Distributionally robust polynomial chance-constraints under mixture ambiguity sets

Abstract: 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.