On Spectral Properties of Steepest Descent Methods

In recent years it has been made more and more clear that the critical issue in gradient methods is the choice of the step length, whereas using the gradient as search direction may lead to very effective algorithms, whose surprising behaviour has been only partially explained, mostly in terms of the spectrum of the Hessian matrix. On the other hand, the convergence of the classical Cauchy steepest descent (SD) method has been extensively analysed and related to the spectral properties of the Hessian matrix, but the connection with the spectrum of the Hessian has been little exploited to modify the method in order to improve its behaviour. In this work we show how, for convex quadratic problems, moving from some theoretical properties of the SD method, second-order information provided by the step length can be exploited to dramatically improve the usually poor practical behaviour of this method. This allows to achieve computational results comparable with those of the Barzilai and Borwein algorithm, with the further advantage of a monotonic behaviour.