Approximate Gauss–Newton methods for solving underdetermined nonlinear least squares problems

We propose several approximate Gauss–Newton methods, i.e., the truncated, perturbed, and truncated-perturbed GN methods, for solving underdetermined nonlinear least squares problems. Under the assumption that the Fréchet derivatives are Lipschitz continuous and of full row rank, Kantorovich-type convergence criteria of the truncated GN method are established and local convergence theorems are presented with the radii of convergence balls obtained. As consequences of the convergence results for the truncated GN method, convergence theorems of the perturbed and truncated-perturbed GN methods are also presented. Finally, numerical experiments are presented where the comparisons with the standard inexact Gauss–Newton method and the inexact trust-region method for bound-constrained least squares problems [23] are made.