This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1,..., xn] the ring of polynomials over R in variables x1,..., xn. For an f ∈ R[x1,..., xn] and a finite subset H of R[x1,..., xn], denote by V(f : H) the set {f( ˉα) | ˉα∈ Rn, and h( ˉα) =0, ? h ∈ H}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum inf V(f : H) of V(f : H) is a root of some polynomial in U whenever inf V(f : H) = ±∞.The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.
By catching the so-called strictly critical points,this paper presents an effective algorithm for computing the global infimum of a polynomial function.For a multivariate real polynomial f ,the algorithm in this paper is able to decide whether or not the global infimum of f is finite.In the case of f having a finite infimum,the global infimum of f can be accurately coded in the Interval Representation.Another usage of our algorithm to decide whether or not the infimum of f is attained when the global infimum of f is finite.In the design of our algorithm,Wu’s well-known method plays an important role.
