Direct, fast and convergent solvers for the non-convex and non-smooth TDoA localization problem

Time-Difference-of-Arrival (TDoA) Source Localization (SL) consists of determining the position of a target given differences of time measurements of several sensors. The TDoA SL problem is formulated as minimizing a non-convex and non-smooth least squares (LS) optimization problem, which is tackled in the existing literature mostly by solving convex relaxations or by taking the square of the measurements. In this work, we tackle the LS problem directly, by first showing that a standard fixed-point (FP) method can be derived. Due to numerical instability and lack of theoretical convergence guarantees of FP, we develop the T-NAM method. This easy-to-implement and novel method uses the Nested Alternating Minimization scheme together with the fast FISTA algorithm. We show that T-NAM converges to critical points of the original LS function – a result that, to the best of our knowledge, is unknown for the TDoA SL problem. Last, we show numerically the advantages of FP and T-NAM relatively to existing works.