Journal of Global Optimization

, Volume 1, Issue 1, pp 15–22

Quadratic programming with one negative eigenvalue is NP-hard


  • Panos M. Pardalos
    • Department of Computer SciencePennsylvania State University
  • Stephen A. Vavasis
    • Department of Computer ScienceCornell University

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 optimizationquadratic programmingNP-hard
Download to read the full article text

Copyright information

© Kluwer Academic Publishers 1991