IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v255y2016i1p308-314.html
   My bibliography  Save this article

Crash start of interior point methods

Author

Listed:
  • Gondzio, Jacek

Abstract

The starting point used by an interior point algorithm for linear and convex quadratic programming may significantly influence the behaviour of the method. A widely used heuristic to construct such a point consists of dropping variable nonnegativity constraints and computing a solution which minimizes the Euclidean norm of the primal (or dual) point while satisfying the appropriate primal (or dual) equality constraints, followed by shifting the variables so that all their components are positive and bounded away from zero. In this Short Communication a new approach for finding a starting point is proposed. It relies on a few inexact Newton steps performed at the start of the solution process. A formal justification of the new heuristic is given and computational results are presented to demonstrate its advantages in practice. Computational experience with a large collection of small- and medium-size test problems reveals that the new starting point is superior to the old one and saves 20–40% of iterations needed by the primal-dual method. For larger and more difficult problems this translates into remarkable savings in the solution time.

Suggested Citation

  • Gondzio, Jacek, 2016. "Crash start of interior point methods," European Journal of Operational Research, Elsevier, vol. 255(1), pages 308-314.
    • Handle: RePEc:eee:ejores:v:255:y:2016:i:1:p:308-314
    DOI: 10.1016/j.ejor.2016.05.030
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221716303502
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2016.05.030?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. J. Hall, 2010. "Towards a practical parallelisation of the simplex method," Computational Management Science, Springer, vol. 7(2), pages 139-170, April.
    2. Robert E. Bixby, 1992. "Implementing the Simplex Method: The Initial Basis," INFORMS Journal on Computing, INFORMS, vol. 4(3), pages 267-284, August.
    3. S. Bellavia, 1998. "Inexact Interior-Point Method," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 109-121, January.
    4. Jacek Gondzio, 2012. "Matrix-free interior point method," Computational Optimization and Applications, Springer, vol. 51(2), pages 457-480, March.
    5. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    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. 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.
    2. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    3. 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.
    4. 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.
    5. Ploskas, Nikolaos & Samaras, Nikolaos, 2015. "Efficient GPU-based implementations of simplex type algorithms," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 552-570.
    6. 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.
    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. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Oliver de Groot & Falk Mazelis & Roberto Motto & Annukka Ristiniemi, "undated". "A Toolkit for Computing Constrained Optimal Policy Projections (COPPs)," Working Papers 202112, University of Liverpool, Department of Economics.
    13. A. Swietanowski, 1995. "A Modular Presolve Procedure for Large Scale Linear Programming," Working Papers wp95113, International Institute for Applied Systems Analysis.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. Sebastiaan Breedveld & Bas Berg & Ben Heijmen, 2017. "An interior-point implementation developed and tuned for radiation therapy treatment planning," Computational Optimization and Applications, Springer, vol. 68(2), pages 209-242, November.
    19. 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.
    20. 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.

    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:ejores:v:255:y:2016:i:1:p:308-314. 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: http://www.elsevier.com/locate/eor .

    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.