IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v318y2018icp196-214.html
   My bibliography  Save this article

Preconditioned Nonlinear Conjugate Gradient methods based on a modified secant equation

Author

Listed:
  • Caliciotti, Andrea
  • Fasano, Giovanni
  • Roma, Massimo

Abstract

This paper includes a twofold result for the Nonlinear Conjugate Gradient (NCG) method, in large scale unconstrained optimization. First we consider a theoretical analysis, where preconditioning is embedded in a strong convergence framework of an NCG method from the literature. Mild conditions to be satisfied by the preconditioners are defined, in order to preserve NCG convergence. As a second task, we also detail the use of novel matrix–free preconditioners for NCG. Our proposals are based on quasi–Newton updates, and either satisfy the secant equation or a secant–like condition at some of the previous iterates. We show that, in some sense, the preconditioners we propose also approximate the inverse of the Hessian matrix. In particular, the structures of our preconditioners depend on low–rank updates used, along with different choices of specific parameters. The low–rank updates are obtained as by–product of NCG iterations. The results of an extended numerical experience using large scale CUTEst problems is reported, showing that our preconditioners can considerably improve the performance of NCG methods.

Suggested Citation

  • Caliciotti, Andrea & Fasano, Giovanni & Roma, Massimo, 2018. "Preconditioned Nonlinear Conjugate Gradient methods based on a modified secant equation," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 196-214.
    • Handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:196-214
    DOI: 10.1016/j.amc.2017.08.029
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300317305805
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2017.08.029?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. Giovanni Fasano & Massimo Roma, 2016. "A novel class of approximate inverse preconditioners for large positive definite linear systems in optimization," Computational Optimization and Applications, Springer, vol. 65(2), pages 399-429, November.
    2. Nicholas Gould & Dominique Orban & Philippe Toint, 2015. "CUTEst: a Constrained and Unconstrained Testing Environment with safe threads for mathematical optimization," Computational Optimization and Applications, Springer, vol. 60(3), pages 545-557, April.
    3. Giovanni Fasano & Massimo Roma, 2013. "Preconditioning Newton–Krylov methods in nonconvex large scale optimization," Computational Optimization and Applications, Springer, vol. 56(2), pages 253-290, October.
    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. Mehiddin Al-Baali & Andrea Caliciotti & Giovanni Fasano & Massimo Roma, 2017. "Exploiting damped techniques for nonlinear conjugate gradient methods," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(3), pages 501-522, December.
    2. Abdul Wahid & Javed Iqbal & Affaq Qamar & Salman Ahmed & Abdul Basit & Haider Ali & Omar M. Aldossary, 2020. "A Novel Power Scheduling Mechanism for Islanded DC Microgrid Cluster," Sustainability, MDPI, vol. 12(17), pages 1-14, August.

    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. Mehiddin Al-Baali & Andrea Caliciotti & Giovanni Fasano & Massimo Roma, 2017. "Exploiting damped techniques for nonlinear conjugate gradient methods," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(3), pages 501-522, December.
    2. Giovanni Fasano & Massimo Roma, 2016. "A novel class of approximate inverse preconditioners for large positive definite linear systems in optimization," Computational Optimization and Applications, Springer, vol. 65(2), pages 399-429, November.
    3. Andrea Caliciotti & Giovanni Fasano & Florian Potra & Massimo Roma, 2020. "Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation," Computational Optimization and Applications, Springer, vol. 77(3), pages 627-651, December.
    4. Matteo Lapucci & Alessio Sortino, 2024. "On the Convergence of Inexact Alternate Minimization in Problems with $$\ell _0$$ ℓ 0 Penalties," SN Operations Research Forum, Springer, vol. 5(2), pages 1-11, June.
    5. Giovanni Fasano, 2015. "A Framework of Conjugate Direction Methods for Symmetric Linear Systems in Optimization," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 883-914, March.
    6. Yutao Zheng & Bing Zheng, 2017. "Two New Dai–Liao-Type Conjugate Gradient Methods for Unconstrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 502-509, November.
    7. Yonggang Pei & Shaofang Song & Detong Zhu, 2023. "A sequential adaptive regularisation using cubics algorithm for solving nonlinear equality constrained optimization," Computational Optimization and Applications, Springer, vol. 84(3), pages 1005-1033, April.
    8. Nicholas I. M. Gould & Daniel P. Robinson, 2017. "A dual gradient-projection method for large-scale strictly convex quadratic problems," Computational Optimization and Applications, Springer, vol. 67(1), pages 1-38, May.
    9. Charles Audet & Kwassi Joseph Dzahini & Michael Kokkolaras & Sébastien Le Digabel, 2021. "Stochastic mesh adaptive direct search for blackbox optimization using probabilistic estimates," Computational Optimization and Applications, Springer, vol. 79(1), pages 1-34, May.
    10. Jianjun Liu & Xiangmin Xu & Xuehui Cui, 2018. "An accelerated nonmonotone trust region method with adaptive trust region for unconstrained optimization," Computational Optimization and Applications, Springer, vol. 69(1), pages 77-97, January.
    11. Rujun Jiang & Man-Chung Yue & Zhishuo Zhou, 2021. "An accelerated first-order method with complexity analysis for solving cubic regularization subproblems," Computational Optimization and Applications, Springer, vol. 79(2), pages 471-506, June.
    12. Yiwen Chen & Warren Hare & Amy Wiebe, 2024. "Q-fully quadratic modeling and its application in a random subspace derivative-free method," Computational Optimization and Applications, Springer, vol. 89(2), pages 317-360, November.
    13. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2017. "A Two-Stage Active-Set Algorithm for Bound-Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 369-401, February.
    14. S. Gratton & C. W. Royer & L. N. Vicente & Z. Zhang, 2019. "Direct search based on probabilistic feasible descent for bound and linearly constrained problems," Computational Optimization and Applications, Springer, vol. 72(3), pages 525-559, April.
    15. Sven Leyffer & Charlie Vanaret, 2020. "An augmented Lagrangian filter method," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(2), pages 343-376, October.
    16. Giovanni Fasano & Raffaele Pesenti, 2017. "Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 764-794, December.
    17. David Ek & Anders Forsgren, 2021. "Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization," Computational Optimization and Applications, Springer, vol. 79(1), pages 155-191, May.
    18. Andrzej Stachurski, 2017. "On a conjugate directions method for solving strictly convex QP problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(3), pages 523-548, December.
    19. David J. Eckman & Shane G. Henderson & Sara Shashaani, 2023. "Diagnostic Tools for Evaluating and Comparing Simulation-Optimization Algorithms," INFORMS Journal on Computing, INFORMS, vol. 35(2), pages 350-367, March.
    20. Brian Irwin & Eldad Haber, 2023. "Secant penalized BFGS: a noise robust quasi-Newton method via penalizing the secant condition," Computational Optimization and Applications, Springer, vol. 84(3), pages 651-702, 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:318:y:2018:i:c:p:196-214. 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.