v2010.10.26 - Convex Optimization

318 CHAPTER 4. SEMIDEFINITE PROGRAMMINGfind {x i , x j }i , j ∈ I(745)subject to d ij ≤ ‖x i − x j ‖ 2 ≤ d ijwhere x i represents sensor location, and where d ij and d ij respectivelyrepresent lower and upper bounds on measured distance-square from i th toj th sensor (or from sensor to anchor). Figure 90 illustrates contours of equalsensor-location uncertainty. By establishing these individual upper and lowerbounds, orientation and eccentricity can effectively be incorporated into theproblem statement.Generally speaking, there can be no unique solution to the sensor-networklocalization problem because there is no unique formulation; that is the art ofOptimization. Any optimal solution obtained depends on whether or how anetwork is partitioned, whether distance data is complete, presence of noise,and how the problem is formulated. When a particular formulation is aconvex optimization problem, then the set of all optimal solutions forms aconvex set containing the actual or true localization. Measurement noiseprecludes equality constraints representing distance. The optimal solutionset is consequently expanded; necessitated by introduction of distanceinequalities admitting more and higher-rank solutions. Even were the optimalsolution set a single point, it is not necessarily the true localization becausethere is little hope of exact localization by any algorithm once significantnoise is introduced.Carter & Jin gauge performance of their heuristics to the SDP formulationof author Biswas whom they regard as vanguard to the art. [14,1] Biswasposed localization as an optimization problem minimizing a distance measure.[47] [45] Intuitively, minimization of any distance measure yields compactedsolutions; (confer6.7.0.0.1) precisely the anomaly motivating Carter & Jin.Their two-dimensional heuristics outperformed Biswas’ localizations bothin execution-time and proximity to the desired result. Perhaps, instead ofheuristics, Biswas’ approach to localization can be improved: [44] [46].The sensor-network localization problem is considered difficult. [14,2]Rank constraints in optimization are considered more difficult. Control ofaffine dimension in Carter & Jin is suboptimal because of implicit projectionon R 2 . In what follows, we present the localization problem as a semidefiniteprogram (equivalent to (745)) having an explicit rank constraint whichcontrols affine dimension of an optimal solution. We show how to achievethat rank constraint only if the feasible set contains a matrix of desired rank.