v2007.09.13 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 259least squares solution to hyperbolic equations; called multilateration. 4.23Indeed, 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 [237] 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 solvingmany little semidefinite programs.Partitioning of large sensor networks is a logical alternative to rapidgrowth of SDP computational complexity with problem size. But whenimpact of noise on distance measurement is of most concern, one is averse toa partitioning scheme because noise-effects vary inversely with problem size.[39,2.2] (5.13.2) Since an individual partition’s solution is not iteratedin Carter & Jin and is interdependent with adjoining partitions, we expecterrors to propagate from one partition to the next; the ultimate partitionsolved, expected to suffer most.Heuristics often fail on real-world data because of unanticipatedcircumstances. When heuristics fail, generally they are repaired by addingmore heuristics. Tenuous is any presumption, for example, that distancemeasurement errors have distribution characterized by circular contours ofequal probability about an unknown sensor-location. That presumptioneffectively appears within Carter & Jin’s optimization problem statementas affine equality constraints relating unknowns to distance measurementsthat are corrupted by noise. Yet in most all urban environments, thismeasurement noise is more aptly characterized by ellipsoids of varying4.23 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.