Distributional robustness analysis for polynomial uncertainty

This paper addresses the problem of computing the worst-case expected value of a polynomial function, over a class of admissible distributions. It is shown that this problem, for the class of distributions considered, is equivalent to a convex optimization problem for which efficient linear matrix inequality (LMI) relaxations are available. In case that the performance function is continuous (not necessarily polynomial), the worst-case expected value can be approximated by using its polynomial approximations. Moreover, the proposed approach is applied to compute hard bounds of the worst-case probability of a polynomial being negative. Numerical examples are presented which illustrate the application of the results in this paper.