Representation of chance-constraints with strong asymptotic guarantees

Given $ϵ\in (0,1)$, a probability measure $\mu$ on $\mathrm{\Omega }\subset {\mathbb{R}}^p$ and a semi-algebraic set $K\subset X\times \mathrm{\Omega }$, we consider the feasible set $X^\epsilon={x\in X:{\rm Prob}[(x,\omega)\in K]\geq 1-\epsilon}$ associated with a chance-constraint. We provide a sequence of outer approximations $X_{ϵ}^d=\{x\in X:h_d(x)\ge 0\}$, $d\in \mathbb{N}$, where $h_d$ is a polynomial of degree $d$ whose vector 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 $\lambda(X^d\epsilon\setminus X^_\epsilon)\to0$as$d$increases,where$\lambda$istheLebesguemeasureon$X$. Inner approximations with same guarantees are also obtained.