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Quadratic programming with one negative eigenvalue is NP-hard

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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.

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Authors and Affiliations

  1. Department of Computer Science, Pennsylvania State University, 16802, University Park, PA, U.S.A.

    Panos M. Pardalos

  2. Department of Computer Science, Cornell University, 14853, Ithaca, NY, U.S.A.

    Stephen A. Vavasis

Authors
  1. Panos M. Pardalos
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  2. Stephen A. Vavasis
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Additional information

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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Pardalos, P.M., Vavasis, S.A. Quadratic programming with one negative eigenvalue is NP-hard. J Glob Optim 1, 15–22 (1991). https://doi.org/10.1007/BF00120662

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  • DOI: https://doi.org/10.1007/BF00120662

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