Separating doubly nonnegative and completely positive matrices
 Hongbo Dong,
 Kurt Anstreicher
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NPHard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then to separate a given DNN but nonCP matrix from the cone of CP matrices. We describe two different constructions for such a separation that apply to 5 × 5 matrices that are DNN but nonCP. We also describe a generalization that applies to larger DNN but nonCP matrices having block structure. Computational results illustrate the applicability of these separation procedures to generate improved bounds on difficult problems.
 Anstreicher, K.M., Burer, S. (2010) Computable representations for convex hulls of lowdimensional quadratic forms. Math. Prog. B 124: pp. 3343 CrossRef
 Burer, S., Anstreicher, K.M., Dür, M. (2009) The difference between 5 × 5 doubly nonnegative and completely positive matrices. Linear Algebra Appl. 431: pp. 15391552 CrossRef
 Barioli, F. (1998) Completely positive matrices with a bookgraph. Linear Algebra Appl. 277: pp. 1131 CrossRef
 Bundfuss, S., Dür, M. (2009) An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20: pp. 3053 CrossRef
 Bomze, I.M., Klerk, E. (2002) Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Global Optim. 24: pp. 163185 CrossRef
 Bomze, I.M., Frommlet, F., Locatelli, M. (2010) Copositivity cuts for improving SDP bounds on the clique number. Math. Prog. B 124: pp. 1332 CrossRef
 Burer, S., Letchford, A.N. (2009) On nonconvex quadratic programming with box constriants. SIAM J. Optim. 20: pp. 10731089 CrossRef
 Bomze, I.M., Locatelli, M., Tardella, F. (2008) New and old bounds for standard quadratic programming: dominance, equivalence and incomparability. Math. Prog. 115: pp. 3164 CrossRef
 Berman, A., ShakedMonderer, N. (2003) Completely Positive Matrices. World Scientific, Singapore CrossRef
 Burer, S. (2009) On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Prog. 120: pp. 479495 CrossRef
 Burer, S. (2010) Optimizing a polyhedralsemidefinite relaxation of completely positive programs. Math. Prog. Comp. 2: pp. 119 CrossRef
 Berman, A., Xu, C. (2004) 5 × 5 Completely positive matrices. Linear Algebra Appl. 393: pp. 5571 CrossRef
 Dong, H., Anstreicher, K. (2010) On ‘5 × 5 Completely positive matrices’. Linear Algebra Appl. 433: pp. 10011004 CrossRef
 Klerk, E., Pasechnik, D.V. (2002) Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12: pp. 875892 CrossRef
 Dür, M., Still, G. (2008) Interior points of the completely positive cone. Electron. J. Linear Algebra 17: pp. 4853
 Hall, M. (1967) Combinatorial Theory. Blaisdell Publishing Company, Waltham
 Hogben, L., Johnson, C.R., Reams, R. (2005) The copositive completion problem. Linear Algebra Appl. 408: pp. 207211 CrossRef
 Kogan, N., Berman, A. (1993) Characterization of completely positive graphs. Discret. Math. 114: pp. 297304 CrossRef
 Peña, J., Vera, J., Zuluaga, L.F. (2007) Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18: pp. 87105 CrossRef
 Väliaho, H. (1989) Almost copositive matrices. Linear Algebra Appl. 116: pp. 121134 CrossRef
 Yajima, Y., Fujie, T. (1998) A polyhedral approach for nonconvex quadratic programming problems with box constraints. J. Global Optim. 13: pp. 151170 CrossRef
 Title
 Separating doubly nonnegative and completely positive matrices
 Journal

Mathematical Programming
Volume 137, Issue 12 , pp 131153
 Cover Date
 20130201
 DOI
 10.1007/s1010701104858
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 90C26
 90C22
 90C20
 15B48
 Industry Sectors
 Authors

 Hongbo Dong ^{(1)}
 Kurt Anstreicher ^{(2)}
 Author Affiliations

 1. Department of Applied Mathematics and Computational Sciences, University of Iowa, Iowa City, IA, 52242, USA
 2. Department of Management Sciences, University of Iowa, Iowa City, IA, 52242, USA