The inexact, inexact perturbed, and quasi-Newton methods are equivalent models

Abstract:

A classical model of Newton iterations which takes into account some error terms is given by the quasi-Newton method, which assumes perturbed Jacobians at each step. Its high convergence orders were characterized by Dennis and Moré [Math. Comp. 28 (1974), 549–560]. The inexact Newton method constitutes another such model, since it assumes that at each step the linear systems are only approximately solved; the high convergence orders of these iterations were characterized by Dembo, Eisenstat and Steihaug [SIAM J. Numer. Anal. 19 (1982), 400–408]. We have recently considered the inexact perturbed Newton method [J. Optim. Theory Appl. 108 (2001), 543–570] which assumes that at each step the linear systems are perturbed and then they are only approximately solved; we have characterized the high convergence orders of these iterates in terms of the perturbations and residuals. In the present paper we show that these three models are in fact equivalent, in the sense that each one may be used to characterize the high convergence orders of the other two. We also study the relationship in the case of linear convergence and we deduce a new convergence result.