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


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.


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

Mathematics Subject Classification

90C30 68W30 12J15 12F10 



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.


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

Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangChina

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