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In this paper we wish to tackle stochastic programs affected by ambiguityabout the probability law that governs their uncertain parameters. Usingoptimal transport theory, we construct an ambiguity set that exploits theknowledge about the distribution of the uncertain parameters, which is providedby: (1) sample data and (2) a-priori information on the order among theprobabilities that the true data-generating distribution assigns to someregions of its support set. This type of order is enforced by means of ordercone constraints and can encode a wide range of information on the shape of theprobability distribution of the uncertain parameters such as informationrelated to monotonicity or multi-modality. We seek decisions that aredistributionally robust. In a number of practical cases, the resultingdistributionally robust optimization (DRO) problem can be reformulated as afinite convex problem where the a-priori information translates into linearconstraints. In addition, our method inherits the finite-sample performanceguarantees of the Wasserstein-metric-based DRO approach proposed by MohajerinEsfahani and Kuhn (Math Program 171(1-2):115-166.https://doi.org/10.1007/s10107-017-1172-1, 2018), while generalizing this andother popular DRO approaches. Finally, we have designed numerical experimentsto analyze the performance of our approach with the newsvendor problem and theproblem of a strategic firm competing \`a la Cournot in a market.

Partition-based Distributionally Robust Optimization via Optimal Transport with Order Cone Constraints