Edge-based semidefinite programming relaxation of sensor network localization with lower bound constraints
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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 ESDPlb (ρ-ESDPlb) 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-ESDPlb 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.
KeywordsSensor network localization Semidefinite programming relaxation Error bound Log-barrier Coordinate gradient descent
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-ESDPlb 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.
- 1.Aspnes, J., Goldenberg, D., Yang, Y.R.: On the computational complexity of sensor network localization. In: ALGOSENSORS 2004, Turku, Finland. Lecture Notes in Comput. Sci., vol. 3121, pp. 32–44. Springer, New York (2004) Google Scholar
- 4.Biswas, P., Ye, Y.: Semidefinite programming for ad hoc wireless sensor network localization. In: Proc. 3rd IPSN, Berkeley, CA, pp. 46–54 (2004) Google Scholar
- 7.Ding, Y., Krislock, N., Qian, J., Wolkowicz, H.: Sensor network localization, Euclidean distance matrix completions, and graph realization. Report, Department of Combinatorics and Optimization, University of Waterloo, Waterloo (November 2008) Google Scholar
- 8.Doherty, L., Pister, K.S.J., El Ghaoui, L.: Convex position estimation in wireless sensor networks. In: Proc. 20th INFOCOM, Los Alamitos, CA, vol. 3, pp. 1655–1663 (2001) Google Scholar
- 14.Krislock, N., Piccialli, V., Wolkowicz, H.: Robust semidefinite programming approaches for sensor network localization with anchors. Report, Department of Combinatorics and Optimization, University of Waterloo, Waterloo (May 2006) Google Scholar
- 15.Liang, T.-C., Wang, T.-C., Ye, Y.: A gradient search method to round the semidefinite programming relaxation solution for ad hoc wireless sensor network localization. Report, Electrical Engineering, Stanford University, Stanford (October 2004). http://serv1.ist.psu.edu:8080/viewdoc/summary?doi=10.1.1.81.7689+
- 22.Savarese, C., Rabaey, J.M., Langendoen, K.: Robust positioning algorithms for distributed ad-hoc wireless sensor networks. In: Proc. USENIX Annual Technical Conference, Monterey, CA, pp. 317–327 (2002) Google Scholar
- 23.Saxe, J.B.: Embeddability of weighted graphs in k-space is strongly NP-hard. In: Proc. 17th Allerton Conference in Communications, Control, and Computing, Monticello, IL, pp. 480–489 (1979) Google Scholar
- 24.Simić, S.N., Sastry, S.: Distributed localization in wireless ad hoc networks. Report, Department of Electrical Engineering and Computer Sciences, University of California, Berkeley (2002); First ACM International Workshop on Wireless Sensor Networks and Applications, Atlanta, GA, 2002, submitted Google Scholar
- 26.Sturm, J.F.: Using SeDuMi 1.02, A Matlab∗ toolbox for optimization over symmetric cones (updated for Version 1.05). Report, Department of Econometrics, Tilburg University, Tilburg, August 1998–October 2001 Google Scholar