Journal of Global Optimization

, Volume 75, Issue 3, pp 683–733

# Solving the equality-constrained minimization problem of polynomial functions

• Shuijing Xiao
• Guangxing Zeng
Article

## 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)
2. 2.
Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, New York (1998)
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)
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)
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)
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)
7. 7.
Heck, A.: Introduction to Maple. Springer, New York (1993)
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)
10. 10.
Nie, J.: Polynomial optimization with real varieties. SIAM J. Optim. 23, 1634–1646 (2013)
11. 11.
Putinar, M.: Positive polynomials on compact semi-algebraic set. Ind. Univ. Math. J. 42, 203–206 (1993)
12. 12.
Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. AAECC 9, 433–461 (1999)
13. 13.
Schweighofer, M.: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17, 920–942 (2006)
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)
16. 16.
Wu, W.T.: Mathematics Mechanization: Mechanical Geometry Theorem-Proving. Mechanical Geometry Problem-Solving and Polynomial Equations-Solving. Science Press, Beijing (2000)
17. 17.
Xiao, S.J., Zeng, G.X.: Equality-constrained minimization of polynomial functions. Sci. China Math. 58, 2181–2204 (2015)
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)
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)