Advertisement

Optimization and Engineering

, Volume 11, Issue 1, pp 45–66 | Cite as

Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization

  • Yichuan Ding
  • Nathan Krislock
  • Jiawei Qian
  • Henry Wolkowicz
Article

Abstract

We study Semidefinite Programming, SDP, relaxations for Sensor Network Localization, SNL, with anchors and with noisy distance information. The main point of the paper is to view SNL as a (nearest) Euclidean Distance Matrix, EDM, completion problem that does not distinguish between the anchors and the sensors. We show that there are advantages for using the well studied EDM model. In fact, the set of anchors simply corresponds to a given fixed clique for the graph of the EDM problem.

We next propose a method of projection when large cliques or dense subgraphs are identified. This projection reduces the size, and improves the stability, of the relaxation. In addition, by viewing the problem as an EDM completion problem, we are able to derive a new approximation scheme for the sensors from the SDP approximation. This yields, on average, better low rank approximations for the low dimensional realizations. This further emphasizes the theme that SNL is in fact just an EDM problem.

We solve the SDP relaxations using a primal-dual interior/exterior-point algorithm based on the Gauss-Newton search direction. By not restricting iterations to the interior, we usually get lower rank optimal solutions and thus, better approximations for the SNL problem. We discuss the relative stability and strength of two formulations and the corresponding algorithms that are used. In particular, we show that the quadratic formulation arising from the SDP relaxation is better conditioned than the linearized form that is used in the literature.

Keywords

Sensor Network Localization Anchors Graph realization Euclidean Distance Matrix completions Semidefinite Programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al-Homidan S, Wolkowicz H (2005) Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming. Linear Algebra Appl 406:109–141 CrossRefMathSciNetzbMATHGoogle Scholar
  2. Alfakih A, Khandani A, Wolkowicz H (1999) Solving Euclidean distance matrix completion problems via semidefinite programming. Comput Optim Appl 12(1–3):13–30. Computational optimization—a tribute to Olvi Mangasarian, Part I CrossRefMathSciNetzbMATHGoogle Scholar
  3. Bakonyi M, Johnson CR (1995) The Euclidean distance matrix completion problem. SIAM J Matrix Anal Appl 16(2):646–654 CrossRefMathSciNetzbMATHGoogle Scholar
  4. Biswas P, Ye Y (2004) Semidefinite programming for ad hoc wireless sensor network localization. In: Information processing in sensor networks, proceedings of the third international symposium on information processing in sensor networks, Berkeley, Calif., 2004, pp 46–54 Google Scholar
  5. Biswas P, Ye Y (2006) A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization. In: Multiscale optimization methods and applications. Nonconvex optim appl, vol 82. Springer, New York, pp 69–84 CrossRefGoogle Scholar
  6. Biswas P, Liang TC, Toh KC, Ye Y (2005) An SDP based approach for anchor-free 3D graph realization. Technical report, Operation Research, Stanford University, Stanford, CA Google Scholar
  7. Biswas P, Liang TC, Toh KC, Wang TC, Ye Y (2006) Semidefinite programming approaches for sensor network localization with noisy distance measurements. IEEE Trans Autom Sci Eng (to appear) Google Scholar
  8. Borwein JM, Wolkowicz H (1980/1981) Facial reduction for a cone-convex programming problem. J Austral Math Soc Ser A 30(3):369–380 CrossRefMathSciNetGoogle Scholar
  9. Chua CB, Tunçel L (2008) Invariance and efficiency of convex representations. Math Program 111(1–2, Ser B):113–140 MathSciNetzbMATHGoogle Scholar
  10. Crippen GM, Havel TF (1988) Distance geometry and molecular conformation. Research Studies Press Ltd, Letchworth zbMATHGoogle Scholar
  11. Eriksson J, Gulliksson ME (2004) Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares. Math Comput 73(248):1865–1883 (electronic) MathSciNetzbMATHGoogle Scholar
  12. Farebrother RW (1987) Three theorems with applications to Euclidean distance matrices. Linear Algebra Appl 95:11–16 CrossRefMathSciNetzbMATHGoogle Scholar
  13. Gower JC (1985) Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra Appl 67:81–97 CrossRefMathSciNetzbMATHGoogle Scholar
  14. Güler O, Tunçel L (1998) Characterization of the barrier parameter of homogeneous convex cones. Math Program 81(1, Ser A):55–76 CrossRefGoogle Scholar
  15. Hayden TL, Wells J, Liu W-M, Tarazaga P (1991) The cone of distance matrices. Linear Algebra Appl 144:153–169 CrossRefMathSciNetzbMATHGoogle Scholar
  16. Jin H (2005) Scalable sensor localization algorithms for wireless sensor networks. PhD thesis, Toronto University, Toronto, Ontario, Canada Google Scholar
  17. Johnson CR, Tarazaga P (1995) Connections between the real positive semidefinite and distance matrix completion problems. Linear Algebra Appl 223/224:375–391. Special issue honoring Miroslav Fiedler and Vlastimil Pták CrossRefMathSciNetGoogle Scholar
  18. Krislock N, Piccialli V, Wolkowicz H (2006) Robust semidefinite programming approaches for sensor network localization with anchors. Technical Report CORR 2006-12, University of Waterloo, Waterloo, Ontario. orion.uwaterloo.ca/~hwolkowi/henry/reports/ABSTRACTS.html#sensorKPW
  19. Kruk S, Muramatsu M, Rendl F, Vanderbei RJ, Wolkowicz H (2001) The Gauss-Newton direction in semidefinite programming. Optim Methods Softw 15(1):1–28 CrossRefMathSciNetzbMATHGoogle Scholar
  20. Laurent M (1998) A tour d’horizon on positive semidefinite and Euclidean distance matrix completion problems. In: Topics in semidefinite and interior-point methods. The Fields Institute for Research in Mathematical Sciences, Communications Series, vol 18, Providence, Rhode Island. American Mathematical Society, Providence, pp 51–76 Google Scholar
  21. Luo Z-Q, Sidiropoulos ND, Tseng P, Zhang S (2007) Approximation bounds for quadratic optimization with homogeneous quadratic constraints. SIAM J Optim 18(1):1–28 MathSciNetzbMATHGoogle Scholar
  22. Schoenberg IJ (1935) Remarks to Maurice Frechet’s article: Sur la definition axiomatique d’une classe d’espaces vectoriels distancies applicables vectoriellement sur l’espace de Hilbert. Ann Math 36:724–732 CrossRefMathSciNetGoogle Scholar
  23. So A, Ye Y, Zhang J (2006) A unified theorem on SDP rank reduction. Technical report, Operation Research, Stanford University, Stanford, CA Google Scholar
  24. So AM, Ye Y (2007) Theory of semidefinite programming for sensor network localization. Math Program 109(2–3, Ser B):367–384 CrossRefMathSciNetzbMATHGoogle Scholar
  25. Srivastav A, Wolf K (1998) Finding dense subgraphs with semidefinite programming. In: Approximation algorithms for combinatorial optimization, Aalborg, 1998. Lecture notes in comput sci, vol 1444. Springer, Berlin, pp 181–191 CrossRefGoogle Scholar
  26. Torgerson WS (1952) Multidimensional scaling. I. Theory and method. Psychometrika 17:401–419 CrossRefMathSciNetzbMATHGoogle Scholar
  27. Tseng P (2004) SOCP relaxation for nonconvex optimization. Technical Report Aug-04, University of Washington, Seattle, WA. Presented at ICCOPT I, RPI, Troy, NY Google Scholar
  28. Tseng P (2007) Second-order cone programming relaxation of sensor network localization. SIAM J Optim 18(1):156–185 CrossRefMathSciNetzbMATHGoogle Scholar
  29. Verbitsky OV (2004) A note on the approximability of the dense subgraph problem. Mat Stud 22(2):198–201 MathSciNetzbMATHGoogle Scholar
  30. Wang Z, Zheng S, Boyd S, Ye Y (2008) Further relaxations of the semidefinite programming approach to sensor network localization. SIAM J Optim 19(2):655–673 CrossRefMathSciNetzbMATHGoogle Scholar
  31. Xu D, Han J, Huang Z, Zhang L (2003) Improved approximation algorithms for MAX \(\frac {n}{2}\) -DIRECTED-BISECTION and MAX \(\frac{n}{2}\) -DENSE-SUBGRAPH. J Global Optim 27(4):399–410 CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Yichuan Ding
    • 1
  • Nathan Krislock
    • 1
  • Jiawei Qian
    • 1
  • Henry Wolkowicz
    • 1
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

Personalised recommendations