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.

在这项工作中，我们研究了给定到达时间测量的源定位问题的最小二乘公式。我们表明，尽管该公式总体上是非凸的，但在锚点和源的几何配置以及测量噪声的特定条件下是全局强凸的。接下来，我们推导出最小二乘公式的临界点的表征，从而在测量噪声的非常温和的假设下得出临界点最大数量的界限。特别是，该结果为分离最小二乘公式的临界点提供了充分的条件。在我们的工作之前，关键点的隔离被视为本地化文献中没有任何理由的假设。所述表征还产生了一种算法，该算法可以通过搜索所有关键点来找到最小二乘公式的全局最优值。然后，我们建立最小二乘估计器的估计误差的上限。最后，我们的数值结果证实了理论结果，并表明我们提出的算法可以获得全局解决方案，无论锚点和源的几何配置如何。