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A new technique is proposed for robust multiuser detection in the presence of non-Gaussian ambient noise. This method is based on minimizing a certain cost function (e.g., the Huber penalty function) over a discrete set of candidate user bit vectors. The set of candidate points are chosen based on the so-called "slowest-descent search." starting from the estimate closest to the unconstrained minimizer of the cost function and along mutually orthogonal directions where this cost function grows the slowest. Simulation results show that this new technique offers substantial performance improvement over the previously proposed robust multiuser detectors with little attendant increase in computational complexity.