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 \[X^*_\varepsilon\,=\,\{x\in X:\:{\rm Prob}_\mu[f(x,\omega)\,>\,0]> 1-\varepsilon,\,\forall\mu\in\mathscr{M}_a\},\] where $\mathscr{M}_a$ is the set of all possibles mixtures of distributions $\mu_a$, $a\in A$. For instance and typically, the family $\mathscr{M}_a$ is the set of all mixtures of Gaussian 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 $X^d_\varepsilon=\{x\in X:w_d(x) <\varepsilon\}$, $d\in \mathbb{N}$, where $w_d$ 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 $\lambda(X^*_\varepsilon\setminus X^d_\varepsilon)\to0$ as $d$ increases, where $\lambda$ is the Lebesgue measure on $X$. Same results are also obtained for the more intricated case of distributionally robust ``joint" chance-constraints.


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