IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v74y2019i1d10.1007_s10589-019-00103-y.html
   My bibliography  Save this article

Implementation of interior-point methods for LP based on Krylov subspace iterative solvers with inner-iteration preconditioning

Author

Listed:
  • Yiran Cui

    (University College London)

  • Keiichi Morikuni

    (University of Tsukuba)

  • Takashi Tsuchiya

    (National Graduate Institute for Policy Studies)

  • Ken Hayami

    (National Institute of Informatics and SOKENDAI (The Graduate University for Advanced Studies))

Abstract

We apply novel inner-iteration preconditioned Krylov subspace methods to the interior-point algorithm for linear programming (LP). Inner-iteration preconditioners recently proposed by Morikuni and Hayami enable us to overcome the severe ill-conditioning of linear equations solved in the final phase of interior-point iterations. The Krylov subspace methods do not suffer from rank-deficiency and therefore no preprocessing is necessary even if rows of the constraint matrix are not linearly independent. By means of these methods, a new interior-point recurrence is proposed in order to omit one matrix-vector product at each step. Extensive numerical experiments are conducted over diverse instances of 140 LP problems including the Netlib, QAPLIB, Mittelmann and Atomizer Basis Pursuit collections. The largest problem has 434,580 unknowns. It turns out that our implementation is more robust than the standard public domain solvers SeDuMi (Self-Dual Minimization), SDPT3 (Semidefinite Programming Toh-Todd-Tütüncü) and the LSMR iterative solver in PDCO (Primal-Dual Barrier Method for Convex Objectives) without increasing CPU time. The proposed interior-point method based on iterative solvers succeeds in solving a fairly large number of LP instances from benchmark libraries under the standard stopping criteria. The work also presents a fairly extensive benchmark test for several renowned solvers including direct and iterative solvers.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:74:y:2019:i:1:d:10.1007_s10589-019-00103-y
    DOI: 10.1007/s10589-019-00103-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-019-00103-y
    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/s10589-019-00103-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. Roland W. Freund & Florian Jarre & Shinji Mizuno, 1999. "Convergence of a Class of Inexact Interior-Point Algorithms for Linear Programs," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 50-71, February.
    2. Irvin J. Lustig & Roy E. Marsten & David F. Shanno, 1994. "Feature Article—Interior Point Methods for Linear Programming: Computational State of the Art," INFORMS Journal on Computing, INFORMS, vol. 6(1), pages 1-14, February.
    3. Sanjay Mehrotra, 1992. "Implementations of Affine Scaling Methods: Approximate Solutions of Systems of Linear Equations Using Preconditioned Conjugate Gradient Methods," INFORMS Journal on Computing, INFORMS, vol. 4(2), pages 103-118, May.
    4. Irvin J. Lustig & Roy E. Marsten & David F. Shanno, 1994. "Rejoinder—The Last Word on Interior Point Methods for Linear Programming—For Now," INFORMS Journal on Computing, INFORMS, vol. 6(1), pages 35-36, February.
    5. Gondzio, Jacek, 1995. "HOPDM (version 2.12) -- A fast LP solver based on a primal-dual interior point method," European Journal of Operational Research, Elsevier, vol. 85(1), pages 221-225, August.
    6. Luca Bergamaschi & Jacek Gondzio & Manolo Venturin & Giovanni Zilli, 2007. "Inexact constraint preconditioners for linear systems arising in interior point methods," Computational Optimization and Applications, Springer, vol. 36(2), pages 137-147, April.
    7. Jacek Gondzio, 2012. "Matrix-free interior point method," Computational Optimization and Applications, Springer, vol. 51(2), pages 457-480, March.
    8. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    9. Jacek Gondzio, 1997. "Presolve Analysis of Linear Programs Prior to Applying an Interior Point Method," INFORMS Journal on Computing, INFORMS, vol. 9(1), pages 73-91, February.
    10. 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)

    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. Jeffery L. Kennington & Karen R. Lewis, 2004. "Generalized Networks: The Theory of Preprocessing and an Empirical Analysis," INFORMS Journal on Computing, INFORMS, vol. 16(2), pages 162-173, May.
    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. Alexandra M. Newman & Martin Weiss, 2013. "A Survey of Linear and Mixed-Integer Optimization Tutorials," INFORMS Transactions on Education, INFORMS, vol. 14(1), pages 26-38, September.
    4. Erling D. Andersen, 1999. "On Exploiting Problem Structure in a Basis Identification Procedure for Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 11(1), pages 95-103, February.
    5. 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.
    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. 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.
    8. 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.
    9. Liu, Yanwu & Tu, Yan & Zhang, Zhongzhen, 2021. "The row pivoting method for linear programming," Omega, Elsevier, vol. 100(C).
    10. 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.
    11. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    12. Silvia Araújo dos Reis & José Eugenio Leal & Antônio Márcio Tavares Thomé, 2023. "A Two-Stage Stochastic Linear Programming Model for Tactical Planning in the Soybean Supply Chain," Logistics, MDPI, vol. 7(3), pages 1-26, August.
    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. M. Xiong & J. Wang & P. Wang, 2002. "Differential-Algebraic Approach to Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 443-470, August.
    15. Gondzio, Jacek, 2016. "Crash start of interior point methods," European Journal of Operational Research, Elsevier, vol. 255(1), pages 308-314.
    16. C. Bruni & R. Bruni & A. De Santis & D. Iacoviello & G. Koch, 2002. "Global Optimal Image Reconstruction from Blurred Noisy Data by a Bayesian Approach," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 67-96, October.
    17. Sturm, J.F., 2002. "Implementation of Interior Point Methods for Mixed Semidefinite and Second Order Cone Optimization Problems," Other publications TiSEM b25faf5d-0142-4e14-b598-a, Tilburg University, School of Economics and Management.
    18. Gondzio, Jacek & González-Brevis, Pablo & Munari, Pedro, 2013. "New developments in the primal–dual column generation technique," European Journal of Operational Research, Elsevier, vol. 224(1), pages 41-51.
    19. 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.
    20. 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.

    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:74:y:2019:i:1:d:10.1007_s10589-019-00103-y. 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.