Computational Optimization and Applications

, Volume 53, Issue 1, pp 23–44 | Cite as

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



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.


Sensor 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. 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
  2. 2.
    Biswas, P., Liang, T.-C., Toh, K.-C., Wang, T.-C., Ye, Y.: Semidefinite programming approaches for sensor network localization with noisy distance measurements. IEEE Trans. Autom. Sci. Eng. 3, 360–371 (2006) CrossRefGoogle Scholar
  3. 3.
    Biswas, P., Liang, T.-C., Wang, T.-C., Ye, Y.: Semidefinite programming based algorithms for sensor network localization. ACM Trans. Sensor Networks 2, 188–220 (2006) CrossRefGoogle Scholar
  4. 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
  5. 5.
    Biswas, P., Ye, Y.: A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization. In: Mutiscale Optimization Methods and Applications. Nonconvex Optim. Appl., vol. 82, pp. 69–84. Springer, New York (2006) CrossRefGoogle Scholar
  6. 6.
    Carter, W., Jin, H.H., Saunders, M.A., Ye, Y.: SpaseLoc: an adaptive subproblem algorithm for scalable wireless sensor network localization. SIAM J. Optim. 17, 1102–1128 (2006) MathSciNetMATHCrossRefGoogle Scholar
  7. 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. 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
  9. 9.
    Fariña, N., Miguez, J., Bugallo, M.F.: Novel decision-fusion algorithms for target tracking using ad hoc networks. In: Proc. 61st Vehicular Technology Conference, vol. 4, pp. 2556–2559 (2005) CrossRefGoogle Scholar
  10. 10.
    Gustafsson, F., Gunnarsson, F., Bergman, N., Forssell, U., Jansson, J., Karlsson, R., Nordlund, P.: Particle filters for positioning, navigation, and tracking. IEEE Trans. Signal Process. 50, 425–437 (2002) CrossRefGoogle Scholar
  11. 11.
    Hightower, J., Borriello, G.: Location systems for ubiquitous computing. Computer 34, 57–66 (2001) CrossRefGoogle Scholar
  12. 12.
    Kim, S., Kojima, M., Waki, H.: Exploiting sparsity in SDP relaxation for sensor network localization. SIAM J. Optim. 17, 192–215 (2009) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM J. Optim. 20, 2679–2708 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 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. 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).
  16. 16.
    Liu, J., Zhang, Y., Zhao, F.: Robust distributed node localization with error management. In: Proc. 7th ACM International Symposium on Mobile Ad Hoc Networking and Computing, Florence, Italy, pp. 250–261 (2006) CrossRefGoogle Scholar
  17. 17.
    Moré, J.J., Wu, Z.: Global continuation for distance geometry problems. SIAM J. Optim. 7, 814–836 (1997) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Nie, J.: Sum of squares method for sensor network localization. Comput. Optim. Appl. 43, 151–179 (2009) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Patwari, N., Ash, J.N., Kyperountas, S., Hero, A.O. III, Moses, R.L., Correal, N.S.: Locating the nodes: cooperative localization in wireless sensor networks. IEEE Signal Process. Mag. 22, 54–69 (2005) CrossRefGoogle Scholar
  20. 20.
    Pong, T.K., Tseng, P.: (Robust) Edge-based semidefinite programming relaxation of sensor network localization. Math. Program. (2010). doi: 10.1007/s10107-009-0338-x Google Scholar
  21. 21.
    Rao, A., Ratnasamy, S., Papadimitriou, C., Shenker, S., Stoica, I.: Geographic routing without location information. In: Proc. 9th Annual International Conference on Mobile Computing and Networking (MobiCom’03), San Diego, CA, pp. 96–108 (2003) CrossRefGoogle Scholar
  22. 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. 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. 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
  25. 25.
    So, A.M.-C., Ye, Y.: Theory of semidefinite programming for sensor network localization. Math. Program. 109, 367–384 (2007) MathSciNetMATHCrossRefGoogle Scholar
  26. 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
  27. 27.
    Tseng, P.: Second-order cone programming relaxation of sensor network localizations. SIAM J. Optim. 18, 156–185 (2007) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Wang, Z., Zheng, S., Ye, Y., Boyd, S.: Further relaxations of the semidefinite programming approach to sensor network localization. SIAM J. Optim. 19, 655–673 (2008) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations