Distributionally robust optimization using optimal transport for Gaussian mixture models

Distributionally robust optimization (DRO) is an increasingly popular approach for optimization under uncertainty when the probability distribution of the uncertain parameter is unknown. Well-explored DRO approaches in literature, such as Wasserstein DRO, do not make any specific assumptions on the nature of the candidate distributions considered in the ambiguity set. However, in many practical applications, the uncertain parameter may be sourced from a distribution that can be well modeled as a Gaussian Mixture Model (GMM) whose components represent the different subpopulations the uncertain parameter may belong to. In this work, we propose a new DRO method based on an ambiguity set constructed around a GMM. The proposed DRO approach is illustrated on a numerical example as well as a portfolio optimization case study for uncertainty sourced from various distributions. The results obtained from the proposed DRO approach are compared with those from Wasserstein DRO, and are shown to be superior in quality with respect to out-of-sample performance.