Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals

Abstract

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.