Convergence of Newton’s Method for Sections on Riemannian Manifolds

Wang, J. H.

doi:10.1007/s10957-010-9748-4

Cited by 26 publications

(10 citation statements)

References 28 publications

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“…Assume now that p k ∈ K(t k ). Thus, using Lemma 6,(38) and (25), we obtain that p k+1 ∈ K(t k+1 ), which completes the induction proof of (40). Using definition of {t k } in (14) ,…”

Section: Convergencementioning

confidence: 52%

“…Early works dealing with the generalization of Newton's methods to Riemannian setting include [10,12,18,33,36,39]. Actually, the generalization of Newton's method to Riemannian setting has been done with several different purposes, including the purpose of finding a zeros of a gradient vector field or, more generally, with the purpose of finding a zero of a differentiable vector field; see [1,2,5,6,7,9,10,12,14,15,24,25,26,27,28,34,35,38,40,41,42,46] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds

Bittencourt¹,

Ferreira²

2016

J Glob Optim

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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.

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“…Assume now that p k ∈ K(t k ). Thus, using Lemma 6,(38) and (25), we obtain that p k+1 ∈ K(t k+1 ), which completes the induction proof of (40). Using definition of {t k } in (14) ,…”

Section: Convergencementioning

confidence: 52%

Section: Introductionmentioning

confidence: 99%

Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds

Bittencourt¹,

Ferreira²

2016

J Glob Optim

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“…da Cruz Neto et al 1998), but most of these proofs required certain curvature conditions on the manifold. Wang (2011), however, provided a curvatureindependent convergence proof.…”

Section: A Review Of the Fieldmentioning

confidence: 99%

Riemannian optimization and multidisciplinary design optimization

Bakker

Parks

2016

Optim Eng

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Riemannian Optimization (RO) generalizes standard optimization methods from Euclidean spaces to Riemannian manifolds. Multidisciplinary Design Optimization (MDO) problems exist on Riemannian manifolds, and with the differential geometry framework which we have previously developed, we can now apply RO techniques to MDO. Here, we provide background theory and a literature review for RO and give the necessary formulae to implement the Steepest Descent Method (SDM), Newton's Method (NM), and the Conjugate Gradient Method (CGM), in Riemannian form, on MDO problems. We then compare the performance of the Riemannian and Euclidean SDM, NM, and CGM algorithms on several test problems (including a satellite design problem from the MDO literature); we use a calculated step size, line search, and geodesic search in our comparisons. With the framework's induced metric, the RO algorithms are generally not as effective as their Euclidean counterparts, and line search is consistently better than geodesic search. In our post-experimental analysis, we also show how the optimization trajectories for the Riemannian SDM and CGM relate to design coupling and thereby provide some explanation for the observed optimization behaviour. This work is only a first step in applying RO to MDO, however, and the use of quasi-Newton methods and different metrics should be explored in future research.

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“…Newton's method has been extended to Riemannian manifolds with many different purposes. In particular, in the last few years, a couple of papers have dealt with the issue of convergence analysis of Newton's method for finding a singularity of a differentiable vector field defined on a complete Riemannian manifold, see [1,4,9,10,15,16,17,22,23,24,25,26,27]. Extensions to Riemannian manifolds of analyses of Newton's method under the γ-condition was given in [4,15,16,17].…”

Section: Introductionmentioning

confidence: 99%

Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds

Bittencourt

Ferreira

2015

Applied Mathematics and Computation

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A local convergence analysis of Inexact Newton's method with relative residual error tolerance for finding a singularity of a differentiable vector field defined on a complete Riemannian manifold, based on majorant principle, is presented in this paper. We prove that under local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q -linearly to a singularity of the vector field under consideration. Using this result we show that the inexact Newton method to find a zero of an analytic vector field can be implemented with a fixed relative residual error tolerance. In the absence of errors, our analysis retrieve the classical local theorem on the Newton method in Riemannian context.

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Convergence of Newton’s Method for Sections on Riemannian Manifolds

Cited by 26 publications

References 28 publications

Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds

Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds

Riemannian optimization and multidisciplinary design optimization

Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds

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