Representation of distributionally robust chance-constraints
Résumé
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 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 $X^d_\varepsilon=\{x\in X: w_d(x) <\varepsilon\}$, $d\in N$,
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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