Online data assimilation in distributionally robust optimization

This paper considers a class of real-time decision making problems to minimize the expected value of a function that depends on a random variable xi under an unknown distribution P. In this process, samples of xi are collected sequentially in real time, and the decisions are made, using the real-time data, to guarantee out-of-sample performance. We approach this problem in a distributionally robust optimization framework and propose a novel ONLINE DATA ASSIMILATION ALGORITHM for this purpose. This algorithm guarantees the out-of-sample performance in high probability, and gradually improves the quality of the data-driven decisions by incorporating the streaming data. We show that the ONLINE DATA ASSIMILATION ALGORITHM guarantees convergence under the streaming data, and a criteria for termination of the algorithm after certain number of data has been collected.