Algorithmic copositivity detection by simplicial partition

Linear Algebra and Its Applications

Volumn 428, Issue 7, 2008, Pages 1511-1523

  Bundfuss, Stefan ^a   Dür, Mirjam ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Conditionally definite matrices; Copositive matrices

Indexed keywords

ALGORITHMS; NUMERICAL METHODS; PROBLEM SOLVING;

CONDITIONALLY DEFINITE MATRICES; COPOSITIVE MATRICES; MAXIMUM CLIQUE PROBLEM;

MATRIX ALGEBRA;

EID: 38849167450     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2007.09.035     Document Type: Article

References (27)

1. Andersson L.E., Chang G., and Elfving T. Criteria for copositive matrices using simplices and barycentric coordinates. Linear Algebra Appl. 220 (1995) 9-30
2. Bomze I.M. Block pivoting and shortcut strategies for detecting copositivity. Linear Algebra Appl. 248 (1996) 161-184
3. Bomze I.M. Linear-time copositivity detection for tridiagonal matrices and extension to block-tridiagonality. SIAM J. Matrix Anal. Appl. 21 (2000) 840-848
4. Bomze I.M., Dür M., de Klerk E., Roos C., Quist A.J., and Terlaky T. On copositive programming and standard quadratic optimization problems. J. Global Optim. 18 (2000) 301-320
5. S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Preprint, available online at .
6. Cottle R.W., Habetler G.J., and Lemke C.E. On classes of copositive matrices. Linear Algebra Appl. 3 (1970) 295-310
7. Danninger G. Role of copositivity in optimality criteria for nonconvex optimization problems. J. Optim. Theory Appl. 75 (1992) 535-558
8. G. Danninger, A recursve algorithm to detect (strict) copositivity of a matrix, in: U. Rieder, A. Peyerimhoff, F.J. Rademacher (Eds.), Methods of Operations Research, vol. 62, 1990, pp. 45-52.
9. DIMACS, Second DIMACS Challenge, Test instances available at .
10. G. Eichfelder, J. Jahn, Set-semidefinite optimization, University of Erlangen-Nuremberg, Preprint No. 316, 2007.
11. Hadeler K.P. On copositive matrices. Linear Algebra Appl. 49 (1983) 79-89
12. Hall Jr. M., and Newman M. Copositive and completely positive quadratic forms. Proc. Cambridge Philos. Soc. 59 (1963) 329-339
13. Horst R. On generalized bisection of n-simplices. Math. Comp. 66 (1997) 691-698
14. Ikramov K.D. Linear-time algorithm for verifying the copositivity of an acyclic matrix. Comput. Math. Math. Phys. 42 (2002) 1701-1703
15. Ikramov K.D., and Savel'eva N.V. Conditionally definite matrices. J. Math. Sci. 98 (2000) 1-50
16. Jüttler B. Arbitrarily weak linear convexity conditions for multivariate polynomials. Stud. Sci. Math. Hungar. 36 (2000) 165-183
17. de Klerk E., and Pasechnik D.V. Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12 (2002) 875-892
18. Kaplan W. A test for copositive matrices. Linear Algebra Appl. 313 (2000) 203-206
19. Kaplan W. A copositivity probe. Linear Algebra Appl. 337 (2001) 237-251
20. Lemke C.E. Bimatrix equilibrium points and mathematical programming. Manage. Sci. 11 (1965) 681-689
21. Martin D.H. Finite criteria for conditional definiteness of quadratic forms. Linear Algebra Appl. 39 (1981) 9-21
22. Martin D.H., and Jacobson D.H. Copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints. Linear Algebra Appl. 35 (1981) 227-258
23. Murty K.G., and Kabadi S.N. Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39 (1987) 117-129
24. Quist A.J., de Klerk E., Roos C., and Terlaky T. Copositive relaxation for general quadratic programming. Optim. Methods Softw. 9 (1998) 185-208
25. Väliaho H. Criteria for copositive matrices. Linear Algebra Appl. 81 (1986) 19-34
26. Väliaho H. Testing the definiteness of matrices on polyhedral cones. Linear Algebra Appl. 101 (1988) 135-165
27. Väliaho H. Quadratic programming criteria for copositive matrices. Linear Algebra Appl. 119 (1989) 163-182