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

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 derivativef(1)/partial derivativex and/or the Jacobian J = partial derivativef/partial derivativex, 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 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.