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.
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References
Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, Singapore (2003)
Bomze, I.M.: Copositive programming—recent developments and applications. Cent. Eur. J. Oper. Res. 216, 509–520 (2012)
Bomze, I.M., Jarre, F., Rendl, F.: Quadratic factorization heuristics for copositive programming. Math. Program. Comput. 3(1), 37–57 (2011)
Bomze, I.M., Schachinger, W., Uchida, G.: Think co(mpletely) positive! matrix properties, examples and a clustered bibliography on copositive optimization. J. Glob. Optim. 52(3), 423–445 (2012)
Bondy, J., Murty, U.: Graph Theory with Applications. Macmillan, London (1976)
Bundfuss, S., Dür, M.: An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20(1), 30–53 (2009)
Burer, S.: Copositive programming. In: Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications, pp. 201–218. Springer, New York (2012)
Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)
de Klerk, E.: The complexity of optimizing over a simplex, hypercube or sphere: a short survey. Cent. Eur. J. Oper. Res. 16(2), 111–125 (2008)
de Klerk, E., Pasechnik, D.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002)
Dickinson, P.J.C.: The copositive cone, the completely positive cone and their generalisations. Ph.D. thesis, University of Groningen (2013)
Dickinson, P.J.C., Dür, M.: Linear-time complete positivity detection and decomposition of sparse matrices. SIAM J. Matrix Anal. Appl. 33(3), 701–720 (2012)
Dür, M.: Copositive programming—a survey. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and Its Applications in Engineering, pp. 3–20. Springer, Berlin (2010)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)
Gvozdenović, N., Laurent, M.: The operator Ψ for the chromatic number of a graph. SIAM J. Optim. 19(2), 572–591 (2008)
Hiriart-Urruty, J.B., Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52(4), 593–629 (2010)
Jarre, F., Schmallowsky, K.: On the computation of certificates. J. Glob. Optim. 45(2), 281–296 (2009)
Kaykobad, M.: On nonnegative factorization of matrices. Linear Algebra Appl. 96, 27–33 (1987)
Murty, K., Kabadi, S.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117–129 (1987)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)
Yudin, D., Nemirovskii, A.: Informational complexity and efficient methods for the solution of convex extremal problems. Èkon. Mat. Metody 12(2), 357–369 (1976). English translation: Matekon 13(3) (1977) 25–45
Acknowledgements
We wish to thank Gerhard J. Woeginger, Mirjam Dür and the anonymous referees for their highly useful advice and comments on this paper. Luuk Gijben was supported by the Netherlands Organisation for Scientific Research (NWO) through grant no.639.033.907. This research was carried out whilst Peter J.C. Dickinson was a PhD student at the Johann Bernoulli Institute, University of Groningen.
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Dickinson, P.J.C., Gijben, L. On the computational complexity of membership problems for the completely positive cone and its dual. Comput Optim Appl 57, 403–415 (2014). https://doi.org/10.1007/s10589-013-9594-z
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DOI: https://doi.org/10.1007/s10589-013-9594-z