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We consider the problem of target localization by a network of passive sensors. When an unknown target emits an acoustic or a radio signal, its position can be localized with multiple sensors using the time difference of arrival (TDOA) information. In this paper, we consider the maximum likelihood formulation of this target localization problem and provide efficient convex relaxations for this nonconvex optimization problem. We also propose a formulation for robust target localization in the presence of sensor location errors. Two Cramer-Rao bounds are derived corresponding to situations with and without sensor node location errors. Simulation results confirm the efficiency and superior performance of the convex relaxation approach as compared to the existing least squares based approach when large sensor node location errors are present.