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.