Robust Target Localization in 2D: A Value-at-Risk Approach

This paper considers the problem of locating a target in 2D from range measurements containing outliers. Assuming the number of outliers is known, we formulate the problem of minimizing inlier losses while ignoring outliers. This leads to a combinatorial, non-convex, non-smooth problem involving the percentile function. Using the framework of risk analysis from Rockafellar et al., we start by interpreting this formulation as a Value-at-Risk (VaR) problem from portfolio optimization. To the best of our knowledge, this is the first time that a localization problem was formulated using risk analysis theory. To solve the VaR formulation, we start by constructing a majorizer set that contains any solution of a general percentile problem. This construction is useful because, when applied to a localization scenario in 2D, it allows to majorize the solution set in terms of singletons, circumferences, ellipses and hyperbolas. Using known parametrizations of these curves, we propose a grid method for the original non-convex problem. So we reduce the task of optimizing the VaR objective to that of efficiently sampling the proposed majorizer set. We compare our algorithm with several benchmarks in target localization. Numerical simulations show that our method is fast while, on average, improving the accuracy of the best benchmarks by at least 100 m in a 1 Km
²
area. Furthermore, our approach is shown to be robust when the number of outliers is over-estimated and the position of the anchors is either fixed or sampled uniformly in the 1 Km
²
area.