Representation of distributionally robust chance-constraints

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)