  We consider the semi-infinite optimization problem: $f^*:=\min_{x\in
X}\:\{f(x): g(x,y)\,\leq \,0,\:\forally\in Y_x\}$, where $f,g$ are polynomials
and $X\subset R^n$ as well as $Y_\x\subset R^p$, $x\in X$, are compact basic
semi-algebraic sets. To approximate $f^*$ we proceed in two steps. First, we
use the "joint+marginal" approach of the author to approximate from above the
function $x\mapsto\Phi(x)=\sup \{g(x,y): y\in Y_x\}$ by a polynomial
$\Phi_d\geq\Phi$, of degree at most $2d$, with the strong property that
$\Phi_d$ converges to $\Phi$ for the $L_1$-norm, as $d\to\infty$ (and in
particular, almost uniformly for some subsequence $(d_\ell)$, $\ell\in\N$).
Then we solve the polynomial optimization problem $f^*_d=\min_{x\in X} \{f(x):
\Phi_d(x)\leq0\}$ via a (by now standard) hierarchy of semidefinite
relaxations. It turns out that the optimal value $f^*_d\geq f^*$ converges to
$f^*$ as $d\to\infty$. In practice we let $d$ be fixed, small, and relax the
constraint $\Phi_d\leq0$ to $\Phi_d(x)\leq\epsilon$ with $\epsilon>0$, allowing
to change $\epsilon$ dynamically.
