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A new branch-and-cut algorithm for non-convex quadratic programming via alternative direction method and semidefinite relaxation


We consider a non-convex quadratic program (QP) with linear and convex quadratic constraints that arises from a broad range of applications and is known to be NP-hard. In this paper, we first prove that the alternative direction method converges to a local solution of the underlying QP problem. We then propose a new branch-and-cut algorithm that finds a globally optimal solution to the underlying QP problem within a pre-specified 𝜖-tolerance by integrating the alternative direction method with semidefinite relaxation and disjunctive cut techniques. We establish the global convergence of the algorithm and estimate its complexity. Preliminary numerical results demonstrate that the proposed algorithm can effectively find a globally optimal solution to medium-scale QP instances in which the number of negative eigenvalues of the Hessian matrix in the objective function is less than or equals 20.

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    All the data used in Section 4 can be downloaded at

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    In our numerical experiments, the SDP subproblem in BB-SDR is solved by the SDP solver SeDuMi [36].


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The authors would like to thank the Associate Editor and the two anonymous referees for the detailed comments and valuable suggestions, which have improved the final presentation of the paper.


The work is supported by the NSFC grants 11871433 and 11371324 and the Zhejiang Provincial NSFC grants LY18A010011 and LZ21A010003.

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Correspondence to Hezhi Luo.

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Luo, H., Chen, S. & Wu, H. A new branch-and-cut algorithm for non-convex quadratic programming via alternative direction method and semidefinite relaxation. Numer Algor 88, 993–1024 (2021).

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  • Quadratic programming
  • Branch-and-cut algorithm
  • Alternative direction method
  • Semidefinite relaxation
  • Computational complexity

Mathematics Subject Classification (2010)

  • 90C20
  • 90C22
  • 90C26