SOS approximation of polynomials nonnegative on a real algebraic set

Let $V\subset R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,j\in J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $\epsilon>0$, there exist nonnegative scalars $\{\lambda_j\}_{j\in J}$ such that, for all $r$ sufficiently large, $f+\epsilon\theta_r+\sum_{j\in J} \lambda_j g_j^2$ is a sum of squares. Here, $\theta_r$ is the (polynomial) truncation up to degree $r$ in the series expansion of $\sum_i exp{x_i^2}$. This representation is an obvious certificate of nonnegativity of $f$ on $V$, and very specific in terms of the $g_j$ that define the set $V$. In particular, it is valid with {\it no} assumption on $V$. Finally, this representation is also useful from a computation point of view, as we can define semidefinite programing relaxations to approximate the global minimum of $f$ on a real algebraic set $V$, or a basic closed semi-algebraic set $K$, and again, with {\it no} assumption on $V$ or $K$.


SIAM J. Optimization 16 (2005), 610--628.