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An Adaptive Infeasible-Interior-Point Method with the One-Norm Wide Neighborhood for Semi-definite Programming

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Abstract

In this paper, we present a Mehrotra-type predictor–corrector infeasible-interior-point method, based on the one-norm wide neighborhood, for semi-definite programming. The proposed algorithm uses Mehrotra’s adaptive updating scheme for the centering parameter, which incorporates a safeguard strategy that keeps the iterates in a prescribed neighborhood and allows to get a reasonably large step size. Moreover, by using an important inequality that is the relationship between the one-norm and the Frobenius-norm, the convergence is shown for a commutative class of search directions. In particular, the complexity bound is \(\mathcal {O}(n\log \varepsilon ^{-1})\) for Nesterov–Todd direction, and \(\mathcal {O}(n^{3/2}\log \varepsilon ^{-1})\) for Helmberg–Kojima–Monteiro directions, where \(\varepsilon \) is the required precision. The derived complexity bounds coincide with the currently best known theoretical complexity bounds obtained so far for the infeasible semi-definite programming. Some preliminary numerical results are provided as well.

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Acknowledgements

We would like to thank the support of National Natural Science Foundation of China (NNSFC) under Grant Nos. 11501180, 11601134 and 11671122, Chinese Postdoctoral Science Foundation No. 2016M590346, 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

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Ximei Yang & Yanqin Bai

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  1. Ximei Yang
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  2. Yanqin Bai
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Correspondence to Ximei Yang.

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Yang, X., Bai, Y. An Adaptive Infeasible-Interior-Point Method with the One-Norm Wide Neighborhood for Semi-definite Programming. J Sci Comput 78, 1790–1810 (2019). https://doi.org/10.1007/s10915-018-0827-2

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  • DOI: https://doi.org/10.1007/s10915-018-0827-2

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