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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. {\it 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 {\it M. K. Kozlov, S. P. Tarasov} and {\it 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.

90C20Quadratic programming
90C60Abstract computational complexity for mathematical programming problems
Full Text: DOI
[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. · Zbl 0411.68039
[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. · Zbl 0434.90071
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[6] Pardalos, P. M. and Rosen, J. B. (1987), Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science 268, Springer-Verlag, Berlin. · Zbl 0638.90064
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[8] Sahni, S. (1974), Computationally Related Prolems, SIAM J. Comput. 3, 262-279. · Zbl 0292.68020 · 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