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Quadratic programming with one negative eigenvalue is NP-hard. (English) Zbl 0755.90065
Summary: We show that the problem of minimizing a concave quadratic function with one concave directions is NP-hard. This result can be interpreted as an attempt to understand exactly what makes nonconvex quadratic programming problems hard. S. Sahni [SIAM J. Computing 3(1974), 262-279 (1975; Zbl 0272.68040)] showed that quadratic programming with a negative definite quadratic term ($n$ negative eigenvalues) is NP-hard, whereas M. K. Kozlov, S. P. Tarasov and L. G. Khachiyan [Sov. Math., Dokl. 20, 1108-1111 (1979); translation from Dokl. Akad. Nauk. SSSR 248, 1049-1051 (1979; Zbl 0434.90071)] showed 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.
##### MSC:
 90C20 Quadratic programming 90C60 Abstract computational complexity for mathematical programming problems
##### Keywords:
global optimization; concave quadratic function; NP-hard
##### References:
 [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. [2] Kozlov, M. K., Tarasov, S. P., and Ha?ijan, L. G. (1979), Polynomial Solvability of Convex Quadratic Programming, Soviet Math. Doklady 20, 1108-111. [3] Murty, K. G. and Kabadi, S. N. (1987), Some NP-Complete Problems in Quadratic and Non-linear Programming, Mathematical Programming 39, 117-129. · Zbl 0637.90078 · doi:10.1007/BF02592948 [4] Pardalos, P. M. (1990), Polynomial Time Algorithms for Some Classes of Nonconvex Quadratic Problems, To appear in Optimization. [5] Pardalos, P. M. and Rosen, J. B. (1986), Global Concave Minimization: A Bibliographic Survey, SIAM Review 28 (3), 367-379. · Zbl 0602.90105 · doi:10.1137/1028106 [6] Pardalos, P. M. and Rosen, J. B. (1987), Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science 268, Springer-Verlag, Berlin. [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. · Zbl 0644.90067 · doi:10.1016/0167-6377(88)90049-1 [8] Sahni, S. (1974), Computationally Related Prolems, SIAM J. Comput. 3, 262-279. · doi:10.1137/0203021 [9] Vavasis, S. A. (1990), Quadratic Programming Is in NP, Inf. Proc. Lett. 36, 73-77. · Zbl 0719.90052 · doi:10.1016/0020-0190(90)90100-C