Local convergence analysis of the Gauss–Newton method under a majorant condition

Abstract

a b s t r a c tThe Gauss-Newton method for solving nonlinear least squares problems is studied in this paper. Under the hypothesis that the derivative of the function associated with the least square problem satisfies a majorant condition, a local convergence analysis is presented. This analysis allows us to obtain the optimal convergence radius and the biggest range for the uniqueness of stationary point, and to unify two previous and unrelated results.

Extended local convergence analysis of inexact Gauss-Newton method for singular systems of equations under weak conditions

“…The results in [6] improved the corresponding ones in [21,22,23,24,25,42,43]. In the present study, we improved the results in [6], since D 1 ⊂ U (x * , r) leading to an at least as tight function h λ,θ than the one used in [6] (see also the Examples).…”

“…The advantages of our analysis over earlier works such as [8,9,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43] are also shown under the same computational cost for the functions and constants involved. These advantages include: a large radius of convergence and more precise error estimates on the distances x n+1 − x * for each n = 0, 1, 2, .…”

“…and a clearer relationship between the majorant function (see (2.8) and the associated least squares problems (1.1)). These advantages are obtained because we use a center-type majorant condition (see (2.11)) for the computation of inverses involved which is more precise that the majorant condition used in [21,22,23,24,25,26,30,31,39,40,41,42,43]. Moreover, these advantages are obtained under the same computational cost, since as we will see in section 3 and section 4, the computation of the majorant function requires the computation of the center-majorant function.…”

Local Convergence of the Gauss-Newton Method for Injective-Overdetermined Systems

“…. , where x 0 ∈ D is an initial point and F ′ (x n ) + is the Moore-Penrose inverse of the linear operator F ′ (x n ) [7,9,12,14,16,18]. In the present paper we use the proximal Gauss-Newton method (to be precised in Section 2, see (2.6)) for solving penalized nonlinear least squares problem (1.1).…”

“…A survey of convergence results under various Lipschitz-type conditions for Gauss-Newton-type methods can be found in [3,9] (see also [7,12,14,18]). The convergence of these methods requires among other hypotheses that F ′ satisfies a Lipschitz condition or F ′′ is bounded in D. Several authors have relaxed these hypotheses.…”