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Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization

Author

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  • G. Al-Jeiroudi

    (University of Edinburgh)

  • J. Gondzio

    (University of Edinburgh)

Abstract

We present the convergence analysis of the inexact infeasible path-following (IIPF) interior-point algorithm. In this algorithm, the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of the interior-point method for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm, prove the global convergence of this method and provide complexity analysis.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:141:y:2009:i:2:d:10.1007_s10957-008-9500-5
    DOI: 10.1007/s10957-008-9500-5
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    References listed on IDEAS

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    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. 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.
    3. S. Bellavia, 1998. "Inexact Interior-Point Method," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 109-121, January.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    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. 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.
    7. Mohammadhossein Mohammadisiahroudi & Ramin Fakhimi & Tamás Terlaky, 2024. "Efficient Use of Quantum Linear System Algorithms in Inexact Infeasible IPMs for Linear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 146-183, July.
    8. Jacek Gondzio, 2012. "Matrix-free interior point method," Computational Optimization and Applications, Springer, vol. 51(2), pages 457-480, March.

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