A typical data-driven stochastic program aims to seek the best decision that minimizes the sum of a deterministic cost function and an expected recourse function under a given distribution. Recently, much success has been witnessed in the development of Distributionally Robust Optimization (DRO), which considers the worst-case expected recourse function under the least favorable probability distribution from a distributional family. However, in the presence of endogenous outlier scenarios such that their corresponding recourse function values are extremely large or even infinite, the commonly-used DRO framework alone tends to over-emphasize these outliers and cause undesirable or even infeasible decisions. On the contrary, Distributionally Favorable Optimization (DFO), concerning the best-case expected recourse function under the most favorable distribution from the distributional family, can serve as a proper measure of the stochastic recourse function and mitigate the effect of endogenous outliers. We show that DFO recovers many robust statistics, echoing that the DFO framework might be appropriate for the stochastic recourse function in the presence of endogenous outliers. While being NP-hard, in general, many DFO models are shown to be mixed-integer convex programming representable (MICP-R). A notion of decision outlier robustness is proposed for selecting a DFO framework for data-driven optimization with outliers. We also provide a unified way to integrate DRO with DFO, where DRO addresses the out-of-sample performances, and DFO properly measures the stochastic recourse function under endogenous outliers. We further extend the proposed DFO framework to solve two-stage stochastic programs without relatively complete recourse. The numerical study confirms the promising of the framework.
Jiang, N., Xie, W. (2021). Distributionally Favorable Optimization: A Framework for Data-driven Decision-making with Endogenous Outliers. Available at Optimization Online.