IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v175y2017i2d10.1007_s10957-017-1170-8.html
   My bibliography  Save this article

Stability and Accuracy of Inexact Interior Point Methods for Convex Quadratic Programming

Author

Listed:
  • Benedetta Morini

    (Università degli Studi di Firenze)

  • Valeria Simoncini

    (Università di Bologna
    IMATI-CNR)

Abstract

We consider primal–dual interior point methods where the linear system arising at each iteration is formulated in the reduced (augmented) form and solved approximately. Focusing on the iterates close to a solution, we analyze the accuracy of the so-called inexact step, i.e., the step that solves the unreduced system, when combining the effects of both different levels of accuracy in the inexact computation and different processes for retrieving the step after block elimination. Our analysis is general and includes as special cases sources of inexactness due either to roundoff and computational errors or to the iterative solution of the augmented system using typical procedures. In the roundoff case, we recover and extend some known results.

Suggested Citation

  • Benedetta Morini & Valeria Simoncini, 2017. "Stability and Accuracy of Inexact Interior Point Methods for Convex Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 450-477, November.
  • Handle: RePEc:spr:joptap:v:175:y:2017:i:2:d:10.1007_s10957-017-1170-8
    DOI: 10.1007/s10957-017-1170-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-017-1170-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10957-017-1170-8?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. Paul Armand & Joël Benoist & Jean-Pierre Dussault, 2012. "Local path-following property of inexact interior methods in nonlinear programming," Computational Optimization and Applications, Springer, vol. 52(1), pages 209-238, May.
    2. Benedetta Morini & Valeria Simoncini & Mattia Tani, 2017. "A comparison of reduced and unreduced KKT systems arising from interior point methods," Computational Optimization and Applications, Springer, vol. 68(1), pages 1-27, September.
    3. C. Durazzi & V. Ruggiero, 2004. "Global Convergence of the Newton Interior-Point Method for Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 199-208, January.
    4. S. Bellavia, 1998. "Inexact Interior-Point Method," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 109-121, January.
    5. G. Al-Jeiroudi & J. Gondzio, 2009. "Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 231-247, May.
    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. David Ek & Anders Forsgren, 2023. "A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming," Computational Optimization and Applications, Springer, vol. 86(1), pages 1-48, September.
    2. 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.

    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. David Ek & Anders Forsgren, 2023. "A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming," Computational Optimization and Applications, Springer, vol. 86(1), pages 1-48, September.
    2. 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.
    3. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    4. Paul Armand & Joël Benoist & Jean-Pierre Dussault, 2012. "Local path-following property of inexact interior methods in nonlinear programming," Computational Optimization and Applications, Springer, vol. 52(1), pages 209-238, May.
    5. Jacek Gondzio, 2012. "Matrix-free interior point method," Computational Optimization and Applications, Springer, vol. 51(2), pages 457-480, March.
    6. Benedetta Morini & Valeria Simoncini & Mattia Tani, 2017. "A comparison of reduced and unreduced KKT systems arising from interior point methods," Computational Optimization and Applications, Springer, vol. 68(1), pages 1-27, September.
    7. 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.
    8. C. Durazzi, 2000. "On the Newton Interior-Point Method for Nonlinear Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 104(1), pages 73-90, January.
    9. Zhenqiu Liu & Dechang Chen & Li Sheng & Amy Y Liu, 2013. "Class Prediction and Feature Selection with Linear Optimization for Metagenomic Count Data," PLOS ONE, Public Library of Science, vol. 8(3), pages 1-7, March.
    10. Paul Armand & Ngoc Nguyen Tran, 2021. "Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 96-116, April.
    11. 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.
    12. Gondzio, Jacek, 2016. "Crash start of interior point methods," European Journal of Operational Research, Elsevier, vol. 255(1), pages 308-314.
    13. G. Al-Jeiroudi & J. Gondzio, 2009. "Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 231-247, May.
    14. C. Durazzi & V. Ruggiero & G. Zanghirati, 2001. "Parallel Interior-Point Method for Linear and Quadratic Programs with Special Structure," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 289-313, August.
    15. J. Gondzio & F. N. C. Sobral, 2019. "Quasi-Newton approaches to interior point methods for quadratic problems," Computational Optimization and Applications, Springer, vol. 74(1), pages 93-120, September.
    16. Yiran Cui & Keiichi Morikuni & Takashi Tsuchiya & Ken Hayami, 2019. "Implementation of interior-point methods for LP based on Krylov subspace iterative solvers with inner-iteration preconditioning," Computational Optimization and Applications, Springer, vol. 74(1), pages 143-176, September.
    17. Md Sarowar Morshed & Md Saiful Islam & Md. Noor-E-Alam, 2020. "Accelerated sampling Kaczmarz Motzkin algorithm for the linear feasibility problem," Journal of Global Optimization, Springer, vol. 77(2), pages 361-382, June.

    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:joptap:v:175:y:2017:i:2:d:10.1007_s10957-017-1170-8. 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.