# Rate of Convergence for (Gauss)-Newton method

I am trying to solve a non-linear least squares problem with newton's and gauss-newton method.

$\min \left\Vert \pmb F( \pmb x) \right\Vert_2^2$ with $F: \mathbb{R}^m \rightarrow \mathbb{R}^n$.

Furthermore, I want to analytically calculate the local rate of convergence for my problem. Because the Hessian and Jacobian of $\pmb F$ may both be indefinite and rank-deficient its most probably not quadratic for Newton's method. What are the criteria for these methods to show (super)-linear or quadratic local convergence?

I know that $\pmb F$ has to Lipschitz continuous, which is the case. But what are the other criteria?

thank you!

• I believe the assumption for quadratic convergence is that the Hessian is invertible in a neighborhood of the optimum. For example, minimizing $x^4$ with Newton's method gives linear convergence to $0$. – p.s. Jul 16 '13 at 10:48
• @p.s. thanks. I even guess that the Hessian has to be positive definite because otherwise the Hessian is not guarantued to be descent direction. But I know, since my problem is non-convex and I will have sometimes also zero eigenvalues, that this will not hold in my case. This is also why I'm very interested in the convergence for Gauss-Newton method. – bonanza Jul 16 '13 at 11:00