Stepsize analysis for descent methods

The convergence rates of descent methods with different stepsize rules are compared. Among the stepsize rules considered are: constant stepsize, minimization along a line, Goldstein-Armijo rules, and stepsize equal to minimum of certain interpolatory polynomials. One of the major results shown is that the rate of convergence of descent methods with the Goldstein-Armijo stepsize rule can be made as close as desired to the rate of convergence of methods that require minimization along a line. Also, a descent algorithm that combines a Goldstein-Armijo stepsize rule with a secant-type step is presented. It is shown that this algorithm has a convergence rate equal to the convergence of descent methods that require minimization along a line and that, eventually (i.e. near the minimum), it does not require a search to determine an acceptable stepsize.