This paper studies the representation of a positive polynomial f (x) on a noncompact semialgebraic set S = {x ∈ R n : g 1 (x) ≥ 0, . . . , g s (x) ≥ 0} modulo its KKT (Karush-Kuhn-Tucker) ideal. Under the assumption that the minimum value of f (x) on S is attained at some KKT point, we show that f (x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f (x) > 0 on S; furthermore, when the KKT ideal is radical, we argue that f (x) can be represented as a sum of squares (SOS) of polynomials modulo the KKT ideal if f (x) ≥ 0 on S. This is a generalization of results in [J. Nie, J. Demmel, B. Sturmfels, Minimizing polynomials via sum of squares over the gradient ideal, Mathematical Programming (in press)], which discusses the SOS representations of nonnegative polynomials over gradient ideals.
Abstract. This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given:. Explicit degree bounds for s and t are given, in terms of the degrees of p and h and the location of the roots of p. This is a special case of Schmüdgen's Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval [0, ∞) are also considered.
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