Journal of Global Optimization

, Volume 1, Issue 1, pp 15-22

First online:

Quadratic programming with one negative eigenvalue is NP-hard

  • Panos M. PardalosAffiliated withDepartment of Computer Science, Pennsylvania State University
  • , Stephen A. VavasisAffiliated withDepartment of Computer Science, Cornell University

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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 [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