In some complex environments, nonline-of-sight (or multipath) usually introduces an additional path delay to the pseudorange, which is known to be one of the most dominant sources of accuracy degradation in global navigation satellite systems (GNSSs). The additional path error is non-Gaussian. Different techniques, including antenna-based, receiver-based, and poster-receiver, have been proposed for mitigating the influence of multipath. However, the estimated additional path delay obtained by classical algorithms, such as maximum likelihood or Bayesian estimation, often has a certain deviation from the real-additional path delay. For more accurate estimation, we first restrict the additional path error in a relatively wide interval; then, the position estimation is converted into a constrained separable nonlinear least squares problem. We propose, in this paper, a two-stage estimation algorithm for global positioning system positioning. At the first stage, we ignore the additional path error and obtain a relatively accurate initial position. Then, based on the result obtained at the first stage, we include the additional path error in the estimation problem and estimate it by variable projection (VP) method. Numerical results show that the proposed algorithm based on the VP method can effectively mitigate multipath interference.