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Novel shortcut strategies in copositivity detection: Decomposition for quicker positive certificates

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  • Zischg, Johannes
  • Bomze, Immanuel

Abstract

Copositivity is a property of symmetric matrices which is NP-hard to check. Nevertheless, it plays a crucial role in tight bounds for conic approaches of several hard optimization problems. In this paper, we present novel promising shortcut strategies to exploit favorable instances in a systematic way, using decomposition strategies based upon the idea to allow for overlapping, smaller blocks, profiting from a beneficial sign structure of the entries of the given matrix. The working hypothesis of this approach is the common empirical observation in the community that for detection of copositivity, a negative certificate is easier to obtain than a positive one. First empirical results on carefully orchestrated randomly generated instances seem to corroborate our approach.

Suggested Citation

  • Zischg, Johannes & Bomze, Immanuel, 2025. "Novel shortcut strategies in copositivity detection: Decomposition for quicker positive certificates," Operations Research Perspectives, Elsevier, vol. 14(C).
    • Handle: RePEc:eee:oprepe:v:14:y:2025:i:c:s2214716024000289
    DOI: 10.1016/j.orp.2024.100324
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    References listed on IDEAS

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    1. Mohammadreza Safi & Seyed Saeed Nabavi & Richard J. Caron, 2021. "A modified simplex partition algorithm to test copositivity," Journal of Global Optimization, Springer, vol. 81(3), pages 645-658, November.
    2. Julia Sponsel & Stefan Bundfuss & Mirjam Dür, 2012. "An improved algorithm to test copositivity," Journal of Global Optimization, Springer, vol. 52(3), pages 537-551, March.
    3. Jacek Gondzio & E. Alper Yıldırım, 2021. "Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations," Journal of Global Optimization, Springer, vol. 81(2), pages 293-321, October.
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