Inexact Gauss-Newton like methods for injective-overdetermined systems of equations under a majorant condition

Abstract

In this paper, inexact Gauss-Newton like methods for solving injective-overdetermined systems of equations are studied. We use a majorant condition, defined by a function whose derivative is not necessarily convex, to extend and improve several existing results on the local convergence of the Gauss-Newton methods. In particular, this analysis guarantees the convergence of the methods for two important new cases.

An inexact Newton-like conditional gradient method for constrained nonlinear systems

“…Therefore, { x k − x * } is a bounded strictly decreasing sequence and hence it converges to some ℓ * ∈ [0, ρ). Moreover, taking into account that n f (·) is continuous in [0, ρ), in particular, from (14) we have…”

“…which, combined with (12), implies that ℓ * = 0 and consequently x k → x * . Now, from (14) we obtain…”

“…For this purpose, let us assume that h3 holds. It follows from (15) and (14) respectively that, for all k ≥ 0, x k − x * < x 0 − x * and…”

“…We now specialize Theorem 1 for two important classes of functions. In the first one, F ′ satisfies a Hölder-like condition [13,14,17], and in the second one, F is an analytic function satisfying a Smale condition [36,37].…”

On the global convergence of an inexact quasi-Newton conditional gradient method for constrained nonlinear systems

“…We refer the reader to [1,6,11,14] where convergence results of the Newton method and its variants have been discussed. Consider now the constrained nonlinear system…”