Distributionally robust chance constraints for non-linear uncertainties

This paper investigates the computational aspects of distributionally robust chance constrained optimization problems. In contrast to previous research that mainly focused on the linear case (with a few exceptions discussed in detail below), we consider the case where the constraints can be non-linear to the decision variable, and in particular to the uncertain parameters. This formulation is of great interest as it can model non-linear uncertainties that are ubiquitous in applications. Our main result shows that distributionally robust chance constrained optimization is tractable, provided that the uncertainty is characterized by its mean and variance, and the constraint function is concave in the decision variables, and quasi-convex in the uncertain parameters. En route, we establish an equivalence relationship between distributionally robust chance constraint and the robust optimization framework that models uncertainty in a deterministic manner. This links two broadly applied paradigms in decision making under uncertainty and extends previous results of the same spirit in the linear case to more general cases. We then consider probabilistic envelope constraints, a generalization of distributionally robust chance constraints first proposed in Xu et al. [40] for the linear case. We extend this framework to the non-linear case, and derive sufficient conditions that guarantee its tractability. Finally, we investigate tractable approximations of joint probabilistic envelope constraints, and provide the conditions when these approximation formulations are tractable.