3 Assuring Convergence

Convergence of the Gauss-Newton method is not guaranteed, and it converges only to a local optimum that depends on the starting parameters. In practice, if the objective function $L(\mathbf{x})$ is locally well-approximated by a quadratic form, then convergence to a local minimum is quadratic. However, the curvature of the error surface of a nonlinear observation model can vary significantly over the parameter space. The Levenberg-Marquardt method is a refinement to the Gauss-Newton procedure that increases the chance of local convergence and prohibits divergence. Note that the results still depend on the starting point.