On the convergence of descent methods (Q1571169)

For many iterative processes of descent methods it is not possible to obtain an estimate of the convergence to solution point \(x_*\) at a linear rate \[ \rho{(x^k,x_*)}\leq{q^k\rho{(x^0,x_*)}} \forall{k}\in\mathbb{N}. \] At the same time, one is often able to obtain the required estimates in terms of a certain function that is more or less naturally defined. This approach, which may be classified as the second Lyapunov method, is used in this paper. Applications to the numerical solution of nonlinear operator equations with approximate initial data are discussed.