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Quadratic convex reformulation for nonconvex binary quadratically constrained quadratic programming via surrogate constraint

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Abstract

We investigate in this paper nonconvex binary quadratically constrained quadratic programming (QCQP) which arises in various real-life fields. We propose a novel approach of getting quadratic convex reformulation (QCR) for this class of optimization problem. Our approach employs quadratic surrogate functions and convexifies all the quadratic inequality constraints to construct QCR. The price of this approach is the introduction of an extra quadratic inequality. The “best” QCR among the proposed family, in terms that the bound of the corresponding continuous relaxation is best, can be found via solving a semidefinite programming problem. Furthermore, we prove that the bound obtained by continuous relaxation of our best QCR is as tight as Lagrangian bound of binary QCQP. Computational experiment is also conducted to illustrate the solution efficiency improvement of our best QCR when applied in off-the-shell software.

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Acknowledgements

The authors would like to thank three anonymous referees for their constructive suggestions and insightful comments, which helped improve the paper substantially.

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Authors and Affiliations

  1. School of Economics and Management, Tongji University, Shanghai, 200092, People’s Republic of China

    Xiaojin Zheng & Yutong Pan

  2. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, People’s Republic of China

    Xueting Cui

Authors
  1. Xiaojin Zheng
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  2. Yutong Pan
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  3. Xueting Cui
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Corresponding author

Correspondence to Xueting Cui.

Additional information

This work was supported by National Natural Science Foundation of China under Grants 11671300, 11371103 and 71501122.

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Zheng, X., Pan, Y. & Cui, X. Quadratic convex reformulation for nonconvex binary quadratically constrained quadratic programming via surrogate constraint. J Glob Optim 70, 719–735 (2018). https://doi.org/10.1007/s10898-017-0591-0

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