v2007.11.26 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 261Generally 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 whatfollows, we present the localization problem as a semidefinite program(equivalent to (636)) having an explicit rank constraint which controlsEuclidean dimension of an optimal solution. We show how to achieve thatrank constraint only if the feasible set contains a matrix of desired rank.Our problem formulation is extensible to any spatial dimension.proposed standardized testJin proposes an academic test in real Euclidean two-dimensional spaceR 2 that we adopt. In essence, this test is a localization of sensors andanchors arranged in a regular triangular lattice. Lattice connectivity issolely determined by sensor radio range; a connectivity graph is assumedincomplete. In the interest of test standardization, we propose adoptionof a few small examples: Figure 65 through Figure 68 and their particularconnectivity represented by matrices (637) through (640) respectively.