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 $X^d\_\varepsilon=\{x\in X: w\_d(x) <\varepsilon\}$, $d\in N$, where $w\_d$ is a polynomial of degree $d$ whosevector of coefficients is an optimal solution of a semidefinite program.The size of the latter increases with the degree $d$. We also obtain the strong and highly desirable asymptotic guarantee that $\lambda(X^*\_\varepsilon\setminus X^d\_\varepsilon)\to0$as $d$ increases, where $\lambda$ is the Lebesgue measure on $X$. Same resultsare also obtained for the more intricated case of distributionally robust "joint" chance-constraints.