Quadratic programming with one negative eigenvalue is NP-hard Authors Panos M. Pardalos Department of Computer Science Pennsylvania State University Stephen A. Vavasis Department of Computer Science Cornell University Article

Received: 06 February 1991 DOI :
10.1007/BF00120662

Cite this article as: Pardalos, P.M. & Vavasis, S.A. J Glob Optim (1991) 1: 15. doi:10.1007/BF00120662
Abstract 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 This author's work supported by the Applied Mathematical Sciences Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013. A000 and in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS 8920550.

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© Kluwer Academic Publishers 1991