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A semidefinite programming method for integer convex quadratic minimization

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Abstract

We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice \({\mathbf{Z}}^n\). We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the problem. By interpreting the solution to the SDP relaxation probabilistically, we obtain a randomized algorithm for finding good suboptimal solutions, and thus an upper bound on the optimal value. The effectiveness of the method is shown for numerical problem instances of various sizes.

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Acknowledgements

We thank three anonymous referees for providing helpful comments and constructive remarks.

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Authors and Affiliations

  1. Stanford University, Packard Building, Room 243, 350 Serra Mall, Stanford, CA, 94305, USA

    Jaehyun Park

  2. Stanford University, Packard Building, Room 254, 350 Serra Mall, Stanford, CA, 94305, USA

    Stephen Boyd

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  1. Jaehyun Park
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  2. Stephen Boyd
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Correspondence to Jaehyun Park.

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Park, J., Boyd, S. A semidefinite programming method for integer convex quadratic minimization. Optim Lett 12, 499–518 (2018). https://doi.org/10.1007/s11590-017-1132-y

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  • DOI: https://doi.org/10.1007/s11590-017-1132-y

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