Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization
 Yichuan Ding,
 Nathan Krislock,
 Jiawei Qian,
 Henry Wolkowicz
 … show all 4 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 primaldual interior/exteriorpoint algorithm based on the GaussNewton 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.
 AlHomidan S, Wolkowicz H (2005) Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming. Linear Algebra Appl 406:109–141 CrossRef
 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 CrossRef
 Bakonyi M, Johnson CR (1995) The Euclidean distance matrix completion problem. SIAM J Matrix Anal Appl 16(2):646–654 CrossRef
 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
 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 CrossRef
 Biswas P, Liang TC, Toh KC, Ye Y (2005) An SDP based approach for anchorfree 3D graph realization. Technical report, Operation Research, Stanford University, Stanford, CA
 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)
 Borwein JM, Wolkowicz H (1980/1981) Facial reduction for a coneconvex programming problem. J Austral Math Soc Ser A 30(3):369–380 CrossRef
 Chua CB, Tunçel L (2008) Invariance and efficiency of convex representations. Math Program 111(1–2, Ser B):113–140
 Crippen GM, Havel TF (1988) Distance geometry and molecular conformation. Research Studies Press Ltd, Letchworth
 Eriksson J, Gulliksson ME (2004) Local results for the GaussNewton method on constrained rankdeficient nonlinear least squares. Math Comput 73(248):1865–1883 (electronic)
 Farebrother RW (1987) Three theorems with applications to Euclidean distance matrices. Linear Algebra Appl 95:11–16 CrossRef
 Gower JC (1985) Properties of Euclidean and nonEuclidean distance matrices. Linear Algebra Appl 67:81–97 CrossRef
 Güler O, Tunçel L (1998) Characterization of the barrier parameter of homogeneous convex cones. Math Program 81(1, Ser A):55–76 CrossRef
 Hayden TL, Wells J, Liu WM, Tarazaga P (1991) The cone of distance matrices. Linear Algebra Appl 144:153–169 CrossRef
 Jin H (2005) Scalable sensor localization algorithms for wireless sensor networks. PhD thesis, Toronto University, Toronto, Ontario, Canada
 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 CrossRef
 Krislock N, Piccialli V, Wolkowicz H (2006) Robust semidefinite programming approaches for sensor network localization with anchors. Technical Report CORR 200612, University of Waterloo, Waterloo, Ontario. orion.uwaterloo.ca/~hwolkowi/henry/reports/ABSTRACTS.html#sensorKPW
 Kruk S, Muramatsu M, Rendl F, Vanderbei RJ, Wolkowicz H (2001) The GaussNewton direction in semidefinite programming. Optim Methods Softw 15(1):1–28 CrossRef
 Laurent M (1998) A tour d’horizon on positive semidefinite and Euclidean distance matrix completion problems. In: Topics in semidefinite and interiorpoint methods. The Fields Institute for Research in Mathematical Sciences, Communications Series, vol 18, Providence, Rhode Island. American Mathematical Society, Providence, pp 51–76
 Luo ZQ, Sidiropoulos ND, Tseng P, Zhang S (2007) Approximation bounds for quadratic optimization with homogeneous quadratic constraints. SIAM J Optim 18(1):1–28
 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 CrossRef
 So A, Ye Y, Zhang J (2006) A unified theorem on SDP rank reduction. Technical report, Operation Research, Stanford University, Stanford, CA
 So AM, Ye Y (2007) Theory of semidefinite programming for sensor network localization. Math Program 109(2–3, Ser B):367–384 CrossRef
 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 CrossRef
 Torgerson WS (1952) Multidimensional scaling. I. Theory and method. Psychometrika 17:401–419 CrossRef
 Tseng P (2004) SOCP relaxation for nonconvex optimization. Technical Report Aug04, University of Washington, Seattle, WA. Presented at ICCOPT I, RPI, Troy, NY
 Tseng P (2007) Secondorder cone programming relaxation of sensor network localization. SIAM J Optim 18(1):156–185 CrossRef
 Verbitsky OV (2004) A note on the approximability of the dense subgraph problem. Mat Stud 22(2):198–201
 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 CrossRef
 Xu D, Han J, Huang Z, Zhang L (2003) Improved approximation algorithms for MAX \(\frac {n}{2}\) DIRECTEDBISECTION and MAX \(\frac{n}{2}\) DENSESUBGRAPH. J Global Optim 27(4):399–410 CrossRef
 Title
 Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization
 Journal

Optimization and Engineering
Volume 11, Issue 1 , pp 4566
 Cover Date
 20100201
 DOI
 10.1007/s1108100890720
 Print ISSN
 13894420
 Online ISSN
 15732924
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

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

 Yichuan Ding ^{(1)}
 Nathan Krislock ^{(1)}
 Jiawei Qian ^{(1)}
 Henry Wolkowicz ^{(1)}
 Author Affiliations

 1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada