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Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach

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Abstract

We present in this paper an integer diagonalization approach for deriving new lower bounds for general quadratic integer programming problems. More specifically, we introduce a semiunimodular transformation in order to diagonalize a symmetric matrix and preserve integral property of the feasible set at the same time. Via the semiunimodular transformation, the resulting separable quadratic integer program is a relaxation of the nonseparable quadratic integer program. We further define the integer diagonalization dual problem to identify the best semiunimodular transformation and analyze some basic properties of the set of semiunimodular transformations for a rational symmetric matrix. In particular, we present a complete characterization of the set of all semiunimodular transformations for a nonsingular 2×2 symmetric matrix. We finally discuss Lagrangian relaxation and convex relaxation schemes for the resulting separable quadratic integer programming problem and compare the tightness of different relaxation schemes.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, P.R. China

    X. J. Zheng

  2. Department of Management Science, School of Management, Fudan University, Shanghai, 200433, P.R. China

    X. J. Zheng & X. L. Sun

  3. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, 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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Correspondence to D. Li.

Additional information

Communicated by P.M. Pardalos.

This work was supported by National Natural Science Foundation of China under Grants 10971034, 70832002, and by Research Grants Council of Hong Kong under Grant 414207.

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Zheng, X.J., Sun, X.L. & Li, D. Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach. J Optim Theory Appl 146, 463–489 (2010). https://doi.org/10.1007/s10957-010-9653-x

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