Distributionally robust polynomial chance-constraints under mixture ambiguity sets

Abstract

Given $X\subset R^n$, $\epsilon \in (0,1)$, a parametrized family of probability distributions $(\mu \_a)\_a\in A$ on $\Omega \subset R^p$, we consider the feasible set $X^{\ast }\_\epsilon \subset X$ associated with the {\em distributionally robust} chance-constraint \[X^*\_\varepsilon\,=\,\{x \in X :\:{\rm Prob}\_μ[f(x,ω)\,>\,0]> 1-\varepsilon,\,\forallμ\in M\_a\},\]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 $X^d\_\epsilon =\{x\in X:w\_d(x)<\epsilon \}$, $d\in N$, 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 $\lambda (X^{\ast }\_\epsilon \setminus X^d\_\epsilon )\to 0$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.