Computational Optimization and Applications

, Volume 43, Issue 2, pp 151–179 | Cite as

Sum of squares method for sensor network localization

  • Jiawang Nie


We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of this polynomial, we propose a new structured SOS relaxation, and discuss its various properties. When distances are given exactly, this SOS relaxation often returns true sensor locations. At each step of interior point methods solving this SOS relaxation, the complexity is \(\mathcal{O}(n^{3})\) , where n is the number of sensors. When the distances have small perturbations, we show that the sensor locations given by this SOS relaxation are accurate within a constant factor of the perturbation error under some technical assumptions. The performance of this SOS relaxation is tested on some randomly generated problems.


Sensor network localization Graph realization Distance geometry Polynomials Semidefinite program (SDP) Sum of squares (SOS) Error bound 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aspnes, J., Goldberg, D., Yang, Y.R.: On the Computational Complexity of Sensor Network Localization. Lecture Notes in Computer Science, vol. 3121, pp. 32–44. Springer, Berlin (2004) Google Scholar
  2. 2.
    Benson, S.J., Ye, Y.: DSDP3: Dual scaling algorithm for general positive semidefinite programming. Tech. Report ANL/MCS-P851-1000, Mathematics and Computer Science Division, Argonne National Laboratory (Feb. 2001) Google Scholar
  3. 3.
    Benson, S.J., Ye, Y.: DSDP5: A software package implementing the dual-scaling algorithm for semidefinite programming. Tech. Report ANL/MCS-TM-255, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL (June 2002) Google Scholar
  4. 4.
    Benson, S.J., Ye, Y., Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10, 443–461 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Blair, J., Peyton, B.: An introduction to chordal graphs and clique trees. In George, J., Gilbert, J., Liu, J. (eds.) Graph Theory and Sparse Matrix Computations, pp. 1–30. Springer, Berlin (1993) Google Scholar
  6. 6.
    Blekherman, G.: Volumes of nonnegative polynomials, sums of squares, and powers of linear forms, preprint, arXiv:math.AG/0402158 Google Scholar
  7. 7.
    Blumenthal, L.: Theory and Applications of Distance Geometry. Chelsea Publishing Company, Bronx (1970) zbMATHGoogle Scholar
  8. 8.
    Biswas, P., Ye, Y.: Semidefinite programming for ad hoc wireless sensor network localization. In: Proc. 3rd IPSN, pp. 46–54 (2004) Google Scholar
  9. 9.
    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. Automat. Sci. Eng. 3(4), 360–371 (2006) CrossRefGoogle Scholar
  10. 10.
    Curto, R.E., Fialkow, L.A.: The truncated complex K-moment problem. Trans. Am. Math. Soc. 352, 2825–2855 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Doherty, L., Pister, K.S.J., El Ghaoui, L.: Convex Position Estimation in Wireless Sensor Networks. Proc. 20th IEEE Infocom 3, 1655–1663 (2001) Google Scholar
  12. 12.
    Gatermann, K., Parrilo, P.: Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra 192(1–3), 95–128 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Henrion, D., Lasserre, J.: GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi. ACM Trans. Math. Soft. 29, 165–194 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Henrion, D., Lasserre, J.: Detecting global optimality and extracting solutions in GloptiPoly. In: Henrion, D., Garulli, A. (eds.) Positive Polynomials in Control. Lecture Notes on Control and Information Sciences. Springer, Berlin (2005) Google Scholar
  15. 15.
    Krislock, N., Piccialli, V., Wolkowicz, H.: Robust semidefinite programming approaches for sensor network localization with anchors. CORR 2006-12, May 2006.
  16. 16.
    Kojima, M., Kim, S., Waki, H.: Sparsity in sums of squares of polynomials. Math. Program. 103(1), 45–62 Google Scholar
  17. 17.
    Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Man-cho So, A., Ye, Y.: The theory of semidefinite programming for sensor network localization. Math. Program. Ser. B 109, 367–384 (2007) zbMATHCrossRefGoogle Scholar
  19. 19.
    Moré, J., Wu, Z.: Global continuation for distance geometry problems. SIAM J. Optim. 7, 814–836 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nie, J., Demmel, J., Sturmfels, B.: Minimizing polynomials via sum of squares over the gradient ideal. Math. Program. Ser. A 106(3), 587–606 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Parrilo, P.: Semidefinite Programming relaxations for semialgebraic problems. Math. Program. Ser. B 96(2), 293–320 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Parrilo, P.: Exploiting structure in sum of squares programs. In: Proceedings for the 42nd IEEE Conference on Decision and Control. Maui, Hawaii (2003) Google Scholar
  23. 23.
    Parrilo, P., Sturmfels, B.: Minimizing polynomial functions. In: Basu, S., Gonzalez-Vega, L. (eds.) Proceedings of the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, March 2001, pp. 83–100. American Mathematical Society, Providence (2003) Google Scholar
  24. 24.
    Prajna, S., Papachristodoulou, A., Parrilo, P.: SOSTOOLS User’s Guide. Website:
  25. 25.
    Reznick, B.: Extremal psd forms with few terms. Duke Math. J. 45, 363–374 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Saxe, J.: 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
  27. 27.
    Sturm, J.F.: SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Soft. 11&12, 625–653 (1999) CrossRefMathSciNetGoogle Scholar
  28. 28.
    Sturmfels, B.: Solving systems of polynomial equations. Am. Math. Soc., CBMS regional conferences series, No. 97, Providence, Rhode Island, 2002 Google Scholar
  29. 29.
    Tseng, P.: Second-order cone programming relaxation of sensor network localization. SIAM J. Optim. 18(1) 156–185 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–242 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: SparsePOP: a sparse semidefinite programming relaxation of polynomial optimization problems.
  32. 32.
    Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.): Handbook of Semidefinite Programming. Kluwer Academic, Dordrecht (2000) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

Personalised recommendations