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On the computational complexity of membership problems for the completely positive cone and its dual

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  • Peter Dickinson
  • Luuk Gijben

Abstract

Copositive programming has become a useful tool in dealing with all sorts of optimisation problems. It has however been shown by Murty and Kabadi (Math. Program. 39(2):117–129, 1987 ) that the strong membership problem for the copositive cone, that is deciding whether or not a given matrix is in the copositive cone, is a co-NP-complete problem. From this it has long been assumed that this implies that the question of whether or not the strong membership problem for the dual of the copositive cone, the completely positive cone, is also an NP-hard problem. However, the technical details for this have not previously been looked at to confirm that this is true. In this paper it is proven that the strong membership problem for the completely positive cone is indeed NP-hard. Furthermore, it is shown that even the weak membership problems for both of these cones are NP-hard. We also present an alternative proof of the NP-hardness of the strong membership problem for the copositive cone. Copyright Springer Science+Business Media New York 2014

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  • 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.
    • Handle: RePEc:spr:coopap:v:57:y:2014:i:2:p:403-415
    DOI: 10.1007/s10589-013-9594-z
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    References listed on IDEAS

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    1. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    2. de Klerk, E., 2006. "The Complexity of Optimizing over a Simplex, Hypercube or Sphere : A Short Survey," Discussion Paper 2006-85, Tilburg University, Center for Economic Research.
    3. Etienne Klerk, 2008. "The complexity of optimizing over a simplex, hypercube or sphere: a short survey," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 111-125, June.
    4. 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.
    5. de Klerk, E., 2008. "The complexity of optimizing over a simplex, hypercube or sphere : A short survey," Other publications TiSEM 485b6860-cf1d-4cad-97b8-2, Tilburg University, School of Economics and Management.
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    Cited by:

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    12. Badenbroek, Riley & de Klerk, Etienne, 2022. "An analytic center cutting plane method to determine complete positivity of a matrix," Other publications TiSEM 088da653-b943-4ed0-9720-6, Tilburg University, School of Economics and Management.
    13. Zhijian Lai & Akiko Yoshise, 2022. "Completely positive factorization by a Riemannian smoothing method," Computational Optimization and Applications, Springer, vol. 83(3), pages 933-966, December.
    14. Riley Badenbroek & Etienne de Klerk, 2022. "An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1115-1125, March.
    15. Haibin Chen & Zheng-Hai Huang & Liqun Qi, 2018. "Copositive tensor detection and its applications in physics and hypergraphs," Computational Optimization and Applications, Springer, vol. 69(1), pages 133-158, January.
    16. Paula Alexandra Amaral & Immanuel M. Bomze, 2019. "Nonconvex min–max fractional quadratic problems under quadratic constraints: copositive relaxations," Journal of Global Optimization, Springer, vol. 75(2), pages 227-245, October.
    17. Anwa Zhou & Jinyan Fan, 2018. "Completely positive tensor recovery with minimal nuclear value," Computational Optimization and Applications, Springer, vol. 70(2), pages 419-441, June.
    18. Gizem Sağol & E. Yıldırım, 2015. "Analysis of copositive optimization based linear programming bounds on standard quadratic optimization," Journal of Global Optimization, Springer, vol. 63(1), pages 37-59, September.
    19. 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.
    20. Jinyan Fan & Anwa Zhou, 2016. "Computing the distance between the linear matrix pencil and the completely positive cone," Computational Optimization and Applications, Springer, vol. 64(3), pages 647-670, July.
    21. Immanuel M. Bomze & Vaithilingam Jeyakumar & Guoyin Li, 2018. "Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations," Journal of Global Optimization, Springer, vol. 71(3), pages 551-569, July.
    22. Yuzhu Wang & Akihiro Tanaka & Akiko Yoshise, 2021. "Polyhedral approximations of the semidefinite cone and their application," Computational Optimization and Applications, Springer, vol. 78(3), pages 893-913, April.
    23. Gabriele Eichfelder & Patrick Groetzner, 2022. "A note on completely positive relaxations of quadratic problems in a multiobjective framework," Journal of Global Optimization, Springer, vol. 82(3), pages 615-626, March.
    24. Anwa Zhou & Jinyan Fan, 2015. "Interiors of completely positive cones," Journal of Global Optimization, Springer, vol. 63(4), pages 653-675, December.

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