v2007.11.26 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 2594.4.1.1.2 Example. Sensor-Network Localization and Wireless Location.Heuristic solution to a sensor-network localization problem, proposed byCarter & Jin in [51], 4.23 is limited to two Euclidean dimensions and appliessemidefinite programming (SDP) to little subproblems. There, a largenetwork is partitioned into smaller subnetworks (as small as one sensor)and then semidefinite programming and heuristics called spaseloc areapplied to localize each and every partition by two-dimensional distancegeometry. Their partitioning procedure is one-pass, yet termed iterative;a term applicable only in so far as adjoining partitions can share localizedsensors and anchors (absolute sensor positions known a priori). But thereis no iteration on the entire network, hence the term “iterative” is perhapsinappropriate. As partitions are selected based on “rule sets” (heuristics, notgeographics), they also term the partitioning adaptive. But no adaptationactually occurs once a partition has been determined.One can reasonably argue that semidefinite programming methods areunnecessary for localization of large sensor networks. In the past, thesenonlinear localization problems were solved algebraically and computed byleast squares solution to hyperbolic equations; called multilateration. 4.24Indeed, practical contemporary numerical methods for global positioning bysatellite (GPS) do not rely on semidefinite programming.The beauty of semidefinite programming as relates to localization lies inconvex expression of classical multilateration: So & Ye showed [241] that theproblem of finding unique solution, to a noiseless nonlinear system describingthe common point of intersection of hyperspheres in real Euclidean vectorspace, can be expressed as a semidefinite program via distance geometry.But the need for SDP methods in Carter & Jin is also a question logicallyconsequent to their reliance on complicated and extensive heuristics forpartitioning a large network and for solving a partition whose intersensormeasurement data is inadequate for localization by distance geometry. Whilepartitions range in size between 2 and 10 sensors, 5 sensors optimally,heuristics provided are only for 2 spatial dimensions (no higher-dimensionalalgorithm is proposed). For these small numbers it remains unclarified as toprecisely what advantage is gained over traditional least squares by solving4.23 The paper constitutes Jin’s dissertation for University of Toronto although her nameappears as second author.4.24 Multilateration − literally, having many sides; shape of a geometric figure formed bynearly intersecting lines of position. In navigation systems, therefore: Obtaining a fix frommultiple lines of position.