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We prove Kantorovich's theorem on Newton's method using a convergence analysis which makes clear, with respect to Newton's Method, the relationship of the majorant function and the non-linear operator under consideration. This approach enable us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results. AMSC: 49M15, 90C30.

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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". AMSC: 47J15, 65H10.

This paper considers optimization problems on Riemannian manifolds and analyzes iteration-complexity for gradient and subgradient methods on manifolds with non-negative curvature. By using tools from the Riemannian convex analysis and exploring directly the tangent space of the manifold, we obtain different iterationcomplexity bounds for the aforementioned methods, complementing and improving related results. Moreover, we also establish iteration-complexity bound for the proximal point method on Hadamard manifolds. keywords: Complexity, gradient method, subgradient method, proximal point method, Riemannian manifold.

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.