Multi-stage distributionally robust convex stochastic optimization with Bayesian-type ambiguity sets

The existent methods for constructing ambiguity sets in distributionally robust optimization often suffer from over-conservativeness and inefficient utilization of available data. To address these limitations and to practically solve multi-stage distributionally robust optimization (MDRO), we propose a data-driven Bayesian-type approach that constructs the ambiguity set of possible distributions from a Bayesian perspective. We demonstrate that our Bayesian-type MDRO problem can be reformulated as a risk-averse multi-stage stochastic programming problem and subsequently investigate its theoretical properties such as consistency, finite sample guarantee, and statistical robustness. Moreover, the reformulation enables us to employ cutting planes algorithms in dynamic settings to solve the Bayesian-type MDRO problem. To illustrate the practicality and advantages of the proposed model and algorithm, we apply it to a distributionally robust inventory control problem and a distributionally robust hydrothermal scheduling problem, and compare it with usual formulations and solution methods to highlight the superior performance of our approach.