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

Author

Listed:
  • Cheng Lu

    (North China Electric Power University)

  • Zhibin Deng

    (Chinese Academy of Sciences)

  • Jing Zhou

    (Zhejiang University of Technology)

  • Xiaoling Guo

    (China University of Mining and Technology)

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.

Suggested Citation

  • Cheng Lu & Zhibin Deng & Jing Zhou & Xiaoling Guo, 2019. "A sensitive-eigenvector based global algorithm for quadratically constrained quadratic programming," Journal of Global Optimization, Springer, vol. 73(2), pages 371-388, February.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:2:d:10.1007_s10898-018-0726-y
    DOI: 10.1007/s10898-018-0726-y
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    References listed on IDEAS

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    1. Jos F. Sturm & Shuzhong Zhang, 2003. "On Cones of Nonnegative Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 246-267, May.
    2. Cheng Lu & Zhibin Deng & Qingwei Jin, 2017. "An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints," Journal of Global Optimization, Springer, vol. 67(3), pages 475-493, March.
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