IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v58y2014i1p225-247.html
   My bibliography  Save this article

Local convergence of quasi-Newton methods under metric regularity

Author

Listed:
  • F. Aragón Artacho
  • A. Belyakov
  • A. Dontchev
  • M. López

Abstract

We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • F. Aragón Artacho & A. Belyakov & A. Dontchev & M. López, 2014. "Local convergence of quasi-Newton methods under metric regularity," Computational Optimization and Applications, Springer, vol. 58(1), pages 225-247, May.
    • Handle: RePEc:spr:coopap:v:58:y:2014:i:1:p:225-247
    DOI: 10.1007/s10589-013-9615-y
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-013-9615-y
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10589-013-9615-y?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. Boubakeur Benahmed & Hocine Mokhtar-Kharroubi & Bruno Malafosse & Adnan Yassine, 2011. "Quasi-Newton methods in infinite-dimensional spaces and application to matrix equations," Journal of Global Optimization, Springer, vol. 49(3), pages 365-379, March.
    2. Wenyu Sun & Ya-Xiang Yuan, 2006. "Optimization Theory and Methods," Springer Optimization and Its Applications, Springer, number 978-0-387-24976-6, June.
    3. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    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. Wei Ouyang & Kui Mei, 2023. "A General Iterative Procedure for Solving Nonsmooth Constrained Generalized Equations," Mathematics, MDPI, vol. 11(22), pages 1-17, November.
    2. Michaël Gaydu & Gilson N. Silva, 2020. "A General Iterative Procedure to Solve Generalized Equations with Differentiable Multifunction," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 207-222, April.
    3. Fabiana R. Oliveira & Orizon P. Ferreira & Gilson N. Silva, 2019. "Newton’s method with feasible inexact projections for solving constrained generalized equations," Computational Optimization and Applications, Springer, vol. 72(1), pages 159-177, January.
    4. Jiaxi Wang & Wei Ouyang, 2022. "Newton’s Method for Solving Generalized Equations Without Lipschitz Condition," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 510-532, February.

    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. M. Durea & R. Strugariu, 2011. "On parametric vector optimization via metric regularity of constraint systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(3), pages 409-425, December.
    2. Yasushi Narushima & Shummin Nakayama & Masashi Takemura & Hiroshi Yabe, 2023. "Memoryless Quasi-Newton Methods Based on the Spectral-Scaling Broyden Family for Riemannian Optimization," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 639-664, May.
    3. Liang Chen & Anping Liao, 2020. "On the Convergence Properties of a Second-Order Augmented Lagrangian Method for Nonlinear Programming Problems with Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 248-265, October.
    4. Saha, Tanay & Rakshit, Suman & Khare, Swanand R., 2023. "Linearly structured quadratic model updating using partial incomplete eigendata," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    5. Guang Li & Paat Rusmevichientong & Huseyin Topaloglu, 2015. "The d -Level Nested Logit Model: Assortment and Price Optimization Problems," Operations Research, INFORMS, vol. 63(2), pages 325-342, April.
    6. Zheng, Sanpeng & Feng, Renzhong, 2023. "A variable projection method for the general radial basis function neural network," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    7. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.
    8. Francisco Aragón Artacho & Boris Mordukhovich, 2011. "Enhanced metric regularity and Lipschitzian properties of variational systems," Journal of Global Optimization, Springer, vol. 50(1), pages 145-167, May.
    9. Jörg Fliege & Andrey Tin & Alain Zemkoho, 2021. "Gauss–Newton-type methods for bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(3), pages 793-824, April.
    10. Fabiana R. Oliveira & Orizon P. Ferreira & Gilson N. Silva, 2019. "Newton’s method with feasible inexact projections for solving constrained generalized equations," Computational Optimization and Applications, Springer, vol. 72(1), pages 159-177, January.
    11. Hai-Jun Wang & Qin Ni, 2010. "A Convex Approximation Method For Large Scale Linear Inequality Constrained Minimization," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 27(01), pages 85-101.
    12. Guo, Qiangqiang & Ban, Xuegang (Jeff), 2023. "A multi-scale control framework for urban traffic control with connected and automated vehicles," Transportation Research Part B: Methodological, Elsevier, vol. 175(C).
    13. Chen, Liang, 2016. "A high-order modified Levenberg–Marquardt method for systems of nonlinear equations with fourth-order convergence," Applied Mathematics and Computation, Elsevier, vol. 285(C), pages 79-93.
    14. Ji, Li-Qun, 2015. "An assessment of agricultural residue resources for liquid biofuel production in China," Renewable and Sustainable Energy Reviews, Elsevier, vol. 44(C), pages 561-575.
    15. Babaie-Kafaki, Saman & Ghanbari, Reza, 2014. "The Dai–Liao nonlinear conjugate gradient method with optimal parameter choices," European Journal of Operational Research, Elsevier, vol. 234(3), pages 625-630.
    16. J. Han & D. Sun, 1997. "Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 659-676, September.
    17. Nguyen Qui, 2014. "Stability for trust-region methods via generalized differentiation," Journal of Global Optimization, Springer, vol. 59(1), pages 139-164, May.
    18. Michael Patriksson & R. Tyrrell Rockafellar, 2003. "Sensitivity Analysis of Aggregated Variational Inequality Problems, with Application to Traffic Equilibria," Transportation Science, INFORMS, vol. 37(1), pages 56-68, February.
    19. U. Felgenhauer, 1999. "Regularity Properties of Optimal Controls with Application to Discrete Approximation," Journal of Optimization Theory and Applications, Springer, vol. 102(1), pages 97-110, July.
    20. Ilker Birbil, S. & Gürkan, G. & Listes, O.L., 2004. "Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints : Sample-Path Analysis," Discussion Paper 2004-25, Tilburg University, Center for Economic Research.

    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:spr:coopap:v:58:y:2014:i:1:p:225-247. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.