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Real-world network applications must cope with failing nodes, maliciousattacks, or, somehow, nodes facing corrupted data --- classified as outliers.One enabling application is the geographic localization of the network nodes.However, despite excellent work on the network localization problem, priorresearch seldom considered outlier data --- even now, when already deployednetworks cry out for robust procedures. We propose robust, fast, anddistributed network localization algorithms, resilient to high-power noise, butalso precise under regular Gaussian noise. We use the Huber M-estimator as adifference measure between the distance of estimated nodes and noisy rangemeasurements, thus obtaining a robust (but nonconvex) optimization problem. Wethen devise a convex underestimator solvable in polynomial time, and tight inthe inter-node terms. We also provide an optimality bound for the convexunderestimator. We put forward a new representation of the Huber functioncomposed with a norm, enabling distributed robust localization algorithms tominimize the proposed underestimator. The synchronous distributed method hasoptimal convergence rate and the asynchronous one converges in finite time, fora given precision. The main highlight of our contribution lies on the fact thatwe pay no price for distributed computation nor in accuracy, nor incommunication cost or convergence speed. Simulations show the advantage ofusing our proposed algorithms, both in the presence of outliers and underregular Gaussian noise: our method exceeds the accuracy of an alternativerobust approach based on L1 norms by at least 100m in an area of 1Km sides.

Dealing with bad apples: Robust range-based network localization via distributed relaxation methods