  We consider the problem of certifying an inequality of the form $f(x)\geq 0$,
$\forall x\in K$, where $f$ is a multivariate transcendental function, and $K$
is a compact semialgebraic set. We introduce a certification method, combining
semialgebraic optimization and max-plus approximation. We assume that $f$ is
given by a syntaxic tree, the constituents of which involve semialgebraic
operations as well as some transcendental functions like $\cos$, $\sin$,
$\exp$, etc. We bound some of these constituents by suprema or infima of
quadratic forms (max-plus approximation method, initially introduced in optimal
control), leading to semialgebraic optimization problems which we solve by
semidefinite relaxations. The max-plus approximation is iteratively refined and
combined with branch and bound techniques to reduce the relaxation gap.
Illustrative examples of application of this algorithm are provided, explaining
how we solved tight inequalities issued from the Flyspeck project (one of the
main purposes of which is to certify numerical inequalities used in the proof
of the Kepler conjecture by Thomas Hales).
