Distributional Robustness Analysis for Nonlinear Uncertainty Structures

One of the main objectives of this note is to address the question: what is the worst-case expected value of a continuous function (worst-case performance) over a class of admissible distributions? In this note, the class of symmetric and non-increasing distributions is considered and results are provided for the class of so-called semi-algebraic functions. The first part of the note shows that, for the class of distributions considered, it suffices to solve a convex optimization problem for which efficient linear matrix inequality (LMI) relaxations are available. Secondly, the proposed approach is applied to estimate hard bounds on the worst-case probability of a semi-algebraic function being negative. Several numerical examples are presented which illustrate the effectiveness of the approach presented.