Abstract
We present several solution techniques for the noisy single source localization problem, i.e. the Euclidean distance matrix completion problem with a single missing node to locate under noisy data. For the case that the sensor locations are fixed, we show that this problem is implicitly convex, and we provide a purification algorithm along with the SDP relaxation to solve it efficiently and accurately. For the case that the sensor locations are relaxed, we study a model based on facial reduction. We present several approaches to solve this problem efficiently, and we compare their performance with existing techniques in the literature. Our tools are semidefinite programming, Euclidean distance matrices, facial reduction, and the generalized trust region subproblem. We include extensive numerical tests.
Highlights
In this paper we consider the noisy, single source localization problem
We show that every extreme point of the semidefinite relaxation of generalized trust region subproblem (GTRS) may be transformed into a solution of GTRS and a solution of the squared least squares (SLS) problem
By homogenizing the quadratic objective and constraint and using the fact that strong duality holds for the standard trust region subproblem [34], we obtain an equivalent formulation of the Lagrangian dual of GTRS as an semidefinite programming (SDP)
Summary
In this paper we consider the noisy, single source localization problem. In an application to cellular networks, the source of the signal is a cellular phone and the Stefan Sremac: Research supported by the Natural Sciences and Engineering Research Council of Canada. The sensor network localization problem is a generalization of our single source problem, where there are multiple sources and only some of the distance estimates are known. We refer the readers for the related nearest Euclidean distance matrix (NEDM) problem to the papers [30,31] where a semismooth Newton approach and a rank majorization approach is presented. We consider two equivalent formulations of SLS: the generalized trust region subproblem (GTRS) formulation; and the nearest Euclidean distance matrix with fixed sensors (NEDMF) formulation. We provide empirical evidence that the solutions to these relaxations may give better prediction for the location of the source
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