Discretization and quantification for distributionally robust optimization with decision-dependent ambiguity sets

In this paper, we investigate the discrete approximation and the quantitative analysis for a class of the distributionally robust optimization problems with decision-dependent ambiguity sets. We establish the local Lipschitz continuity of the decision-dependent ambiguity set, measured by the Hausdorff distance, under a broad class of metrics known as zeta-structure and the Slater condition. Furthermore, we employ Lagrange duality and first-order growth conditions to derive quantitative analysis for the optimal value and optimal solution. We also examine the application of a classical and widely-used ambiguity set within the theoretical framework of this paper. Finally, we conduct experiments to demonstrate the computational times and variations in the optimal value.