Summary: We present an efficient iterative approach to solving separable nonlinear least squares problems that arise in large-scale inverse problems. A variable projection Gauss-Newton method is used to solve the nonlinear least squares problem, and Tikhonov regularization is incorporated using an iterative hybrid scheme. Regularization parameters are chosen automatically using a weighted generalized cross validation method, thus providing a nonlinear solver that requires very little input from the user. Applications from image deblurring and digital tomosynthesis illustrate the effectiveness of the resulting numerical scheme.