early version of the manuscript; and Ewout van den Berg for discussion about mex files.

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