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A sensitive-eigenvector based global algorithm for quadratically constrained quadratic programming

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Abstract

In this paper, we design an eigenvalue decomposition based branch-and-bound algorithm for finding global solutions of quadratically constrained quadratic programming (QCQP) problems. The hardness of nonconvex QCQP problems roots in the nonconvex components of quadratic terms, which are represented by the negative eigenvalues and the corresponding eigenvectors in the eigenvalue decomposition. For certain types of QCQP problems, only very few eigenvectors, defined as sensitive-eigenvectors, determine the relaxation gaps. We propose a semidefinite relaxation based branch-and-bound algorithm to solve QCQP. The proposed algorithm, which branches on the directions of the sensitive-eigenvectors, is very efficient for solving certain types of QCQP problems.

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Notes

  1. The terms \(s_i\) and \(t_i\) are computed for all \(i\in \{1,\ldots ,r\}\), rather than only for \(i\in \mathcal {A}_{a}\). The computational cost is very low, even when r is large.

  2. The solvers BARON, COUENNE and SCIP have provided interfaces to control their error tolerances.

  3. As discussed in [31], the complex Gaussian distribution is a reasonable assumption.

  4. The problem size \((n,m)=(4,8)\) is a typical setting, as discussed in [31].

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Acknowledgements

The authors would like to thank the two anonymous reviewers, whose invaluable comments have significantly improved the quality of this paper.

Author information

Authors and Affiliations

  1. School of Economics and Management, North China Electric Power University, Beijing, 102206, China

    Cheng Lu

  2. School of Economics and Management, University of Chinese Academy of Sciences, Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing, 100190, China

    Zhibin Deng

  3. Department of Applied Mathematics, College of Science, Zhejiang University of Technology, Hangzhou, 310023, China

    Jing Zhou

  4. Departmemt of Mathematics, China University of Mining and Technology, Beijing, 100083, China

    Xiaoling Guo

Authors
  1. Cheng Lu
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  2. Zhibin Deng
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  3. Jing Zhou
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  4. Xiaoling Guo
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Corresponding author

Correspondence to Zhibin Deng.

Additional information

Lu’s research has been supported by National Natural Science Foundation of China Grant Nos. 11701177 and 11771243, and Fundamental Research Funds for the Central Universities Grant Nos. 2017MS058 and 2018ZD14. Deng’s research has been supported by National Natural Science Foundation of China Grant No. 11501543. Zhou’s research has been supported by National Natural Science Foundation of China Grant No. 11701512, and the Zhejiang Provincial Natural Science Foundation of China Grant No. LQ16A010010. Guo’s research has been supported by National Natural Science Foundation of China Grant No. 11801557, and Fundamental Research Funds for the Central Universities Grant No. 800015LF.

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Lu, C., Deng, Z., Zhou, J. et al. A sensitive-eigenvector based global algorithm for quadratically constrained quadratic programming. J Glob Optim 73, 371–388 (2019). https://doi.org/10.1007/s10898-018-0726-y

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  • DOI: https://doi.org/10.1007/s10898-018-0726-y

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