Exploiting sparsity in SDP relaxation for sensor network localization

SIAM Journal on Optimization

Volumn 20, Issue 1, 2009, Pages 192-215

  Kim, Sunyoung ^a   Kojima, Masakazu ^b   Waki, Hayato ^c  

^a Ewha Womans University   (South Korea)

^b TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

^c UNIVERSITY OF ELECTRO COMMUNICATIONS   (Japan)

Author keywords

Polynomial optimization problem; Semidefinite relaxation; Sensor network localization problem; Sparsity

Indexed keywords

EDGE-BASED; POLYNOMIAL OPTIMIZATION PROBLEM; QUADRATIC OPTIMIZATION PROBLEMS; SDP RELAXATION; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE RELAXATION; SENSOR NETWORK LOCALIZATION;

MATHEMATICAL PROGRAMMING; POLYNOMIALS;

SENSOR NETWORKS;

EID: 70450210946     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/080713380     Document Type: Article

References (35)

1. A. Y. ALFAKIH, A. KHANDANI, AND H. WOLKOWICZ, Solving Euclidean matrix completion problem via semidefinite programming, Comput. Optim. Appl., 12 (1999), pp. 13-30.
2. P. BISWAS AND Y. YE, Semidefinite programming for ad hoc wireless sensor network localization, in Proceedings of the Third International Symposium on Information Processing in Sensor Networks, ACM, New York, 2004, pp. 46-54.
3. P. BISWAS AND Y. YE, A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization, in Multiscale Optimization Methods and Applications, Springer-Varlag, New York, 2006, pp. 69-84.
4. P. BISWAS, T.-C. LIANG, T.-C. WANG, AND Y. YE, Semidefinite programming based algorithms for sensor network localization, ACM Trans. Sensor Networks, 2 (2006), pp. 188-220.
5. P. BISWAS, T.-C. LIANG, K.-C. TOH, T.-C. WANG, AND Y. YE, Semidefinite programming approaches for sensor network localization with noisy distance measurements, IEEE Trans. Automat. Sci. Engrg., 3 (2006), pp. 360-371.
6. J. R. S. BLAIR AND B. PEYTON, An introduction to chordal graphs and clique trees, in Graph Theory and Sparse Matrix Computation, A. George, J. R. Gilbert, and J. W. H. Liu, eds., Springer-Verlag, New York, 1993, pp. 1-29.
7. T. F. COLEMAN AND Y. LI, On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds, Math. Programming, 67 (1994), pp. 189-224.
8. T. F. COLEMAN AND Y. LI, An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. Optim., 6 (1996), pp. 418-445.
9. L. DOHERTY, K. S. J. PISTER, AND L. EL GHAOUI, Convex position estimation in wireless sensor networks, in Proceedings of the 20th INFOCOM, IEEE Computer Society, Washington D. C., 2001, pp. 1655-1663.
10. T. EREN, D. K. GOLDENBERG, W. WHITELEY, Y. R. WANG, A. S. MORSE, B. D. O. ANDERSON, AND P. N. BELHUMEUR, Rigidity, computation, and randomization in network localization, in Proceedings of IEEE INFOCOM, IEEE Computer Society, Washington, D. C., 2004, pp. 2673-2684.
11. T. FUJIE AND M. KOJIMA, Semidefinite relaxation for nonconvex programs, J. Global Optim., 10 (1997), pp. 367-380.
12. M. FUKUDA, M. KOJIMA, K. MUROTA, AND K. NAKATA, Exploiting sparsity in semidefinite programming via matrix completion I: General framework, SIAM J. Optim., 11 (2000), pp. 647-674.
13. D. GANESAN, B. KRISHNAMACHARI, A. WOO, D. CULLER, D. ESTRIN, AND S. WICKER, An empirical study of epidemic algorithms in large scale multihop wireless network, 2002.
14. A. HOWARD, M. MATARIĆ, AND G. SUKHATME, Relaxation on a mesh: A formalism for generalized localization, in the IEEE/RSJ International Conference on Intelligent Robots and Systems, Wailea, Hawaii, 2001, pp. 1055-1060.
15. K. KOBAYASHI, S. KIM, AND M. KOJIMA, Correlative sparsity in primal-dual interior-point methods for LP, SDP and SOCP, Appl. Math. Optim., 58 (2008), pp. 69-88.
16. K. KOBAYASHI, K. NAKATA, AND M. KOJIMA, A conversion of an SDP having free variables into the standard form SDP, Comput. Optim. Appl., 36 (2007), pp. 289-307.
17. M. KOJIMA, Conversion methods for large scale SDPs to exploit their structured sparsity, in The MIT Computation for Design and Optimization Program Distinguished Speaker Series, Massachusetts Institute of Technology, Boston, MA, 2008; also available online from http://www.is.titech.ac.jp/~kojima/articles/ MIT2008.pdf.
18. M. KOJIMA, S. KIM, AND H. WAKI, Elimination of free variables for solving linear optimization problems efficiently, in SIAM Conference on Optimization, Boston, MA, 2008; also available online from http://www.is.titech.ac.jp/~kojima/ articles/SIOPT2008.pdf.
19. J. B. LASSERRE, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796-817.
20. J. B. LASSERRE, Convergent SDP-relaxations in polynomial optimization with sparsity, SIAM J. Optim., 17 (2006), pp. 822-843.
21. T.-C. LIAN, T.-C. WANG, AND Y. YE, A gradient search method to round the semidefinite programming relaxation solution for ad hoc wireless sensor network localization, Technical report, Department of Management Science and Engineering, Stanford University, Stanford, CA, 2004.
22. J. J. MORÉ AND Z. WU, Global continuation for distance geometry problems, SIAM J. Optim., 7 (1997), pp. 814-836. (Pubitemid 127108820)
23. K. NAKATA, K. FUJISAWA, M. FUKUDA, M. KOJIMA, AND K. MUROTA, Exploiting sparsity in semidefinite programming via matrix completion II: Implementation and numerical results, Math. Program., 95 (2003), pp. 303-327.
24. K. NAKATA, M. YAMASHITA, K. FUJISAWA, AND M. KOJIMA, A parallel primal-dual interior-point method for semidefinite programs using positive definite matrix completion, Parallel Comput., 32 (2006), pp. 24-43.
25. J. NIE, Sum of squares method for sensor network localization, in Comput. Optim. Appl., to appear.
26. N. Z. SHOR, Quadratic optimization problems, Soviet J. Comput. Systems Sci., 25 (1987), pp. 1-11.
27. N. Z. SHOR, Dual quadratic estimates in polynomial and boolean programming, Ann. Oper. Res., 25 (1990), pp. 163-168.
28. A. M. SO AND Y. YE, Theory of semidefinite programming for sensor network localization, Math. Program. Ser. B, 109 (2007), pp. 367-384.
29. J. F. STRUM, SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11 & 12 (1999), pp. 625-653.
30. P. TSENG, Second-order cone programming relaxation of sensor network localization, SIAM J. Optim., 18 (2007), pp. 156-185.
31. H. WAKI, S. KIM, M. KOJIMA, AND M. MURAMATSU, Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity, SIAM J. Optim., 17 (2006), pp. 218-242.
32. H. WAKI, S. KIM, M. KOJIMA, M. MURAMATSU, AND H. SUGIMOTO, Algorithm 883: SparsePOP: A sparse semidefinite programming relaxation of polynomial optimization problems, ACM Trans. Math. Software, 35 (2008), pp. 1-13.
33. Z. WANG, S. ZHENG, Y. YE, AND S. BOYD, Further relaxations of the semidefinite programming approach to sensor network localization, SIAM J. Optim., 19 (2008), pp. 655-673.
34. M. YAMASHITA, K. FUJISAWA, AND M. KOJIMA, Implementation and evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0), Optim. Methods Softw., 18 (2003), pp. 491-505.
35. Y. YE, http://www.stanford.edu/~yyye.