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.
Al-Homidan S, Wolkowicz H (2005) Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming. Linear Algebra Appl 406:109–141
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
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
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
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 cone-convex programming problem. J Austral Math Soc Ser A 30(3):369–380
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 Gauss-Newton method on constrained rank-deficient nonlinear least squares. Math Comput 73(248):1865–1883 (electronic)
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
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
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
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
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
So A, Ye Y, Zhang J (2006) A unified theorem on SDP rank reduction. Technical report, Operation Research, Stanford University, Stanford, CA
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
Xu D, Han J, Huang Z, Zhang L (2003) Improved approximation algorithms for MAX
-DIRECTED-BISECTION and MAX
-DENSE-SUBGRAPH. J Global Optim 27(4):399–410