On Newton's method for Huber's robust M-estimation problems in linear regression (Q1279695)

The Newton-type method for computing Huber's robust \(M\)-estimator in the linear regression model of \textit{K. Madsen} and \textit{H. B. Nielsen} [BIT 30, No. 4, 682-699 (1990; Zbl 0717.65118)] is reconsidered. The following enhancements to the original algorithm are studied: \(\bullet\) Finite termination property of the algorithm is proved without any assumption on the \(M\)-estimation problem. \(\bullet\) It is not necessary to make restrictions on the choice of the search direction and the step length in some degenerate cases. Detailed algorithms are given and it is proved that the modified Newton-type algorithm finds a minimizer in a finite number of iterations.