Decision Making with Side Information: A Causal Transport Robust Approach

We consider stochastic optimization with side information where, prior to decision making, covariate data are available to inform better decisions. To hedge against data uncertainty while capturing the information structure revealed from the conditional distribution of random problem parameters given the covariate values, we propose a distributionally robust formulation based on causal transport distance. We derive a dual reformulation for evaluating the worst-case expected cost and show that the worst-case distribution in a causal transport distance ball preserves the conditional information structure from the nominal distribution. When optimizing over affine decision rules, we identify cases in which the overall problem can be solved via convex programming. When optimizing over all (nonparametric) decision rules, we identify a new class of robust optimal decision rules when the cost function is convex with respect to a one-dimensional decision variable.

Funding: N. Chen’s research is partially supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant] and the Institute for Management & Innovation [Research Grant]. M. Hu’s research is supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-04295].

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2024.0997.