Further Relaxations of the SDP Approach to Sensor Network Localization

Recently, a semi-deflnite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating sensor’s locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we propose methods to further relax the SDP relaxation; more precisely, to relax the single semi-deflnite matrix cone into a set of small-size semideflnite matrix cones, which we call the smaller SDP (SSDP) approach. We present two such relaxations; and they are, although weaker than the original SDP relaxation, retaining the key theoretical property and tested to be both e‐cient and accurate in computation. The speed of the SSDP is even faster than that of other further weaker approaches. The SSDP approach may also pave a way to e‐ciently solve general SDP relaxations without sacriflcing their solution quality.

[1]  Charles R. Johnson,et al.  Positive definite completions of partial Hermitian matrices , 1984 .

[2]  B. Peyton,et al.  An Introduction to Chordal Graphs and Clique Trees , 1993 .

[3]  Yinyu Ye,et al.  Convergence behavior of interior-point algorithms , 1993, Math. Program..

[4]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[5]  Bruce Hendrickson,et al.  The Molecule Problem: Exploiting Structure in Global Optimization , 1995, SIAM J. Optim..

[6]  J. J. Moré,et al.  Global continuation for distance geometry problems , 1995 .

[7]  Alexander I. Barvinok,et al.  Problems of distance geometry and convex properties of quadratic maps , 1995, Discret. Comput. Geom..

[8]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[9]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[10]  Katya Scheinberg,et al.  Interior Point Trajectories in Semidefinite Programming , 1998, SIAM J. Optim..

[11]  Gábor Pataki,et al.  On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues , 1998, Math. Oper. Res..

[12]  Henry Wolkowicz,et al.  Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming , 1999, Comput. Optim. Appl..

[13]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[14]  Xiong Zhang,et al.  Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization , 1999, SIAM J. Optim..

[15]  Charles R. Johnson,et al.  The positive definite completion problem relative to a subspace , 2000 .

[16]  A. Alfakih.  Graph rigidity via Euclidean distance matrices , 2000 .

[17]  Kazuo Murota,et al.  Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework , 2000, SIAM J. Optim..

[18]  L. Hogben.  Graph theoretic methods for matrix completion problems , 2001 .

[19]  L. El Ghaoui,et al.  Convex position estimation in wireless sensor networks , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[20]  A. Alfakih.  On rigidity and realizability of weighted graphs , 2001 .

[21]  Mihai Badoiu,et al.  Approximation algorithm for embedding metrics into a two-dimensional space , 2003, SODA '03.

[22]  Yinyu Ye,et al.  Semidefinite programming for ad hoc wireless sensor network localization , 2004, Third International Symposium on Information Processing in Sensor Networks, 2004. IPSN 2004.

[23]  Brian D. O. Anderson,et al.  Rigidity, computation, and randomization in network localization , 2004, IEEE INFOCOM 2004.

[24]  Kilian Q. Weinberger,et al.  Learning a kernel matrix for nonlinear dimensionality reduction , 2004, ICML.

[25]  Neal Patwari,et al.  Distributed Multidimensional Scaling with Adaptive Weighting for Node Localization in Sensor Networks , 2004 .

[26]  James Aspnes,et al.  On the Computational Complexity of Sensor Network Localization , 2004, ALGOSENSORS.

[27]  Piotr Indyk,et al.  Low-Dimensional Embedding with Extra Information , 2004, SCG '04.

[28]  Stephen P. Boyd,et al.  Fastest Mixing Markov Chain on a Graph , 2004, SIAM Rev..

[29]  Masakazu Kojima,et al.  Sparsity in sums of squares of polynomials , 2005, Math. Program..

[30]  Anthony Man-Cho So,et al.  A semidefinite programming approach to tensegrity theory and realizability of graphs , 2006, SODA '06.

[31]  Michael A. Saunders,et al.  SpaseLoc: An Adaptive Subproblem Algorithm for Scalable Wireless Sensor Network Localization , 2006, SIAM J. Optim..

[32]  Piotr Indyk,et al.  Low-Dimensional Embedding with Extra Information , 2006, Discret. Comput. Geom..

[33]  Kim-Chuan Toh,et al.  Semidefinite Programming Approaches for Sensor Network Localization With Noisy Distance Measurements , 2006, IEEE Transactions on Automation Science and Engineering.

[34]  Yinyu Ye,et al.  Semidefinite programming based algorithms for sensor network localization , 2006, TOSN.

[35]  Paul Tseng,et al.  Second-Order Cone Programming Relaxation of Sensor Network Localization , 2007, SIAM J. Optim..

[36]  Nathan Krislock,et al.  Sensor network localization, euclidean distance matrix completions, and graph realization , 2008, MELT '08.

[37]  Jiawang Nie,et al.  Sum of squares method for sensor network localization , 2006, Comput. Optim. Appl..

[38]  Monique Laurent,et al.  Matrix Completion Problems , 2009, Encyclopedia of Optimization.