Mathematical Programming

, Volume 39, Issue 2, pp 117–129

Some NP-complete problems in quadratic and nonlinear programming

  • Katta G. Murty
  • Santosh N. Kabadi

DOI: 10.1007/BF02592948

Cite this article as:
Murty, K.G. & Kabadi, S.N. Mathematical Programming (1987) 39: 117. doi:10.1007/BF02592948


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.

Key words

Nonconvex nonlinear programminglocal minimumglobal minimumcopositive matricesNP-complete

Copyright information

© The Mathematical Programming Society, Inc. 1987

Authors and Affiliations

  • Katta G. Murty
    • 1
  • Santosh N. Kabadi
    • 2
  1. 1.Department of Industrial and Operations EngineeringThe University of MichiganAnn ArborUSA
  2. 2.Faculty of AdministrationUniversity of New BrunswickFrederictonCanada