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Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations

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Abstract

We propose in this paper a general D.C. decomposition scheme for constructing SDP relaxation formulations for a class of nonconvex quadratic programs with a nonconvex quadratic objective function and convex quadratic constraints. More specifically, we use rank-one matrices and constraint matrices to decompose the indefinite quadratic objective into a D.C. form and underestimate the concave terms in the D.C. decomposition formulation in order to get a convex relaxation of the original problem. We show that the best D.C. decomposition can be identified by solving an SDP problem. By suitably choosing the rank-one matrices and the linear underestimation, we are able to construct convex relaxations that dominate Shor’s SDP relaxation and the strengthened SDP relaxation. We then propose an extension of the D.C. decomposition to generate an SDP bound that is tighter than the SDP+RLT bound when additional box constraints are present. We demonstrate via computational results that the optimal D.C. decomposition schemes can generate both tight SDP bounds and feasible solutions with good approximation ratio for nonconvex quadratically constrained quadratic problems.

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Authors and Affiliations

  1. Department of Management Science, School of Management, Fudan University, 200433, Shanghai, People’s Republic of China

    X. J. Zheng & X. L. Sun

  2. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    D. Li

Authors
  1. X. J. Zheng
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  2. X. L. Sun
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  3. D. Li
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Corresponding author

Correspondence to X. L. Sun.

Additional information

This research is partially supported by National Science Foundation of China Grants 10971034 and 70832002, by the Joint NSFC/RGC Grants under Grant N_HKUST626/10 and by Research Grants Council of Hong Kong Grant 414207.

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Zheng, X.J., Sun, X.L. & Li, D. Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations. J Glob Optim 50, 695–712 (2011). https://doi.org/10.1007/s10898-010-9630-9

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  • DOI: https://doi.org/10.1007/s10898-010-9630-9

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