Variational methods for non-linear least-squares.

*(English)*Zbl 0578.65064The authors consider numerical methods for solving the unconstrained nonlinear least squares problem, and try to find a compromise between the Gauss-Newton-method and an appropriate quasi-Newton-method with the aim to get an algorithm suitable for small and large residual problems. Various alternatives are discussed to find an updating formula for least squares problems proceeding either from the BFGS- or the DFP-formula.

Based on numerical experience,the final proposal of the authors is to use a simple hybrid method without optimal scaling combining the Gauss-Newton and the BFGS-method, where the difference vector of successive gradients is modified to take advantage of the special problem structure. The usage of a line search is recommended and details of the numerical implementation are presented.

The new code is compared experimentally with a pure Gauss-Newton, a pure BFGS- and a standard method of the IMSL library (NL2S1, Dennis et al.). The numerical tests are based on classical test problems and some new large residual test problems. They indicate that the proposed method is more robust and slightly more efficient with respect to residual evaluations.

Based on numerical experience,the final proposal of the authors is to use a simple hybrid method without optimal scaling combining the Gauss-Newton and the BFGS-method, where the difference vector of successive gradients is modified to take advantage of the special problem structure. The usage of a line search is recommended and details of the numerical implementation are presented.

The new code is compared experimentally with a pure Gauss-Newton, a pure BFGS- and a standard method of the IMSL library (NL2S1, Dennis et al.). The numerical tests are based on classical test problems and some new large residual test problems. They indicate that the proposed method is more robust and slightly more efficient with respect to residual evaluations.

Reviewer: K.Schittkowski