Quadratic programming with one negative eigenvalue is NP-hard
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We show that the problem of minimizing a concave quadratic function with one concave direction is NP-hard. This result can be interpreted as an attempt to understand exactly what makes nonconvex quadratic programming problems hard. Sahni in 1974  showed that quadratic programming with a negative definite quadratic term (n negative eigenvalues) is NP-hard, whereas Kozlov, Tarasov and Hačijan  showed in 1979 that the ellipsoid algorithm solves the convex quadratic problem (no negative eigenvalues) in polynomial time. This report shows that even one negative eigenvalue makes the problem NP-hard.
Key wordsGlobal optimization quadratic programming NP-hard
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- 1.Garey, M. R. and Johnson, D. S. (1979), Computers and Intractability, A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, San Francisco.Google Scholar
- 2.Kozlov, M. K., Tarasov, S. P., and Hačijan, L. G. (1979), Polynomial Solvability of Convex Quadratic Programming, Soviet Math. Doklady 20, 1108–111.Google Scholar
- 3.Murty, K. G. and Kabadi, S. N. (1987), Some NP-Complete Problems in Quadratic and Non-linear Programming, Mathematical Programming 39, 117–129.Google Scholar
- 4.Pardalos, P. M. (1990), Polynomial Time Algorithms for Some Classes of Nonconvex Quadratic Problems, To appear in Optimization.Google Scholar
- 5.Pardalos, P. M. and Rosen, J. B. (1986), Global Concave Minimization: A Bibliographic Survey, SIAM Review 28 (3), 367–379.Google Scholar
- 6.Pardalos, P. M. and Rosen, J. B. (1987), Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science 268, Springer-Verlag, Berlin.Google Scholar
- 7.Pardalos, P. M. and Schnitger, G. (1988), Checking Local Optimality in Constrained Quadratic Programming is NP-hard, Operations Research Letters 7 (1), 33–35.Google Scholar
- 8.Sahni, S. (1974), Computationally Related Prolems, SIAM J. Comput. 3, 262–279.Google Scholar
- 9.Vavasis, S. A. (1990), Quadratic Programming Is in NP, Inf. Proc. Lett. 36, 73–77.Google Scholar