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On duality gap in binary quadratic programming

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Abstract

We investigate in this paper the duality gap between the binary quadratic optimization problem and its semidefinite programming relaxation. We show that the duality gap can be underestimated by \({\xi_{r+1}\delta^2}\), where δ is the distance between {−1, 1}n and certain affine subspace, and ξ r+1 is the smallest positive eigenvalue of a perturbed matrix. We also establish the connection between the computation of δ and the cell enumeration of hyperplane arrangement in discrete geometry.

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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. L. Sun

  2. Department of Applied Mathematics, Shanghai University of Finance and Economics, 200433, Shanghai, People’s Republic of China

    C. L. Liu

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

    D. Li & J. J. Gao

Authors
  1. X. L. Sun
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  2. C. L. Liu
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  3. D. Li
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  4. J. J. Gao
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Corresponding author

Correspondence to D. Li.

Additional information

This work was supported by National Natural Science Foundation of China under grants 10971034 and 70832002, by the Joint NSFC/RGC grants under grant 71061160506, and by Research Grants Council of Hong Kong under grant 414207.

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Sun, X.L., Liu, C.L., Li, D. et al. On duality gap in binary quadratic programming. J Glob Optim 53, 255–269 (2012). https://doi.org/10.1007/s10898-011-9683-4

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