IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v261y2015icp28-38.html
   My bibliography  Save this article

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

Author

Listed:
  • Bittencourt, Tiberio
  • Ferreira, Orizon Pereira

Abstract

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 retrieves the classical local theorem on the Newton method in Riemannian context.

Suggested Citation

  • Bittencourt, Tiberio & Ferreira, Orizon Pereira, 2015. "Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 28-38.
    • Handle: RePEc:eee:apmaco:v:261:y:2015:i:c:p:28-38
    DOI: 10.1016/j.amc.2015.03.080
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S009630031500394X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2015.03.080?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Jin-Hua Wang & Jen-Chih Yao & Chong Li, 2012. "Gauss–Newton method for convex composite optimizations on Riemannian manifolds," Journal of Global Optimization, Springer, vol. 53(1), pages 5-28, May.
    2. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Davide Cuccato & Alessandro Saccon & Antonello Ortolan & Alessandro Beghi, 2016. "Computing Laser Beam Paths in Optical Cavities: An Approach Based on Geometric Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 297-315, October.
    2. Petre Birtea & Dan Comănescu, 2017. "Newton Algorithm on Constraint Manifolds and the 5-Electron Thomson Problem," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 563-583, May.
    3. Fabiana R. Oliveira & Fabrícia R. Oliveira, 2021. "A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 259-273, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Castro, Jordi & Escudero, Laureano F. & Monge, Juan F., 2023. "On solving large-scale multistage stochastic optimization problems with a new specialized interior-point approach," European Journal of Operational Research, Elsevier, vol. 310(1), pages 268-285.
    2. Luciana Casacio & Aurelio R. L. Oliveira & Christiano Lyra, 2018. "Using groups in the splitting preconditioner computation for interior point methods," 4OR, Springer, vol. 16(4), pages 401-410, December.
    3. Stefano Cipolla & Jacek Gondzio, 2023. "Proximal Stabilized Interior Point Methods and Low-Frequency-Update Preconditioning Techniques," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1061-1103, June.
    4. Fatemeh Marzbani & Akmal Abdelfatah, 2024. "Economic Dispatch Optimization Strategies and Problem Formulation: A Comprehensive Review," Energies, MDPI, vol. 17(3), pages 1-31, January.
    5. T. Bittencourt & O. P. Ferreira, 2017. "Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds," Journal of Global Optimization, Springer, vol. 68(2), pages 387-411, June.
    6. Stefania Bellavia & Valentina De Simone & Daniela di Serafino & Benedetta Morini, 2016. "On the update of constraint preconditioners for regularized KKT systems," Computational Optimization and Applications, Springer, vol. 65(2), pages 339-360, November.
    7. Yu, Jianxi & Liu, Pei & Li, Zheng, 2021. "Data reconciliation of the thermal system of a double reheat power plant for thermal calculation," Renewable and Sustainable Energy Reviews, Elsevier, vol. 148(C).
    8. Oliver de Groot & Falk Mazelis & Roberto Motto & Annukka Ristiniemi, "undated". "A Toolkit for Computing Constrained Optimal Policy Projections (COPPs)," Working Papers 202112, University of Liverpool, Department of Economics.
    9. Martijn H. H. Schoot Uiterkamp & Marco E. T. Gerards & Johann L. Hurink, 2022. "On a Reduction for a Class of Resource Allocation Problems," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1387-1402, May.
    10. Cecilia Orellana Castro & Manolo Rodriguez Heredia & Aurelio R. L. Oliveira, 2023. "Recycling basic columns of the splitting preconditioner in interior point methods," Computational Optimization and Applications, Springer, vol. 86(1), pages 49-78, September.
    11. Michaël Allouche & Emmanuel Gobet & Clara Lage & Edwin Mangin, 2023. "Structured Dictionary Learning of Rating Migration Matrices for Credit Risk Modeling," Working Papers hal-03715954, HAL.
    12. Enrico Bettiol & Lucas Létocart & Francesco Rinaldi & Emiliano Traversi, 2020. "A conjugate direction based simplicial decomposition framework for solving a specific class of dense convex quadratic programs," Computational Optimization and Applications, Springer, vol. 75(2), pages 321-360, March.
    13. Manolo Rodriguez Heredia & Aurelio Ribeiro Leite Oliveira, 2020. "A new proposal to improve the early iterations in the interior point method," Annals of Operations Research, Springer, vol. 287(1), pages 185-208, April.
    14. Dominik Garmatter & Margherita Porcelli & Francesco Rinaldi & Martin Stoll, 2023. "An improved penalty algorithm using model order reduction for MIPDECO problems with partial observations," Computational Optimization and Applications, Springer, vol. 84(1), pages 191-223, January.
    15. Fabio Vitor & Todd Easton, 2018. "The double pivot simplex method," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(1), pages 109-137, February.
    16. Gondzio, Jacek, 2016. "Crash start of interior point methods," European Journal of Operational Research, Elsevier, vol. 255(1), pages 308-314.
    17. Coralia Cartis & Yiming Yan, 2016. "Active-set prediction for interior point methods using controlled perturbations," Computational Optimization and Applications, Springer, vol. 63(3), pages 639-684, April.
    18. Belli, Edoardo, 2022. "Smoothly adaptively centered ridge estimator," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    19. Quoc Tran-Dinh & Anastasios Kyrillidis & Volkan Cevher, 2018. "A Single-Phase, Proximal Path-Following Framework," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1326-1347, November.
    20. Coralia Cartis & Yiming Yan, 2016. "Active-set prediction for interior point methods using controlled perturbations," Computational Optimization and Applications, Springer, vol. 63(3), pages 639-684, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:261:y:2015:i:c:p:28-38. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.