  The variable projection (VarPro) method is an efficient method to solve
separable nonlinear least squares problems. In this paper, we propose a
modified VarPro for large-scale separable nonlinear inverse problems that
promotes edge-preserving and sparsity properties on the desired solution and
enhances the convergence of the parameters that define the forward problem. We
adopt a majorization minimization method that relies on constructing a
quadratic tangent majorant to approximate a general $\ell_p$ regularized
problem by an $\ell_2$ regularized problem that can be solved by the aid of
generalized Krylov subspace methods at a relatively low cost compared to the
original unprojected problem. In addition, we can use more potential general
regularizers including total variation (TV), framelet, and wavelets operators.
The regularization parameter can be defined automatically at each iteration by
means of generalized cross validation. Numerical examples on large-scale
two-dimensional imaging problems arising from blind deconvolution are used to
highlight the performance of the proposed approach in both quality of the
reconstructed image as well as the reconstructed forward operator.
