Solving the equality-constrained minimization problem of polynomial functions

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.