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Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood

Author

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  • Zsolt Darvay

    (Babeş-Bolyai University)

  • Petra Renáta Takács

    (Babeş-Bolyai University
    Budapest University of Technology and Economics)

Abstract

The interior-point algorithms can be classified in multiple ways. One of these takes into consideration the length of the step. In this way, we can speak about large-step and short-step methods, that work in different neighbourhoods of the central path. The large-step algorithms work in a wide neighbourhood, while the short-step ones determine the new iterates that are in a smaller neighbourhood. In spite of the fact that the large-step algorithms are more efficient in practice, the theoretical complexity of the short-step ones is generally better. Ai and Zhang introduced a large-step interior-point method for linear complementarity problems using a wide neighbourhood of the central path, which has the same complexity as the best short-step methods. We present a new wide neighbourhood of the central path. We prove that the obtained large-step primal–dual interior-point method for linear programming has the same complexity as the best short-step algorithms.

Suggested Citation

  • Zsolt Darvay & Petra Renáta Takács, 2018. "Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(3), pages 551-563, September.
    • Handle: RePEc:spr:cejnor:v:26:y:2018:i:3:d:10.1007_s10100-018-0524-0
    DOI: 10.1007/s10100-018-0524-0
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    References listed on IDEAS

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    1. Ximei Yang & Hongwei Liu & Yinkui Zhang, 2015. "A New Strategy in the Complexity Analysis of an Infeasible-Interior-Point Method for Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 572-587, August.
    2. Y. Q. Bai & G. Lesaja & C. Roos & G. Q. Wang & M. El Ghami, 2008. "A Class of Large-Update and Small-Update Primal-Dual Interior-Point Algorithms for Linear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 341-359, September.
    3. T. Illés & M. Nagy & T. Terlaky, 2009. "EP Theorem for Dual Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 233-238, February.
    4. Illes, Tibor & Nagy, Marianna, 2007. "A Mizuno-Todd-Ye type predictor-corrector algorithm for sufficient linear complementarity problems," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1097-1111, September.
    5. Péter Tar & Bálint Stágel & István Maros, 2017. "Parallel search paths for the simplex algorithm," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 25(4), pages 967-984, December.
    6. Hongwei Liu & Ximei Yang & Changhe Liu, 2013. "A New Wide Neighborhood Primal–Dual Infeasible-Interior-Point Method for Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 796-815, September.
    7. Potra, Florian A., 2002. "The Mizuno-Todd-Ye algorithm in a larger neighborhood of the central path," European Journal of Operational Research, Elsevier, vol. 143(2), pages 257-267, December.
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    Cited by:

    1. Botond Bertók & Tibor Csendes & Tibor Jordán, 2019. "Editorial," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 325-327, June.
    2. Marianna E.-Nagy & Anita Varga, 2023. "A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 31(3), pages 691-711, September.
    3. Marianna E.-Nagy & Anita Varga, 2023. "A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 31(3), pages 691-711, September.

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