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New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation

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Abstract

We consider a quadratic program with a few negative eigenvalues (QP-r-NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one negative eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified \(\epsilon \)-tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within \(\epsilon \)-tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP-r-NE instances when \(r\le 10\), and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.1344739.

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Notes

  1. It is worth mentioning that in [8], the authors developed a global algorithm for QPs with a small number of negative eigenvalues. However, the algorithm was tested only on small scale instances.

  2. All the quadratic convex optimization problems solved in the GSA algorithm are in form of (36), (37) or (44).

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Acknowledgements

We would like to thank all the anonymous reviewers and the associate editor for their useful suggestions that has helped to substantially improve the presentation of this work.

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Authors and Affiliations

  1. Department of Management Science & Engineering, College of Economics and Management, Zhejiang University of Technology, Hangzhou, 310032, Zhejiang, P. R. China

    Hezhi Luo

  2. Department of Applied Mathematics, College of Science, Zhejiang University of Technology, Hangzhou, 310032, Zhejiang, P. R. China

    Xiaodi Bai

  3. Department of Industrial Engineering, University of Houston, Houston, TX, 77204, USA

    Gino Lim & Jiming Peng

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  1. Hezhi Luo
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  2. Xiaodi Bai
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Correspondence to Jiming Peng.

Additional information

The research of Hezhi Luo is jointly supported by NSFC Grants 11871433 and 11371324 and the Zhejiang Provincial NSFC Grants LY17A010023 and LY18A010011.

The research of Xiaodi Bai is supported by NSFC Grants 11371103 and 11701511.

The research of Jiming Peng is supported by NSF Grants CMMI-1131690 and CMMI-1537712.

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Luo, H., Bai, X., Lim, G. et al. New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation . Math. Prog. Comp. 11, 119–171 (2019). https://doi.org/10.1007/s12532-018-0142-9

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