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

  • Cheng Lu
  • Zhibin DengEmail author
  • Jing Zhou
  • Xiaoling Guo


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.


Quadratically constrained quadratic programming Semidefinite relaxation Branch-and-bound algorithm Global optimization 



The authors would like to thank the two anonymous reviewers, whose invaluable comments have significantly improved the quality of this paper.


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Authors and Affiliations

  1. 1.School of Economics and ManagementNorth China Electric Power UniversityBeijingChina
  2. 2.School of Economics and Management, University of Chinese Academy of Sciences, Key Laboratory of Big Data Mining and Knowledge ManagementChinese Academy of SciencesBeijingChina
  3. 3.Department of Applied Mathematics, College of ScienceZhejiang University of TechnologyHangzhouChina
  4. 4.Departmemt of MathematicsChina University of Mining and TechnologyBeijingChina

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