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.
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Acknowledgements
We thank the editor and the anonymous reviewers for the valuable suggestions that improved the presentation of the paper. The authors are also thankful for the support of the Babeş-Bolyai University and the Budapest University of Technology and Economics. Moreover, this work was supported by a grant of Ministry of Research and Innovation, CNCS—UEFISCDI, Project Number PN-III-P4-ID-PCE-2016-0190, within PNCDI III.
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The paper was presented at the Special Section on IPM and Related Topics in Honor of Goran Lesaja at the 16th International Conference on Operational Research KOI 2016, Osijek, Croatia.
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Darvay, Z., Takács, P.R. Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood. Cent Eur J Oper Res 26, 551–563 (2018). https://doi.org/10.1007/s10100-018-0524-0
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DOI: https://doi.org/10.1007/s10100-018-0524-0