A study of data-driven distributionally robust optimization with incomplete joint data under finite support

Missing data is a common issue for many practical data-driven stochastic programming problems. The state-of-the-art approaches first estimate the missing data values and then separately solve the corresponding stochastic programming. Accurate estimation of missing values is typically inaccessible as it requires enormous data and sophisticated statistical methods. Therefore, this paper proposes an integrated approach, a distributionally robust optimization (DRO) framework, that simultaneously tackles the missing data problem and data-driven stochastic optimization by hedging against the uncertainties of the missing values. This paper adds to the DRO literature by considering the practical scenario where the data can be incomplete and partially observable; it particularly focuses on data distributions with finite support. We construct several classes of ambiguity sets for our DRO model utilizing the incomplete data sets, maximum likelihood estimation method, and different metrics. We prove the statistical consistency and finite sample guarantees of the corresponding models and provide tractable reformulations of our model for different scenarios. We perform computational studies on the multi-item inventory control problem and portfolio optimization using synthetic and real-world data. We validate that our method outperforms the traditional estimate-then-optimized approaches.