  We consider a property of positive polynomials on a compact set with a small
perturbation. When applied to a Polynomial Optimization Problem (POP), the
property implies that the optimal value of the corresponding SemiDefinite
Programming (SDP) relaxation with sufficiently large relaxation order is
bounded from below by $(f^\ast - \epsilon)$ and from above by $f^\ast +
\epsilon (n+1)$, where $f^\ast$ is the optimal value of the POP. We propose new
SDP relaxations for POP based on modifications of existing sums-of-squares
representation theorems. An advantage of our SDP relaxations is that in many
cases they are of considerably smaller dimension than those originally proposed
by Lasserre. We present some applications and the results of our computational
experiments.
