Pratik Biswas
Abstract
An SDP relaxation based method is developed to solve the localization problem in sensor networks using incomplete and inaccurate distance information. The problem is set up to find a set of sensor positions such that given distance constraints are satisfied. The nonconvex constraints in the formulation are then relaxed in order to yield a semidefinite program that can be solved efficiently.The basic model is extended in order to account for noisy distance information. In particular, a maximum likelihood based formulation and an interval based formulation are discussed. The SDP solution can then also be used as a starting point for steepest descent based local optimization techniques that can further refine the SDP solution.We also describe the extension of the basic method to develop an iterative distributed SDP method for solving very large scale semidefinite programs that arise out of localization problems for large dense networks and are intractable using centralized methods.The performance evaluation of the technique with regard to estimation accuracy and computation time is also presented by the means of extensive simulations.Our SDP scheme also seems to be applicable to solving other Euclidean geometry problems where points are locally connected.
- Alfakih, A. Y., Khandani, A., and Wolkowicz, H. 1999. Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12, 1--3, 13--30. Google Scholar
- Benson, S. J., Ye, Y., and Zhang, X. 2000. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10, 2, 443--461. Google Scholar
- Biswas, P., Aghajan, H., and Ye, Y. 2005a. Integration of angle of arrival information for multimodal sensor network localization using semidefinite programming. Tech. rep., Wireless Sensor Networks Lab, Stanford University, May.Google Scholar
- Biswas, P., Liang, T.-C., Toh, K.-C., and Ye, Y. 2005b. An SDP based approach for anchor-free 3d graph realization. Tech. rep., Dept of Management Science and Engineering, Stanford University. Submitted to SIAM Journal on Scientific Computing. March.Google Scholar
- Biswas, P. and Ye, Y. 2003. A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization. Tech. rep., Dept of Management Science and Engineering, Stanford University, October.Google Scholar
- Biswas, P. and Ye, Y. 2004. Semidefinite programming for ad hoc wireless sensor network localization. In Proceedings of the Third International Symposium on Information Processing in Sensor Networks. ACM Press, 46--54. Google Scholar
- Boyd, S., Ghaoui, L. E., Feron, E., and Balakrishnan, V. 1994. Linear Matrix Inequalities in System and Control Theory. SIAM.Google Scholar
- Bulusu, N., Heidemann, J., and Estrin, D. 2000. Gps-less low cost outdoor localization for very small devices. Tech. rep., Computer science department, University of Southern California, April.Google Scholar
- Carter, M., Jin, H., Saunders, M., and Ye, Y. 2005. Spaseloc: An adaptable subproblem algorithm for scalable wireless network localization. Submitted to the SIAM J. on Optimization. Google Scholar
- Costa, J., Patwari, N., and HeroIII, A. O. 2005. Distributed multidimensional scaling with adaptive weighting for node localization in sensor networks. To appear in ACM Trans. Sens. Netw. Google Scholar
- Cox, T. and Cox, M. A. A. 2001. Multidimensional Scaling. Chapman Hall/CRC, London.Google Scholar
- Doherty, L., Ghaoui, L. E., and Pister, K. S. J. 2001. Convex position estimation in wireless sensor networks. In Proceedings of IEEE Infocom. Anchorage, Alaska, 1655--1663.Google Scholar
- Ganesan, D., Krishnamachari, B., Woo, A., Culler, D., Estrin, D., and Wicker, S. 2002. An empirical study of epidemic algorithms in large scale multihop wireless networks.Google Scholar
- Groenen, P. 1993. The Majorization Approach to Multidimensional Scaling: Some Problems and Extensions. DSWO Press, Leiden University.Google Scholar
- Hightower, J. and Borriello, G. 2001. Location systems for ubiquitous computing. Computer 34, 8, 57--66. Google Scholar
- Howard, A., Matarić, M., and Sukhatme, G. 2001. Relaxation on a mesh: a formalism for generalized localization. In IEEE/RSJ International Conference on Intelligent Robots and Systems. Wailea, Hawaii, 1055--1060.Google Scholar
- Jin, H. H. 2005. Scalable sensor localization algorithms for wireless sensor networks. Ph.D. thesis, Department of Mechanical and Industrial Engineering, University of Toronto. Google Scholar
- Laurent, M. 2001. Matrix completion problems. The Encyclopedia of Optimization 3, 221--229.Google Scholar
- Liang, T.-C., Wang, T.-C., and Ye, Y. 2004. A gradient search method to round the semidefinite programming relaxation solution for ad hoc wireless sensor network localization. Tech. rep., Dept of Management Science and Engineering, Stanford University, August.Google Scholar
- Moré, J. and Wu, Z. 1997. Global continuation for distance geometry problems. SIAM J. Optim. 7, 814--836. Google Scholar
- Niculescu, D. and Nath, B. 2001. Ad hoc positioning system (APS). In GLOBECOM (1). 2926--2931.Google Scholar
- Niculescu, D. and Nath, B. 2003. Ad hoc positioning system (APS) using AoA. In INFOCOM. San Francisco, CA.Google Scholar
- Patwari, N. and Hero III, A. O. 2002. Location estimation accuracy in wireless sensor networks. In Proceedings of Asilomar Conference on Signals and Systems. Pacific Grove, CA.Google Scholar
- Priyantha, N. B., Miu, A. K. L., Balakrishnan, H., and Teller, S. 2001. The cricket compass for context-aware mobile applications. In Mobile Computing and Networking. 1--14. Google Scholar
- Savarese, C., Rabaey, J. M., and Langendoen, K. 2002. Robust positioning algorithms for distributed ad hoc wireless sensor networks. In Proceedings of the General Track: 2002 USENIX Annual Technical Conference. USENIX Association, 317--327. Google Scholar
- Savvides, A., Han, C.-C., and Strivastava, M. B. 2001. Dynamic fine-grained localization in ad hoc networks of sensors. In Mobile Computing and Networking. 166--179. Google Scholar
- Savvides, A., Park, H., and Srivastava, M. B. 2002. The bits and flops of the n-hop multilateration primitive for node localization problems. In Proceedings of the 1st ACM International Workshop on Wireless Sensor Networks and Applications. ACM Press, 112--121. Google Scholar
- Shang, Y. and Ruml, W. 2004. Improved MDS-based localization. In Proceedings of the 23rd Conference of the IEEE Communicatons Society (Infocom 2004).Google Scholar
- Shang, Y., Ruml, W., Zhang, Y., and Fromherz, M. P. 2003. Localization from mere connectivity. In Proceedings of the fourth ACM International Symposium on Mobile Ad Hoc Networking and Computing. ACM Press, 201--212. Google Scholar
- Shang, Y., Ruml, W., Zhang, Y., and Fromherz, M. P. J. 2004. Localization from connectivity in sensor networks. IEEE Trans. Para. Distrib. Syst. 15, 11, 961--974. Google Scholar
- So, A. M.-C. and Ye, Y. 2004. Theory of semidefinite programming relaxation for sensor network localization. Tech. rep., Dept of Management Science and Engineering, Stanford University, April.Google Scholar
- Sturm, J. F. 1999. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 11 & 12, 625--633.Google Scholar
- Xue, G. and Ye, Y. 1997. An efficient algorithm for minimizing a sum of Euclidean norms with applications. SIAM J. Optim. 7, 4, 1017--1036. Google Scholar
Index Terms
- Semidefinite programming based algorithms for sensor network localization
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