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

Author

Listed:
  • Xiaojin Zheng

    (Tongji University)

  • Yutong Pan

    (Tongji University)

  • Xueting Cui

    (Shanghai University of Finance and Economics)

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.

Suggested Citation

  • Xiaojin Zheng & Yutong Pan & Xueting Cui, 2018. "Quadratic convex reformulation for nonconvex binary quadratically constrained quadratic programming via surrogate constraint," Journal of Global Optimization, Springer, vol. 70(4), pages 719-735, April.
    • Handle: RePEc:spr:jglopt:v:70:y:2018:i:4:d:10.1007_s10898-017-0591-0
    DOI: 10.1007/s10898-017-0591-0
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    References listed on IDEAS

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    1. X. Zheng & X. Sun & D. Li, 2011. "Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations," Journal of Global Optimization, Springer, vol. 50(4), pages 695-712, August.
    2. Klose, Andreas & Drexl, Andreas, 2005. "Facility location models for distribution system design," European Journal of Operational Research, Elsevier, vol. 162(1), pages 4-29, April.
    3. X. Cui & X. Zheng & S. Zhu & X. Sun, 2013. "Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems," Journal of Global Optimization, Springer, vol. 56(4), pages 1409-1423, August.
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