Global strong convexity and characterization of critical points of time-of-arrival-based source localization

In this work, we study a least-squares formulation of the source localization problem given time-of-arrival measurements. We show that the formulation, albeit non-convex in general, is globally strongly convex under certain condition on the geometric configuration of the anchors and the source and on the measurement noise. Next, we derive a characterization of the critical points of the least-squares formulation, leading to a bound on the maximum number of critical points under a very mild assumption on the measurement noise. In particular, the result provides a sufficient condition for the critical points of the least-squares formulation to be isolated. Prior to our work, the isolation of the critical points is treated as an assumption without any justification in the localization literature. The said characterization also leads to an algorithm that can find a global optimum of the least-squares formulation by searching through all critical points. We then establish an upper bound of the estimation error of the least-squares estimator. Finally, our numerical results corroborate the theoretical findings and show that our proposed algorithm can obtain a global solution regardless of the geometric configuration of the anchors and the source.