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Notoriously hard (mixed-)binary QPs: empirical evidence on new completely positive approaches

Author

Listed:
  • Immanuel M. Bomze

    (Universität Wien)

  • Jianqiang Cheng

    (University of Arizona)

  • Peter J. C. Dickinson

    (University of Twente)

  • Abdel Lisser

    (Université Paris Sud - XI)

  • Jia Liu

    (Xi’an Jiaotong University)

Abstract

By now, many copositive reformulations of mixed-binary QPs have been discussed, triggered by Burer’s seminal characterization from 2009. In conic optimization, it is very common to use approximation hierarchies based on positive-semidefinite (psd) matrices where the order increases with the level of the approximation. Our purpose is to keep the psd matrix orders relatively small to avoid memory size problems in interior point solvers. Based upon on a recent discussion on various variants of completely positive reformulations and their relaxations (Bomze et al. in Math Program 166(1–2):159–184, 2017), we here present a small study of the notoriously hard multidimensional quadratic knapsack problem and quadratic assignment problem. Our observations add some empirical evidence on performance differences among the above mentioned variants. We also propose an alternative approach using penalization of various classes of (aggregated) constraints, along with some theoretical convergence analysis. This approach is in some sense similar in spirit to the alternating projection method proposed in Burer (Math Program Comput 2:1–19, 2010) which completely avoids SDPs, but for which no convergence proof is available yet.

Suggested Citation

  • Immanuel M. Bomze & Jianqiang Cheng & Peter J. C. Dickinson & Abdel Lisser & Jia Liu, 2019. "Notoriously hard (mixed-)binary QPs: empirical evidence on new completely positive approaches," Computational Management Science, Springer, vol. 16(4), pages 593-619, October.
    • Handle: RePEc:spr:comgts:v:16:y:2019:i:4:d:10.1007_s10287-018-0337-6
    DOI: 10.1007/s10287-018-0337-6
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    References listed on IDEAS

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    1. Immanuel Bomze & Werner Schachinger & Gabriele Uchida, 2012. "Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 423-445, March.
    2. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
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