Abstract
We show how to separate a doubly nonnegative matrix, which is not completely positive and has a triangle-free graph, from the completely positive cone. This method can be used to compute cutting planes for semidefinite relaxations of combinatorial problems. We illustrate our approach by numerical tests on the stable set problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific Publishing, Cleveland (2003)
Bomze, I.M., Frommlet, F., Locatelli, M.: Copositivity cuts for improving SDP bounds on the clique number. Math. Program. 124, 13–32 (2010)
Bomze, I.M., Locatelli, M., Tardella, F.: New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. Math. Program. 115, 31–64 (2008)
Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)
Burer, S., Anstreicher, K., Dür, M.: The difference between \(5\times 5\) doubly nonnegative and completely positive matrices. Linear Algebra Appl. 431, 1539–1552 (2009)
Burer, S., Dong, H.: Separation and relaxation for cones of quadratic forms. Math. Program. 137, 343–370 (2013)
Cottle, R.W., Habetler, G.J., Lemke, C.E.: On classes of copositive matrices. Linear Algebra Appl. 3, 295–310 (1970)
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)
Dickinson, P.J.C.: Geometry of the copositive and completely positive cones. J. Math. Anal. Appl. 380, 377–395 (2011)
Dong, H., Anstreicher, K.: Separating doubly nonnegative and completely positive matrices. Math. Program. 137, 131–153 (2013)
Drew, J.H., Johnson, C.R., Loewy, R.: Completely positive matrices associated with M-matrices. Linear Multilinear Algebra 37, 303–310 (1994)
Hadeler, K.-P.: On copositive matrices. Linear Algebra Appl. 49, 79–89 (1983)
Haynsworth, E., Hoffman, A.J.: Two remarks on copositive matrices. Linear Algebra Appl. 2, 387–392 (1969)
Hildebrand, R.: The extreme rays of the \(5\times 5\) copositive cone. Linear Algebra Appl. 437, 1538–1547 (2012)
Hoffman, A.J., Pereira, F.: On copositive matrices with \(-1, 0, 1\) entries. J. Comb. Theory (A) 14, 302–309 (1973)
Kaplan, W.: A test for copositive matrices. Linear Algebra Appl. 313, 203–206 (2000)
de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875–892 (2002)
Peña, J., Vera, J., Zuluaga, L.F.: Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18, 87–105 (2007)
Sponsel, J., Dür, M.: Factorization and cutting planes for completely positive matrices by copositive projection. Math. Program. 143, 211–229 (2014)
Väliaho, H.: Criteria for copositive matrices. Linear Algebra Appl. 81, 19–34 (1986)
Acknowledgments
We are grateful to the referees for their careful reading and helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by Grant no. G-18-304.2/2011 by the German-Israeli Foundation for Scientific Research and Development (GIF).
Rights and permissions
About this article
Cite this article
Berman, A., Dür, M., Shaked-Monderer, N. et al. Cutting planes for semidefinite relaxations based on triangle-free subgraphs. Optim Lett 10, 433–446 (2016). https://doi.org/10.1007/s11590-015-0922-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-015-0922-3