Convexifying positive polynomials and a proximity algorithm

Résumé

We prove that if $f$ is a positive $C^2$ function on a convex compact set $X$ then it becomes strongly convex when multiplied by $(1+|x|^2)^N$ with $N$ large enough. For $f$ polynomial we give an explicit estimate for $N$, which depends on the size of the coefficients of $f$ and on the lower bound of $f$ on$X$. As an application of our convexification method we propose an algorithm which for a given polynomial $f$ on a convex compact semialgebraic set $X$ produces a sequence (starting from an arbitrary point in $X$) which converges to a (lower) critical point of $f$ on $X$. The convergence is based on the method of talweg which is a generalization of the Lojasiewicz gradient inequality. (Joint work with S. Spodzieja).