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Polynomial convergence of two higher order interior-point methods for \(P_*(\kappa )\)-LCP in a wide neighborhood of the central path

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Abstract

In this paper, we propose two interior-point methods for solving \(P_*(\kappa )\)-linear complementarity problems (\(P_*(\kappa )\)-LCPs): a high order large update path following method and a high order corrector–predictor method. Both algorithms generate sequences of iterates in the wide neighborhood \((\mathcal {N}_{2,\tau }^-(\alpha ))\) of the central path introduced by Ai and Zhang. The methods do not depend on the handicap \(\kappa \) of the problem so that they work for any \(P_*(\kappa )\)-LCP . They have \(O((1 +\kappa )\sqrt{n}L)\) iteration complexity, the best-known iteration complexity obtained so far by any interior-point method for solving \(P_*(\kappa )\)-LCP. The high order corrector–predictor algorithm is superlinearly convergent with Q-order \((m_p+1)\) for problems that admit a strict complementarity solution and \((m_p+1)/2\) for general problems, where \(m_p\) is the order of the predictor step.

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Acknowledgements

The authors would like to thank the Editors and the anonymous referees for their useful comments and suggestions, which helped to improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

    Behrouz Kheirfam & Maryam Chitsaz

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  1. Behrouz Kheirfam
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  2. Maryam Chitsaz
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Correspondence to Behrouz Kheirfam.

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Kheirfam, B., Chitsaz, M. Polynomial convergence of two higher order interior-point methods for \(P_*(\kappa )\)-LCP in a wide neighborhood of the central path. Period Math Hung 76, 243–264 (2018). https://doi.org/10.1007/s10998-017-0231-y

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