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An arc-search infeasible-interior-point method for symmetric optimization in a wide neighborhood of the central path

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Abstract

In this paper, we propose a new arc-search infeasible-interior-point method for symmetric optimization using a wide neighborhood of the central path. The proposed algorithm searches for optimizers along the ellipses that approximate the central path. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. The complexity bound is \(\mathcal {O}(r^{5/4}\log \varepsilon ^{-1})\) for the Nesterov–Todd direction, and \(\mathcal {O}(r^{7/4}\log \varepsilon ^{-1})\) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and \(\varepsilon \) is the required precision. The obtained complexity bounds coincide with the currently best known theoretical complexity bounds for the short step path-following algorithm. Some limited encouraging computational results are reported.

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Acknowledgments

The authors are very grateful to the editor and the anonymous referees for their valuable suggestions which helped to improve the paper. We would also like to thank the support of National Natural Science Foundation of China (NNSFC) under Grant No.11501180, 61179040 and U1404105, Henan Normal University Doctoral Startup Issues No. qd14150 and Young Scientists Foundation No.2014QK03, and Innovative Research Team (in Science and Technology) in University of Henan Province No. 14IRTSTHN023.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People’s Republic of China

    Ximei Yang & Yinkui Zhang

  2. Department of Mathematics, Xidian University, Xi’an, 710071, People’s Republic of China

    Ximei Yang & Hongwei Liu

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  1. Ximei Yang
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  2. Hongwei Liu
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  3. Yinkui Zhang
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Correspondence to Hongwei Liu.

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Yang, X., Liu, H. & Zhang, Y. An arc-search infeasible-interior-point method for symmetric optimization in a wide neighborhood of the central path. Optim Lett 11, 135–152 (2017). https://doi.org/10.1007/s11590-016-0997-5

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  • DOI: https://doi.org/10.1007/s11590-016-0997-5

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