On solving nonlinear least-squares problems in case of rankdeficient Jacobians
This paper is concerned with the nonlinear least squares problem min g(x)\(=1/2F^ T(x)F(x)\), where \(F: R^ n\to R^ m\), \(m\geq n\), is a sufficiently smooth nonlinear mapping, in the case of a rank-deficient Jacobian \(F'(x^*)\) at a solution point x. The method presented here for solving the above problem defines an auxiliary function \(T: R^ N\to R^ M\), \(N=n(r+1)\), \(M=(m+r/2)(r+1)\), where n-r, \(r\geq 1\), is the rank of \(F'(x^*)\), and constructs a least squares problem min G(u)\(=1/2T^ T(u)T(u)\), \(u=\left( \begin{matrix} x\\ d\end{matrix} \right)\) of higher dimension which is shown to be a well-posed one, if the rank deficiency r of \(F'(x^*)\) is small. As is also proved, when \(g(x^*)=0\), the Gauss- Newton sequence \(\{x^ k\}\) converges at least q-linearly with asymptotic rate 1/2.
scientific article; zbMATH DE number 3926118
scientific article