Advertisement

Journal of Global Optimization

, Volume 75, Issue 3, pp 683–733 | Cite as

Solving the equality-constrained minimization problem of polynomial functions

  • Shuijing Xiao
  • Guangxing ZengEmail author
Article
  • 92 Downloads

Abstract

The purpose of this paper is to solve the equality-constrained minimization problem of polynomial functions. Let \({\mathbb {R}}\) be the field of real numbers, and \({\mathbb {R}}[x_1,\ldots ,x_n]\) the ring of polynomials over \({\mathbb {R}}\) in variables \(x_1\), ..., \(x_n\). For an \(f\in {\mathbb {R}}[x_1,\ldots ,x_n]\) and a finite subset H of \({\mathbb {R}}[x_1,\ldots ,x_n]\), denote by \({\mathscr {V}}(f:H)\) the set \(\{f({\bar{\alpha }})\mid {\bar{\alpha }}\in {\mathbb {R}}^n, \hbox { and }h({\bar{\alpha }})=0,\,\forall h\in H\}\). In this paper, we provide some effective algorithms for computing the accurate value of the infimum \(\inf {\mathscr {V}}(f:H)\) of \({\mathscr {V}}(f:H)\), deciding whether or not the constrained infimum \(\inf {\mathscr {V}}(f:H)\) is attained when \(\inf {\mathscr {V}}(f:H)\ne \pm \infty \), and finding a point for the constrained minimum \(\min {\mathscr {V}}(f:H)\) if \(\inf {\mathscr {V}}(f:H)\) is attained. With the aid of the computer algebraic system Maple, our algorithms have been compiled into a general program to treat the equality-constrained minimization of polynomial functions with rational coefficients.

Keywords

Polynomial function Equality-constrained minimization Infimum Attainability Minimum point Triangular decomposition Revised resultant Transfer principle 

Mathematics Subject Classification

90C30 68W30 12J15 12F10 

Notes

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11561046, 11161034). The authors are very grateful to the referees for their valuable suggestions that helped to improve this paper.

References

  1. 1.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Math. 10. Springer, Berlin (2003)CrossRefGoogle Scholar
  2. 2.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, New York (1998)CrossRefGoogle Scholar
  3. 3.
    Gonzales-Vega, L., Rouillier, F., Roy, M.-F.: Symbolic recipes for polynomial system solving. In: Cohen, A.M., Cuypers, H., Sterk, H. (eds.) Some Tapas of Computer Algebra, pp. 34–65. Springer, New York (1999)CrossRefGoogle Scholar
  4. 4.
    Greuet, A., Guo, F., El Din, S.M., Zhi, L.: Global optimization of polynomials restricted to a smooth variety using sums of squares. J. Symb. Comput. 47, 503–518 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Greuet, A., El Din, S.M.: Probabilistic algorithm for the global optimization of a polynomial over a real algebraic set. SIAM J. Optim. 24, 1313–1343 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Há, H.V., Pham, T.S.: Solving polynomial optimization problems via the truncated tangency variety and sums of squares. J. Pure Appl. Algebra 213, 2167–2176 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Heck, A.: Introduction to Maple. Springer, New York (1993)CrossRefGoogle Scholar
  8. 8.
    Mishra, B.: Algorithmic Algebra. Texts and Monographs in Computer Science. Springer, New York (1993)Google Scholar
  9. 9.
    Nie, J.: An exact Jacobian SDP relaxation for polynomial optimization. Math. Program. Ser. A 137, 225–255 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nie, J.: Polynomial optimization with real varieties. SIAM J. Optim. 23, 1634–1646 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Putinar, M.: Positive polynomials on compact semi-algebraic set. Ind. Univ. Math. J. 42, 203–206 (1993)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. AAECC 9, 433–461 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Schweighofer, M.: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17, 920–942 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wang, D.K.: The software wsolve: a Maple package for solving system of polynomial equations (2008). http://www.mmrc.iss.ac.cn/~dwang/wsolve.html
  15. 15.
    Wu, W.T.: On zeros of algebraic equations—an application of Ritt principle. Kexue Tongbao 31, 1–5 (1986)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Wu, W.T.: Mathematics Mechanization: Mechanical Geometry Theorem-Proving. Mechanical Geometry Problem-Solving and Polynomial Equations-Solving. Science Press, Beijing (2000)zbMATHGoogle Scholar
  17. 17.
    Xiao, S.J., Zeng, G.X.: Equality-constrained minimization of polynomial functions. Sci. China Math. 58, 2181–2204 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Xiao, S.J., Zeng, G.X.: Algorithms for computing the global infimum and minimum of a polynomial function. Sci. China Math. 55, 881–891 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zeng, G.X., Xiao, S.J.: Computing the rational univariate representations for zero-dimensional systems by Wu’s method (in Chinese). Sci. Sin. Math. 40, 999–1016 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangChina

Personalised recommendations