Curvature steps and geodesic moves for nonlinear least squares descent algorithms

We address in this article the choice of both the step and the curve of the parameter space to be used in the line search part of descent algorithms for the minimization of least squares objective functions. Our analysis is based on the curvature of the path of the data space followed during the line search. We define first a new and easy to compute "maximum curvature step", which gives a guaranteed value to the residual at the next iterate, and satisfies a linear decrease condition with omega = 1/2. Then we optimize the "worst possible situation", by moving from one iterate to the next along a geodesic or the output set. Preliminary numerical comparisons of the proposed algorithm with the Gauss-Newton algorithm are presented.