Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals

This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set image 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.