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

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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.

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Acknowledgments

We would like to thank three anonymous reviewers for their invaluable suggestions and comments. This work was supported by NSFC No. 11301479, NSFC No. 11501543, Research Foundation for Young Faculty of University of Chinese Academy of Sciences No. Y551037Y00, and the project supported by Zhejiang Provincial Natural Science Foundation of China No. LQ13A010001.

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Authors and Affiliations

  1. School of Economics and Management, North China Electric Power University, Beijing, China

    Cheng Lu

  2. School of Economics and Management, University of Chinese Academy of Sciences, Beijing, China

    Zhibin Deng

  3. Department of Management Science and Engineering, Zhejiang University, Hangzhou, China

    Qingwei Jin

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  1. Cheng Lu
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  2. Zhibin Deng
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  3. Qingwei Jin
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Correspondence to Qingwei Jin.

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Lu, C., Deng, Z. & Jin, Q. An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints. J Glob Optim 67, 475–493 (2017). https://doi.org/10.1007/s10898-016-0436-2

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  • DOI: https://doi.org/10.1007/s10898-016-0436-2

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