IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v67y2017i1d10.1007_s10589-016-9887-0.html
   My bibliography  Save this article

A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods

Author

Listed:
  • Porfirio Suñagua

    (San Andres University of La Paz)

  • Aurelio R. L. Oliveira

    (University of Campinas)

Abstract

The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra’s predictor-corrector method for large scale linear programming problems needs to find a basis through a sophisticated process based on the application of a rectangular LU factorization. This class of splitting preconditioners works better near a solution of the linear programming problem when the matrices are highly ill-conditioned. In this study, we develop and implement a new approach to find a basis for the splitting preconditioner, based on standard rectangular LU factorization with partial permutation of the scaled transpose linear programming constraint matrix. In most cases, this basis is better conditioned than the existing one. In addition, we include a penalty parameter in Mehrotra’s predictor-corrector method in order to reduce ill-conditioning of the normal equations matrix. Computational experiments show a reduction in the average number of iterations of the preconditioned conjugate gradient method. Also, the increased efficiency and robustness of the new approach become evident by the performance profile.

Suggested Citation

  • Porfirio Suñagua & Aurelio R. L. Oliveira, 2017. "A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods," Computational Optimization and Applications, Springer, vol. 67(1), pages 111-127, May.
    • Handle: RePEc:spr:coopap:v:67:y:2017:i:1:d:10.1007_s10589-016-9887-0
    DOI: 10.1007/s10589-016-9887-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-016-9887-0
    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/s10589-016-9887-0?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. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    2. Harry M. Markowitz, 1957. "The Elimination form of the Inverse and its Application to Linear Programming," Management Science, INFORMS, vol. 3(3), pages 255-269, April.
    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. 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.
    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.

    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. 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.
    2. 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.
    3. Harry M. Markowitz, 2004. "Trains of Thought," Chapters, in: Michael Szenberg & Lall Ramrattan (ed.), Reflections of Eminent Economists, chapter 17, Edward Elgar Publishing.
    4. 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.
    5. de Groot, Oliver & Mazelis, Falk & Motto, Roberto & Ristiniemi, Annukka, 2021. "A toolkit for computing Constrained Optimal Policy Projections (COPPs)," Working Paper Series 2555, European Central Bank.
    6. Defeng Liu & Andrea Lodi & Mathieu Tanneau, 2021. "Learning chordal extensions," Journal of Global Optimization, Springer, vol. 81(1), pages 3-22, September.
    7. 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.
    8. Gass, Saul I., 1997. "The Washington operations research connection: the rest of the story," Socio-Economic Planning Sciences, Elsevier, vol. 31(4), pages 245-255, December.
    9. 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.
    10. Keeping, B.R. & Pantelides, C.C., 1998. "A distributed memory parallel algorithm for the efficient computation of sensitivities of differential-algebraic systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 44(6), pages 545-558.
    11. 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.
    12. 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.
    13. 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.
    14. Gondzio, Jacek, 2016. "Crash start of interior point methods," European Journal of Operational Research, Elsevier, vol. 255(1), pages 308-314.
    15. Belli, Edoardo, 2022. "Smoothly adaptively centered ridge estimator," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    16. 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.
    17. Kirschner, Felix, 2023. "Conic optimization with applications in finance and approximation theory," Other publications TiSEM e9bef4a5-ee46-45be-90d7-9, Tilburg University, School of Economics and Management.
    18. Alberto Marchi, 2022. "On a primal-dual Newton proximal method for convex quadratic programs," Computational Optimization and Applications, Springer, vol. 81(2), pages 369-395, March.
    19. Pedro Munari & Alfredo Moreno & Jonathan De La Vega & Douglas Alem & Jacek Gondzio & Reinaldo Morabito, 2019. "The Robust Vehicle Routing Problem with Time Windows: Compact Formulation and Branch-Price-and-Cut Method," Transportation Science, INFORMS, vol. 53(4), pages 1043-1066, July.
    20. Milan Dražić & Rade Lazović & Vera Kovačević-Vujčić, 2015. "Sparsity preserving preconditioners for linear systems in interior-point methods," Computational Optimization and Applications, Springer, vol. 61(3), pages 557-570, July.

    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:67:y:2017:i:1:d:10.1007_s10589-016-9887-0. 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.