Sensor network localization, euclidean distance matrix completions, and graph realization

Selected Open Problems in Discrete Geometry and Optimization

“…On the other hand, by dropping the nonconvex constraint rank.Z/ Ä d , one immediately obtains an SDP relaxation of the fixed-dimensional localization problem. Such a relaxation and its variants have been extensively studied in recent years (see, e.g., [3,4,7,9,10,12,16,19,21,22,24]) and are very natural as far as polynomial-time solvability is concerned. Moreover, they have the added advantage that in many cases, localization accuracy guarantees can be established; see, e.g., [10,[20][21][22]24].…”

Euclidean distance matrices, semidefinite programming and sensor network localization

“…In fact, see [69,35,36,70], a set of anchors simply corresponds to a given fixed clique for the graph of the EDM problem. It can be advantageous to delay using the difference between anchors and sensors and instead solve the resulting EDM problem.…”

“…The B −Y relaxation can be derived directly from the definitions, e.g., [15]. Alternatively, we can use the approach in [69,35,70] and derive this relaxation from the EDM framework. In fact, we now show that the B −Y relaxation can also be obtained as a restricted second step in facial reduction for the EDM relaxation, following on the one for centering in Remark 3.1.…”

Novel Mechanisms for Location-Tracking Systems

“…In the localization problem posed as in equation (5), several challenges are met and the one that has attracted a large research community is the design of efficient minimization techniques Costa et al (2006), , Ding et al (2008), Destino & Abreu (2009c), Wymeersch et al (2009). In the sequel, this issue will be addressed and the most effective state-of-the-art solutions will be described in details.…”