Globalized Distributionally Robust Optimization with Multi Core Sets

It is essential to capture the true probability distribution of uncertain data in the distributionally robust optimization (DRO). The uncertain data presents multimodality in numerous application scenarios, in the sense that the probability density function of the uncertain data has two or more modes (local maximums). In this study, we propose a globalized distributionally robust optimization framework with multiple core sets (MGDRO) to handle the complicated situation when the uncertain data is multimodal. This framework captures the multimodal structure, via a penalty function composed of the minimum distance from the random vector to all core sets and penalty coefficients. Multiple core sets are constructed to capture all clustering regions of the sample points. The penalty item weakens the impact of the regions outside of the core sets on the expectation of the objective function, thereby highlighting the impact of the multimodality. Under some assumptions, the MGDRO model can be reformulated as tractable semi-definite programs, for both moment-based and metric-based ambiguity sets. We apply the MGDRO models to a multi-product newsvendor problem with multimodal demands. The numerical results turn out that the MGDRO models greatly outperform traditional DRO models and other multimodal models.