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 R be the field of real numbers, and R[x(1),...,x(n)] the ring of polynomials over R in variables x(1),...,x(n). For an f is an element of R[x(1),...,x(n)] and a finite subset H of R[x(1),...,x(n)], denote by V (f : H) the set {f ((a) over bar) vertical bar (a) over bar is an element of R-n, and h((alpha) over bar) = 0, for all h is an element of H}. In this paper, we provide some effective algorithms for computing the accurate value of the infimum inf V (f : H) of V (f : H), deciding whether or not the constrained infimum inf V (f : H) is attained when inf V (f : H) not equal +/-infinity, and finding a point for the constrained minimum min V (f : H) if 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.