Distributionally robust polynomial chance-constraints under mixture ambiguity sets

Given $X \subset R^n$, $\varepsilon \in (0,1)$, a parametrized family of probability distributions $(\mu\_{a})\_{a\in A}$ on $\Omega\subset R^p$, we consider the feasible set $X^*\_\varepsilon\subset X$ associated with the {\em distributionally robust} chance-constraint \[X^*\_\varepsilon\,=\,\{x \in X :\:{\rm Prob}\_\mu[f(x,\omega)\,>\,0]> 1-\varepsilon,\,\forall\mu\in M\_a\},\]where $M\_a$ is the set of all possibles mixtures of distributions $\mu\_a$, $a\in A$.For instance and typically, the family$M\_a$ is the set of all mixtures ofGaussian distributions on $R$ with mean and standard deviation $a=(a,\sigma)$ in some compact set $A\subset R^2$.We provide a sequence of inner approximations $X^d\_\varepsilon=\{x\in X: w\_d(x)