Journal of Systems Science and Complexity

, Volume 26, Issue 6, pp 1028–1046 | Cite as

Semi-algebraically connected components of minimum points of a polynomial function

  • Shuijing XiaoEmail author
  • Guangxing Zeng


In a recent article, the authors provided an effective algorithm for both computing the global infimum of f and deciding whether or not the infimum of f is attained, where f is a multivariate polynomial over the field R of real numbers. As a complement, the authors investigate the semi-algebraically connected components of minimum points of a polynomial function in this paper. For a given multivariate polynomial f over R, it is shown that the above-mentioned algorithm can find at least one point in each semi-algebraically connected component of minimum points of f whenever f has its global minimum.

Key words

Global minimum minimum point polynomial optimization rational univariate representation (RUR) semi-algebraically connected component strictly critical point 


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  1. [1]
    Hägglöf K, Lindberg P O, and Stevenson L, Computing global minima to polynomial optimization problems using Gröbner bases, J. Global Optimization, 1995, 7: 115–125.CrossRefzbMATHGoogle Scholar
  2. [2]
    Uteshev A Y and Cherkasov T M, The search for the maximum of a polynomial, J. Symbolic. Comput., 1998, 25: 587–618.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Parrilo P A and Sturmfels B, Minimizing polynomial functions, Algorithmic and Quantitative Real Algebraic Geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 60, Amer. Math. Soc., Providence, 2003, 83–100.Google Scholar
  4. [4]
    Hanzon B and Jibetean D, Global minimization of a multivariate polynomial using matrix methods, J. Global Optimization, 2003, 27: 1–23.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Nie J, Demmel J, and Sturmfels B, Minimizing polynomials via sums of squares over the gradient ideal, Math. Program Ser. A, 2006, 106: 587–606.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Guo F, EI Din M S, and Zhi L H, Global optimization of polynomials using generalized critical values and sums of squares, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ACM, Munich, Germany, 2010.Google Scholar
  7. [7]
    Xiao S J and Zeng G X, Algorithms for computing the global infimum and minimum of a polynomial function, Sci. Sin. Math., 2011, 41(9): 759–788 (in Chinese).CrossRefGoogle Scholar
  8. [8]
    Xiao S J and Zeng G X, Algorithms for computing the global infimum and minimum of a polynomial function, Sci. China Math., 2012, 55(4): 881–891.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Wu W T, Mathematics Mechanization: Mechanical Geometry Theorem-Proving, Mechanical Geometry Problem-Solving and Polynomial Equations-Solving, Science Press/Kluwer Academic Publishers, Beijing/Dordrecht-Boston-London, 2000.Google Scholar
  10. [10]
    Zeng G X and Xiao S J, Computing the rational univariate representations for zero-dimensional systems by Wu’s method, Sci. Sin. Math., 2010, 40(10): 999–1016 (in Chinese).MathSciNetGoogle Scholar
  11. [11]
    Bochnak J, Coste M, and Roy M F, Real Algebraic Geometry, Springer-Verlag, New York-Berlin-Heidelberg, 1998.zbMATHGoogle Scholar
  12. [12]
    Basu S, Pollack R, and Roy M F, Algorithms in Real Algebraic Geometry, Algorithms and Computation in Math., Vol. 10, Springer-Verlag, Berlin, 2003.CrossRefzbMATHGoogle Scholar
  13. [13]
    Zeng G X and Zeng X N, An effective decision method for semidefinite polynomials, J. Symbolic Comput., 2004, 37: 83–99.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Mishra B, Algorithmic Algebra, Texts and Monographs in Computer Science, Springer-Verlag, New York-Berlin-Heidelberg, 1993.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangChina

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