Convergence criteria of Newton’s method on Lie groups

Jinsu He,Jen-Chih Yao,Jinhua Wang

doi:10.1186/1687-1812-2013-293

Abstract

In the present paper, we study Newton’s method on Lie groups (independent of affine connections) for finding zeros of a mapping f from a Lie group to its Lie algebra. Under a generalized L-average Lipschitz condition of the differential of f, we establish a unified convergence criterion of Newton’s method. As applications, we get the convergence criteria under the Kantorovich’s condition and the γ-condition, respectively. Moreover, applications to optimization problems are also provided. MSC:65H10, 65D99.

Highlights

Newton’s method is one of the most important methods for finding the approximation solution of the equation f (x) =, where f is an operator from some domain D in a real or complex Banach space X to another Y
There are a lot of works on the weakness and/or the extension of the Lipschitz continuity made on the mappings; see, for example, [ – ] and references therein
Zabrejko-Nguen parametrized in [ ] the classical Lipschitz continuity

Summary

Introduction

In [ ], Li and Wang extended the generalized L-average Lipschitz condition (introduced in [ ]) to Riemannian manifolds and established a unified convergence criterion of Newton’s method on Riemannian manifolds. Under the assumption that the differential of f satisfies the Lipschitz condition around the initial point (which is in terms of one-parameter semigroups and independent of the metric), the convergence criterion of Newton’s method is presented. The purpose of the present paper is to establish a unified convergence criterion for Newton’s method (independent of the connection) on Lie groups under a generalized L-average Lipschitz condition.

Objectives

Results

Conclusion