Representations of positive polynomials on non-compact semialgebraic sets via KKT ideals
نویسندگان
چکیده
This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R : g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (Karush-KuhnTucker) 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 have that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x) ≥ 0 on S. This is a generalization of results in [19], which discuss the SOS representations of nonnegative polynomials over gradient ideals.
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