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An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints

Author

Listed:
  • Cheng Lu

    (North China Electric Power University)

  • Zhibin Deng

    (University of Chinese Academy of Sciences)

  • Qingwei Jin

    (Zhejiang University)

Abstract

In this paper, we propose a branch-and-bound algorithm for finding a global optimal solution for a nonconvex quadratic program with convex quadratic constraints (NQPCQC). We first reformulate NQPCQC by adding some nonconvex quadratic constraints induced by eigenvectors of negative eigenvalues associated with the nonconvex quadratic objective function to Shor’s semidefinite relaxation. Under the assumption of having a bounded feasible domain, these nonconvex quadratic constraints can be further relaxed into linear ones to form a special semidefinite programming relaxation. Then an efficient branch-and-bound algorithm branching along the eigendirections of negative eigenvalues is designed. The theoretic convergence property and the worst-case complexity of the proposed algorithm are proved. Numerical experiments are conducted on several types of quadratic programs to show the efficiency of the proposed method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:67:y:2017:i:3:d:10.1007_s10898-016-0436-2
    DOI: 10.1007/s10898-016-0436-2
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    References listed on IDEAS

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    1. Deng, Zhibin & Fang, Shu-Cherng & Jin, Qingwei & Xing, Wenxun, 2013. "Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme," European Journal of Operational Research, Elsevier, vol. 229(1), pages 21-28.
    2. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    3. Jos F. Sturm & Shuzhong Zhang, 2003. "On Cones of Nonnegative Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 246-267, May.
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    Cited by:

    1. 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.
    2. Bo Zhang & YueLin Gao & Xia Liu & XiaoLi Huang, 2022. "An Outcome-Space-Based Branch-and-Bound Algorithm for a Class of Sum-of-Fractions Problems," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 830-855, March.
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    4. Xiaoli Cen & Yong Xia, 2021. "A New Global Optimization Scheme for Quadratic Programs with Low-Rank Nonconvexity," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1368-1383, October.

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