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

Computational Optimization and Applications

Volumn 57, Issue 2, 2014, Pages 403-415

  Dickinson, Peter J C ^a   Gijben, Luuk ^b  

^a UNIVERSITY OF VIENNA   (Austria)

^b UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

Completely positive; Copositive; NP hard; Stable set

Indexed keywords

EID: 84894450105     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-013-9594-z     Document Type: Article

References (22)

1. Berman, A.; Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, Singapore (2003)
2. Bomze, I.M.: Copositive programming - recent developments and applications. Cent. Eur. J. Oper. Res. 216, 509-520 (2012)
3. Bomze, I.M.; Jarre, F.; Rendl, F.: Quadratic factorization heuristics for copositive programming. Math. Program. Comput. 3(1), 37-57 (2011)
4. 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)
5. Bondy, J.; Murty, U.: Graph Theory with Applications. Macmillan, London (1976)
6. Bundfuss, S.; Dür, M.: An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20(1), 30-53 (2009)
7. Burer, S.: Copositive programming. In: Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications, pp. 201-218. Springer, New York (2012)
8. Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479-495 (2009)
9. 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)
10. de Klerk, E.; Pasechnik, D.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875-892 (2002)
11. Dickinson, P.J.C.: The copositive cone, the completely positive cone and their generalisations. Ph.D. thesis, University of Groningen (2013)
12. 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)
13. 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)
14. Garey, M.; Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
15. Grötschel, M.; Lovász, L.; Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)
16. Gvozdenović, N.; Laurent, M.: The operator Ψ for the chromatic number of a graph. SIAM J. Optim. 19(2), 572-591 (2008)
17. Hiriart-Urruty, J.B.; Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52(4), 593-629 (2010)
18. Jarre, F.; Schmallowsky, K.: On the computation of [InlineEquation not available: see fulltext.] certificates. J. Glob. Optim. 45(2), 281-296 (2009)
19. Kaykobad, M.: On nonnegative factorization of matrices. Linear Algebra Appl. 96, 27-33 (1987)
20. Murty, K.; Kabadi, S.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117-129 (1987)
21. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)
22. 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