v2007.09.13 - Convex Optimization

260 CHAPTER 4. SEMIDEFINITE PROGRAMMINGorientation and eccentricity as one recedes from a sensor. Each unknownsensor must therefore instead be bound to its own particular range ofdistance, primarily determined by the terrain. 4.24 The nonconvex problemwe must instead solve is:find {x i , x j }i , j ∈ I(634)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 from i th to j thsensor (or from sensor to anchor). Figure 67 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 thenetwork is partitioned and how the problem is formulated. When a particularformulation is a convex optimization problem, then the set of all optimalsolutions forms a convex set containing the actual or true localization.Measurement noise precludes equality constraints representing distance. Theoptimal solution set is consequently expanded; necessitated by introductionof distance inequalities admitting more and higher-rank solutions. Evenwere the optimal solution set a single point, it is not necessarily the truelocalization because there is little hope of exact localization by any algorithmonce significant noise is introduced.Carter & Jin gauge performance of their heuristics to the SDP formulationof author Biswas whom they regard as vanguard to the art. [12,1] Biswasposed localization as an optimization problem minimizing a distance measure.[35] [33] Intuitively, minimization of any distance measure yields compactedsolutions; (confer6.4.0.0.1) 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: [32] [34].The sensor-network localization problem is considered difficult. [12,2]Rank constraints in optimization are considered more difficult. In what4.24 A distinct contour map corresponding to each anchor is required in practice.