Journal of Global Optimization

, Volume 1, Issue 1, pp 15–22

Quadratic programming with one negative eigenvalue is NP-hard

  • Panos M. Pardalos
  • Stephen A. Vavasis

DOI: 10.1007/BF00120662

Cite this article as:
Pardalos, P.M. & Vavasis, S.A. J Glob Optim (1991) 1: 15. doi:10.1007/BF00120662


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 [8] showed that quadratic programming with a negative definite quadratic term (n negative eigenvalues) is NP-hard, whereas Kozlov, Tarasov and Hačijan [2] 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 words

Global optimization quadratic programming NP-hard 

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Panos M. Pardalos
    • 1
  • Stephen A. Vavasis
    • 2
  1. 1.Department of Computer SciencePennsylvania State UniversityUniversity ParkU.S.A.
  2. 2.Department of Computer ScienceCornell UniversityIthacaU.S.A.

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