Quadratic programming with one negative eigenvalue is NP-hard
- Cite this article as:
- Pardalos, P.M. & Vavasis, S.A. J Glob Optim (1991) 1: 15. doi:10.1007/BF00120662
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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.