Distributionally robust optimization through the lens of submodularity (2312.04890v2)

Abstract: Distributionally robust optimization is used to tackle decision making problems under uncertainty where the distribution of the uncertain data is ambiguous. Many ambiguity sets have been proposed for continuous uncertainty that build on convexity and for which the resulting formulations scale polynomially in the number of random variables. However fewer ambiguity sets have been proposed for discrete uncertainty where the exact formulations scale polynomially in the number of random variables. Towards this, we define a submodular ambiguity set and showcase its expressive power in modeling both discrete and continuous uncertainty. With discrete uncertainty, we show that a class of distributionally robust optimization problems is solvable in polynomial time by viewing it through the lens of submodularity. With continuous uncertainty, we show that it is solvable approximately up to an additive error in pseudo-polynomial time. We then focus on a specific class of submodular ambiguity sets where univariate marginal information and bivariate dependence information on the random vector is specified and provide an exact reformulation as a polynomial sized linear program when the uncertainty is discrete and as a polynomial sized semidefinite program when the uncertainty is continuous. We provide numerical evidence of the modeling flexibility and expressive power of the submodular ambiguity set and demonstrate its applicability in two examples: project networks and multi-newsvendor problems. The paper highlights that the submodular ambiguity set is the natural discrete counterpart of the convex ambiguity set and supplements it for continuous uncertainty, both in modeling and computation.