Dealing with bad apples: Robust range-based network localization via   distributed relaxation methods
  Real-world network applications must cope with failing nodes, malicious attacks, 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, prior research seldom considered outlier data --- even now, when already deployed networks cry out for robust procedures. We propose robust, fast, and distributed network localization algorithms, resilient to high-power noise, but also precise under regular Gaussian noise. We use the Huber M-estimator as a difference measure between the distance of estimated nodes and noisy range measurements, thus obtaining a robust (but nonconvex) optimization problem. We then devise a convex underestimator solvable in polynomial time, and tight in the inter-node terms. We also provide an optimality bound for the convex underestimator. We put forward a new representation of the Huber function composed with a norm, enabling distributed robust localization algorithms to minimize the proposed underestimator. The synchronous distributed method has optimal convergence rate and the asynchronous one converges in finite time, for a given precision. The main highlight of our contribution lies on the fact that we pay no price for distributed computation nor in accuracy, nor in communication cost or convergence speed. Simulations show the advantage of using our proposed algorithms, both in the presence of outliers and under regular Gaussian noise: our method exceeds the accuracy of an alternative robust approach based on L1 norms by at least 100m in an area of 1Km sides. 
Cl\'audia Soares and Jo\~ao Gomes
math.OC