Hybrid method for nonlinear least-square problems without calculating derivatives
This paper presents a no-derivative modification of the hybrid Gauss- Newton-BFGS method for nonlinear least-square problems suggested initially by \textit{M. Al-Baali} and \textit{R. Fletcher} [J. Oper. Res. Soc. 36, 405-421 (1985; Zbl 0578.65064)] and modified later by \textit{R. Fletcher} and \textit{C. Xu} [IMA J. Numer. Anal. 7, 371-389 (1987; Zbl 0648.65051)]. The modification is made in such a way that, in a Gauss- Newton step, the Broyden's rank-one updating formula is used to obtain an approximate Jacobian and, in a BFGS step, the Jacobian is estimated using difference formulas. A set of numerical comparisons among the new hybrid method, the Gauss-Newton-Broyden method, and the finite-difference BFGS method is made and shows that the new hybrid method combines the better features of the Gauss-Newton-Broyden method and the finite-difference BFGS method. This paper also extends to the least-square problem the finite-termination property of the Broyden method, proved for a nonsingular system of equations by \textit{D. M. Gay} [SIAM J. Numer. Anal. 16, 623-630 (1979; Zbl 0453.65034)] and for the full-rank rectangular system of equations by \textit{R. R. Gerber} and \textit{F. T. Luk} [ibid. 18, 882-890 (1981; Zbl 0472.65036)].
scientific article; zbMATH DE number 4110475
scientific article