A summary of the paper can be given by the authors’ conclusion as follows.
“Computing or even approximating the feasible set associated with a distributionally robust chance-constraint is a challenging problem. We have described a systematic numerical scheme which provides a monotone sequence (a hierarchy) of inner approximations, all in the form \(\{\mathbf{x} \in \mathbf{X} : w_d (\mathbf{x}) < \varepsilon\}\) for some polynomial of increasing degree \(d\), with strong asymptotic guarantees as \(d\) increases. To the best of our knowledge it is the first result of this type at this level of generality. Of course this comes with a price as the polynomial which defines each approximation is obtained by solving a semidefinite program whose size increases with its degree. Therefore and so far, this approach is limited to problems of small dimension (except perhaps if some sparsity can be exploited). So in its present form this contribution should be considered as complementary (rather than a competitor) to other algorithmic approaches where scalability is of primary importance. However it may also provide useful insights and a benchmark (for small dimension problems) for the latter approaches.”
With some ad-hoc adjustments, the framework presented in this paper can be extended to consider problems with only first- and second-order moments knowledge (no information about the distributions contributing to the mixture) and/or distributionally robust joint chance-constraints.
The approach is a non trivial extension of the numerical scheme proposed in [J. B. Lasserre, “Representation of chance-constraints with strong asymptotic properties”, IEEE Control Syst. Lett. 1, 50–55 (2017)].