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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
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