Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals

This paper studies the representation of a positive polynomial f(x)f(x) on a noncompact semialgebraic set S={x∈Rn:g1(x)≥0,…,gs(x)≥0}S={x∈Rn:g1(x)≥0,…,gs(x)≥0} modulo its KKT (Karush–Kuhn–Tucker) ideal. Under the assumption that the minimum value of f(x)f(x) on SS is attained at some KKT point, we show that f(x)f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)>0f(x)>0 on SS; furthermore, when the KKT ideal is radical, we argue that f(x)f(x) can be represented as a sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)≥0f(x)≥0 on SS. 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.