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Some NP-complete problems in quadratic and nonlinear programming

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Abstract

In continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether

  1. (a)

    a given feasible solution is not a local minimum, and

  2. (b)

    the objective function is not bounded below on the set of feasible solutions.

We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.

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Author information

Authors and Affiliations

  1. Department of Industrial and Operations Engineering, The University of Michigan, 1205 Beal Avenue, 48109-2117, Ann Arbor, MI, USA

    Katta G. Murty

  2. Faculty of Administration, University of New Brunswick, E3B 5A6, Fredericton, NB, Canada

    Santosh N. Kabadi

Authors
  1. Katta G. Murty
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  2. Santosh N. Kabadi
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Additional information

Research partially supported by NSF Grants No. ECS-8401081 and ECS-8521183.

Research partially supported by NSERC (Canada) Grant No. A8085.

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Murty, K.G., Kabadi, S.N. Some NP-complete problems in quadratic and nonlinear programming. Mathematical Programming 39, 117–129 (1987). https://doi.org/10.1007/BF02592948

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  • DOI: https://doi.org/10.1007/BF02592948

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