Solving the equality-constrained minimization problem of polynomial functions

The purpose of this paper is to solve the equality-constrained minimization problem of polynomial functions. Let $${\mathbb {R}}$$ R be the field of real numbers, and $${\mathbb {R}}[x_1,\ldots ,x_n]$$ R [ x 1 , … , x n ] the ring of polynomials over $${\mathbb {R}}$$ R in variables $$x_1$$ x 1 , ..., $$x_n$$ x n . For an $$f\in {\mathbb {R}}[x_1,\ldots ,x_n]$$ f ∈ R [ x 1 , … , x n ] and a finite subset H of $${\mathbb {R}}[x_1,\ldots ,x_n]$$ R [ x 1 , … , x n ] , denote by $${\mathscr {V}}(f:H)$$ V ( f : H ) the set $$\{f({\bar{\alpha }})\mid {\bar{\alpha }}\in {\mathbb {R}}^n, \hbox { and }h({\bar{\alpha }})=0,\,\forall h\in H\}$$ { f ( α ¯ ) ∣ α ¯ ∈ R n , and h ( α ¯ ) = 0 , ∀ h ∈ H } . In this paper, we provide some effective algorithms for computing the accurate value of the infimum $$\inf {\mathscr {V}}(f:H)$$ inf V ( f : H ) of $${\mathscr {V}}(f:H)$$ V ( f : H ) , deciding whether or not the constrained infimum $$\inf {\mathscr {V}}(f:H)$$ inf V ( f : H ) is attained when $$\inf {\mathscr {V}}(f:H)\ne \pm \infty $$ inf V ( f : H ) ≠ ± ∞ , and finding a point for the constrained minimum $$\min {\mathscr {V}}(f:H)$$ min V ( f : H ) if $$\inf {\mathscr {V}}(f:H)$$ inf 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.