- SOS approximation of polynomials nonnegative on a real algebraic set Jean B. Lasserre (lasserrelaas.fr) Abstract: 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 $K$. Keywords: Real algebraic geometry; positive polynomials; sum of squares; semidefinite programming. Category 1: Global Optimization (Theory ) Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization ) Citation: SIAM J. Optimization 16 (2005), 610--628. Download: Entry Submitted: 09/23/2005Entry Accepted: 09/24/2005Entry Last Modified: 09/18/2006Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.