On the Gauss–Newton method

Abstract

We provide a new semilocal convergence analysis of the Gauss-Newton method (GNM) for solving nonlinear equation in the Euclidean space.Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, finer convergence analysis. The results can be extended in case outer or generalized inverses are used.Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel,

Local convergence analysis of inexact Gauss–Newton method for singular systems of equations under majorant and center-majorant condition

“…This way the advantages as stated in the Introduction of this study can be obtained. In order to achieve these advantages we introduce the following notion [2][3][4][5][6][7][8][9][10][11].…”

“…Moreover, the radius of the convergence balls under the corresponding conditions were estimated in these two papers. The preceding results were improved by Argyros et al [2][3][4][5][6][7][8][9][10][11] using the concept of the center Lipschitz condition (see also (2.11) and the numerical examples) under the same computational cost on the parameters and functions involved.…”

Improved semi-local convergence of the Gauss-Newton method for systems of equations

“…In the present paper, we are motivated by the work of Goncalves and Oliveira in [14] (see also [12], [13]) and our works in [1,2,3,4,6,7,8]. These authors presented a semi-local convergence analysis for the Gauss-Newton method (1.2) for systems of nonlinear equations where the function F satisfies…”

Expanding the applicability of the Gauss–Newton method for convex optimization under a majorant condition

“…Many problems in Mathematical Programming such as convex inclusion, minimax problems, penalization methods, goal programming, constrained optimization and other problems can be formulated like composite optimizations problem (see, e.g., [1,4,6,7,10,11,13,19,23,24,26]). …”