On the Separable Nonlinear Least Squares Problems (Journal of Computational Mathematics, 2018)
Separable nonlinear least squares problems are a special class of nonlinear least squares problems, where the objective functions are linear and nonlinear on different parts of variables. Such problems have broad applications in practice. Most existing algorithms for this kind of problems are derived from the variable projection method proposed by Golub and Pereyra, which utilizes the separability under a separate framework. However, the methods based on variable projection strategy would be invalid if there exist some constraints to the variables, as the real problems always do, even if the constraint is simply the ball constraint. We present a new algorithm which is based on a special approximation to the Hessian by noticing the fact that certain terms of the Hessian can be derived from the gradient. Our method maintains all the advantages of variable projection based methods, and moreover, it can be combined with trust region methods easily and can be applied to general constrained separable nonlinear problems. Convergence analysis of our method is presented and numerical results are also reported.
Keywords: Separable nonlinear least squares problem; Variable projection method; Gauss-Newton method; Levenberg-Marquardt method; Trust region method; Asymptotical convergence rate; Data fitting.