Abstract. Let f, f 1 , . . . , fs be n-variate polynomials with rational coefficients of maximum degree D and let V be the set of common complex solutions of F = (f 1 , . . . , fs). We give an algorithm which, up to some regularity assumptions on F, computes an exact representation of the global infimum f ⋆ of the restriction of the map x → f (x) to V ∩ R n , i.e. a univariate polynomial vanishing at f ⋆ and an isolating interval for f ⋆ . Furthermore, it decides whether f ⋆ is reached and if so, it returnsThis algorithm is probabilistic. It makes use of the notion of polar varieties. Its complexity is essentially cubic in (sD) n and linear in the complexity of evaluating the input. This fits within the best known deterministic complexity class D O(n) .We report on some practical experiments of a first implementation that is available as a Maple package. It appears that it can tackle global optimization problems that were unreachable by previous exact algorithms and can manage instances that are hard to solve with purely numeric techniques. As far as we know, even under the extra genericity assumptions on the input, it is the first probabilistic algorithm that combines practical efficiency with good control of complexity for this problem.

International audienceLet $f_1,\dots,f_p$ be in $\Q[\bfX]$, where $\bfX=(X_1,\dots,X_n)^t$, that generate a radical ideal and let $V$ be their complex zero-set. Assume that $V$ is smooth and equidimensional. Given $f\in\Q[\bfX]$ bounded below, consider the optimization problem of computing $f^\star=\inf_{x\in V\cap\R^n} f(x)$. For $\bfA\in GL_n(\C)$, we denote by $f^\bfA$ the polynomial $f(\bfA\bfX)$ and by $V^\bfA$ the complex zero-set of $f_1^\bfA,\ldots,f_p^\bfA$. We construct families of polynomials ${\sf M}^\bfA_0, \ldots, {\sf M}^\bfA_d$ in $\Q[\bfX]$: each ${\sf M}_i^\bfA$ is related to the section of a linear subspace with the critical locus of a linear projection. We prove that there exists a non-empty Zariski-open set $\mathscr{O}\subset GL_n(\C)$ such that for all $\bfA\in \mathscr{O}\cap GL_n(\Q)$, $f(x)$ is non-negative for all $x\in V\cap\R^n$ if, and only if, $f^\bfA$ can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal $\langle {\sf M}^\bfA_i\rangle$, for $0\leq i \leq d$. Hence, we can obtain algebraic certificates for lower bounds on $f^\star$ using semidefinite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in ${\sf M}^\bfA_i$