Quantitative Stability Analysis for Minimax Distributionally Robust RiskOptimization

Abstract

This paper considers distributionally robust formulations of a two stage stochastic programmingproblem with the objective of minimizing a distortion risk of the minimal cost incurred at the secondstage.We carry out stability analysis by looking into variations of the ambiguity set under theWassersteinmetric, decision spaces at both stages and the support set of the random variables. In the case when itis risk neutral, the stability result is presented with the variation of the ambiguity set being measuredby generic metrics of ζ-structure, which provides a unified framework for quantitative stability analysisunder various metrics including total variation metric and Kantorovich metric. When the ambiguity set isstructured by a ζ-ball, we find that the Hausdorff distance between two ζ-balls is bounded by the distanceof their centres and difference of their radius. The findings allow us to strengthen some recent convergenceresults on distributionally robust optimization where the centre of the Wasserstein ball is constructed bythe empirical probability distribution.