Nonstandard scaling matrices for trust region Gauss-Newton methods

For solving large nonlinear least-squares problems via trust region Gauss–Newton methods, nonstandard scaling matrices are proposed for scaling the norm of the step. The scaling matrices are rectangular, of full rank, and contain a block of the Jacobian matrix of the residual function.Three types of such matrices are investigated. The corresponding trust region methods are shown to have qualitatively the same convergence properties as the standard method. Nonstandard scaling matrices are especially intended for solving large and structured problems such as orthogonal distance regression or surface fitting. Initial computational experience suggests that for such problems the proposed scaling implies sometimes a modest increase in the number of iterations but reduces the overall computational costs.