Journal of Global Optimization

, Volume 7, Issue 2, pp 115–125 | Cite as

Computing global minima to polynomial optimization problems using Gröbner bases

  • K. Hägglöf
  • P. O. Lindberg
  • L. Svensson


The local optimality conditions to polynomial optimization problems are a set of polynomial equations (plus some inequality conditions). With the recent techniques of Gröbner bases one can find all solutions to such systems, and hence also find global optima. We give a short survey of these methods. We also apply them to a set of problems termed ‘with exact solutions unknown’ in the problem sets of Hock and Schittkowski. To these problems we give exact solutions.

Key words

Global optima Gröbner bases polynomial optimization problems 


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  1. 1.
    Bazaraa, M. S., Sherali, H. D. and Shetty, C. M.,Nonlinear Programming, Theory and Algorithms, 2nd Edition, John Wiley and Sons, 1993.Google Scholar
  2. 2.
    Cox, D., Little, J. and O'Shea, D.,Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag, 1992.Google Scholar
  3. 3.
    Czapor, S. R.,Saving Algebraic Equations: Combining Buchberger's Algorithm with Multivariate Factorization, J. Symbolic Computation 7, pp. 49–53, Academic Press Limited, 1989.Google Scholar
  4. 4.
    Hock, W. and Schittkowski, K.,Test Examples for Nonlinear Programming Codes, Springer-Verlag, 1981.Google Scholar
  5. 5.
    Hollman, J.,Theory and Applications of Gröbner Bases, PhD thesis, Royal Institute of Technology, Stockholm, Sweden, 1992.Google Scholar
  6. 6.
    Schittkowski, K., MoreTest Examples for NonLinear Programming Codes, Springer-Verlag, 1987.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • K. Hägglöf
    • 1
  • P. O. Lindberg
    • 1
  • L. Svensson
    • 1
  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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