Semidefinite Programming Approaches for Sensor Network Localization With Noisy Distance Measurements

Abstract

Abstract-A sensor network localization problem is to determine the positions of the sensor nodes in a network given incomplete and inaccurate pairwise distance measurements. Such distance data may be acquired by a sensor node by communicating with its neighbors. We describe a general semidefinite programming (SDP) based approach for solving the graph realization problem, of which the sensor network localization problems is a special case. We investigate the performance of this method on problems with noisy distance data. Error bounds are derived from the SDP formulation. The sources of estimation error in the SDP formulation are identified.The SDP solution usually has a rank higher than the underlying physical space, which when projected onto the lower dimensional space generally results in high estimation error. We describe two improvements to ameliorate such a difficulty. First, we propose a regularization term in the objective function that can help to reduce the rank of the SDP solution. Second, we use the points estimated from the SDP solution as the initial iterate for a gradient-descent method to further refine the estimated points. A lower bound obtained from the optimal SDP objective value can be used to check the solution quality. Experimental results are presented to validate our methods and show that they outperform existing SDP methods.

Tackling the flip ambiguity in wireless sensor network localization and beyond

“…We set pars.objSW = 3 to add the regularization term (the trace term) to the SDP objective function. It is reported in [17] that the use of regularization in SDP provides notable improvement for the networks with random anchor distribution. We believe in these settings, SFSDP would have the best performance in terms of localization accuracy.…”

“…Another important feature of SFSDP is its refinement step, which was previously used by [17]. The refinement is a heuristic step that uses the steepest gradient method to improve the quality of the final localization.…”

“…In fact, the term i,j x i − x j 2 (called the variance in [32]) is maximized. Biswas et al also used the similar idea in [17] to add a regularization term in their SDP model. One benefit of maximizing this term is to alleviate the drawback of those optimization models in that they often lead to "crowding embedding" around their geometric center.…”

“…Using the above fact, the minimization SDP models in [32,17] add the term −tr(Y ) to their objective. In other words, they try to maximize the trace.…”

“…Important techniques have been introduced in order to improve the quality and/or computational efficiency of various relaxations. For example, techniques of regularization and refinement have been employed in [17]. Edgebased SDP (ESDP) and node-based SDP (NSDP) relaxations of the full SDP (FSDP) [15] are studied in [18].…”