Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares

Abstract:

A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deficient in two ways. Considering the problem formulated as $\min _{x} 1/2\| f_{2}(x) \|_{2}^{2}$ subject to the constraints $f_{1}(x) = 0$, the Jacobian $J_{1} = \partial f_{1}/ \partial x$ and/or the Jacobian $J = \partial f/ \partial x$, $f = [f_{1};f_{2}]$, may be ill conditioned at the solution. We analyze the important special case when $J_{1}$ and/or $J$ do not have full rank at the solution. In order to solve such a problem, we formulate a nonlinear least norm problem. Next we describe a truncated Gauss-Newton method. We show that the local convergence rate is determined by the maximum of three independent Rayleigh quotients related to three different spaces in $\mathbb {R}^{n}$. Another way of solving an ill-posed nonlinear least squares problem is to regularize the problem with some parameter that is reduced as the iterates converge to the minimum. Our approach is a Tikhonov based local linear regularization that converges to a minimum norm problem. This approach may be used both for almost and rank-deficient Jacobians. Finally we present computational tests on constructed problems verifying the local analysis.