A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

Abstract

Abstract:We consider the problem of finding a singularity of a vector field X on a complete Riemannian manifold. In this regard we prove a unified result for local convergence of Newton's method. Inspired by previous work of Zabrejko and Nguen on Kantorovich's majorant method, our approach relies on the introduction of an abstract one-dimensional Newton's method obtained using an adequate Lipschitz-type radial function of the covariant derivative of X. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector fields an application of the central result leads to improvements of some recent developments in this area.

Kantorovich’s majorants principle for Newton’s method

“…x 0 ) be a Banach space. Use Lemma 14 to prove that f : [0, 1) → R defined by f (t) = t/(1 − t) − 2t + β, is a majorant function to g in x 0 , with roots equal to t * and t * * , see [2]. So, the result follows from Theorem 2.…”

“…Kantorovich's Theorem was used in [10] to prove Smale's Theorem [9], and it was used in [2] to prove Nesterov-Nemirovskii's Theorem [6]. We present these proofs here, for the sake of ilustration.…”

A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

“…x > 1, [1,2], x = 1 {1} 0 < x < 1 [−1, 1], x = 0 and ∂f (x) = −∂f (−x) for x < 0. Define t 1 = λ 2 + 1 log 2, t 2 = λ 2 + 2 log 2, t 3 = λ 2 + 2 log 2 + λ.…”

“…One may consult [1], [13], [15], [21], [27] and references therein for an overview on such Newton-like methods. In [23], one can find a survey on the rich connections between continuous evolution equations generated by maximal monotone operators and their discrete time versions.…”

Local convergence analysis of inexact Newton-like methods under majorant condition

“…In addition to improving the convergence theory (this allows us to estimate the convergence radius and to enlarge the range of application) some modifications of the Lipschitz condition also permit us to unify several results. Works dealing with this subject include [1,8,9,17] and [18].…”