Convergence of the Gauss-Newton method for convex composite optimization under a majorant condition

Under the hypothesis that an initial point is a quasi-regular point, we use a majorant condition to present a new semi-local convergence analysis of an extension of the Gauss-Newton method for solving convex composite optimization problems. In this analysis the conditions and proof of convergence are simplified by using a simple majorant condition to define regions where a Gauss-Newton sequence is "well behaved"

On the Kantorovich's theorem for Newton's method for solving generalized equations under the majorant condition

In this paper we consider a version of the Kantorovich's theorem for solving the generalized equation $F(x)+T(x)\ni 0$, where $F$ is a Fr\'echet derivative function and $T$ is a set-valued and maximal monotone acting between Hilbert spaces. We show that this method is quadratically convergent to a solution of $F(x)+T(x)\ni 0$. We have used the idea of majorant function, which relaxes the Lipschitz continuity of the derivative $F'$. It allows us to obtain the optimal convergence radius, uniqueness of solution and also to solving generalized equations under Smale's condition.Comment: 16 pages, 0 figur

Convergence of the Gauss-Newton method for a special class of systems of equations under a majorant condition

In this paper, we study the Gauss-Newton method for a special class of systems of nonlinear equation. Under the hypothesis that the derivative of the function under consideration satisfies a majorant condition, semi-local convergence analysis is presented. In this analysis the conditions and proof of convergence are simplified by using a simple majorant condition to define regions where the Gauss-Newton sequence is "well behaved". Moreover, special cases of the general theory are presented as applications

Inexact Newton's method to nonlinear functions with values in a cone

The problem of finding a solution of nonlinear inclusion problems in Banach space is considered in this paper. Using convex optimization techniques introduced by Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), a robust convergence theorem for inexact Newton's method is proved. As an application, an affine invariant version of Kantorovich's theorem and Smale's \alpha-theorem for inexact Newton's method is obtained.Comment: 15 pages, 0 figure. arXiv admin note: text overlap with arXiv:1403.246

Local convergence analysis of Newton's method for solving strongly regular generalized equations

In this paper we study Newton's method for solving generalized equations in Banach spaces. We show that under strong regularity of the generalized equation, the method is locally convergent to a solution with superlinear/quadratic rate. The presented analysis is based on Banach Perturbation Lemma for generalized equation and the classical Lipschitz condition on the derivative is relaxed by using a general majorant function, which enables obtaining the optimal convergence radius, uniqueness of solution as well as unifies earlier results pertaining to Newton's method theory.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1604.0456

Kantorovich's theorem on Newton's method for solving strongly regular generalized equation

In this paper we consider the Newton's method for solving the generalized equation of the form $f(x) +F(x) \ni 0,$ where $f:{\Omega}\to Y$ is a continuously differentiable mapping, $X$ and $Y$ are Banach spaces, $\Omega\subseteq X$ an open set and $F:X \rightrightarrows Y$ be a set-valued mapping with nonempty closed graph. We show that, under strong regularity of the generalized equation, concept introduced by S.M.Robinson in [27], and starting point satisfying the Kantorovich's assumptions, the Newton's method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. The analysis presented based on Banach Perturbation Lemma for generalized equation and the majorant technique, allow to unify some results pertaining the Newton's method theory.Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv: 1604.04568, arXiv:1603.0478

A robust semi-local convergence analysis of Newton's method for cone inclusion problems in Banach spaces under affine invariant majorant condition

A semi-local analysis of Newton's method for solving nonlinear inclusion problems in Banach space is presented in this paper. Under a affine majorant condition on the nonlinear function which is associated to the inclusion problem, the robust convergence of the method and results on the convergence rate are established. Using this result we show that the robust analysis of the Newton's method for solving nonlinear inclusion problems under affine Lipschitz-like and affine Smale's conditions can be obtained as a special case of the general theory. Besides for the degenerate cone, which the nonlinear inclusion becomes a nonlinear equation, ours analysis retrieve the classical results on local analysis of Newton's method

Local convergence analysis of a proximal Gauss-Newton method under a majorant condition

In this paper, the proximal Gauss-Newton method for solving penalized nonlinear least squares problems is studied. A local convergence analysis is obtained under the assumption that the derivative of the function associated with the penalized least square problem satisfies a majorant condition. Our analysis provides a clear relationship between the majorant function and the function associated with the penalized least squares problem. The convergence for two important special cases is also derived

Robust Kantorovich's theorem on Newton's method under majorant condition in Riemannian Manifolds

A robust affine invariant version of Kantorovich's theorem on Newton's method, for finding a zero of a differentiable vector field defined on a complete Riemannian manifold, is presented in this paper. In the analysis presented, the classical Lipschitz condition is relaxed by using a general majorant function, which allow to establish existence and local uniqueness of the solution as well as unifying previously results pertaining Newton's method. The most important in our analysis is the robustness, namely, is given a prescribed ball, around the point satisfying Kantorovich's assumptions, ensuring convergence of the method for any starting point in this ball. Moreover, bounds for $Q$-quadratic convergence of the method which depend on the majorant function is obtained.Comment: 25 page

Relaxed Gauss-Newton methods with applications to electrical impedance tomography

As second-order methods, Gauss--Newton-type methods can be more effective than first-order methods for the solution of nonsmooth optimization problems with expensive-to-evaluate smooth components. Such methods, however, often do not converge. Motivated by nonlinear inverse problems with nonsmooth regularization, we propose a new Gauss--Newton-type method with inexact relaxed steps. We prove that the method converges to a set of disjoint critical points given that the linearisation of the forward operator for the inverse problem is sufficiently precise. We extensively evaluate the performance of the method on electrical impedance tomography (EIT).Comment: 43 pages, 29 figure