VARPROX: a primal-dual variable projection method for the minimization of penalized separable non-linear least squares

Abstract We present a methodology for solving non-linear least squares problems that extends the variable projection. We propose to add a non-smooth convex regularization on the non-linear variable in order to handle instability when it is high-dimensional. While conserving the variable projection structure, our method relies on a primal-dual proximal method and consequently, benefits from the full splitting property of all the involved operators. We then extend our methodology to the common case where box constraints are set on the non-linear variables. We report numerical application of our method to image texture analysis where our methodology shows significant improvement over the standard variable projection.