Abstract. Robust cost optimization is the challenging task of fitting a large number of parameters to data points containing a significant and unknown fraction of outliers. In this work we identify three classes of deterministic second-order algorithms that are able to tackle this type of optimization problem: direct approaches that aim to optimize the robust cost directly with a second order method, lifting-based approaches that add so called lifting variables to embed the given robust cost function into a higher dimensional space, and graduated optimization methods that solve a sequence of smoothed cost functions. We study each of these classes of algorithms and propose improvements either to reduce their computational time or to make them find better local minima. Finally, we experimentally demonstrate the superiority of our improved graduated optimization method over the state of the art algorithms both on synthetic and real data for four different problems