Cone Ordering in Distributionally Robust Optimization with Set-Valued Probabilities

We extend the classical distributionally robust optimization framework by introducing set valued probabilities along with an ordering between sets based on convex, pointed cones where we define aligned A ≤ C B A ⊆ B - C, aligned A ≤ C B ⟺ A ⊆ B - C , with C a closed convex pointed cone. This ordering generalizes inclusion and allows for the modeling of directional preferences and asymmetries. Within this framework, we redefine robustness, convexity, and minimizers; we establish scalarization results, derive optimality conditions, and prove stability theorems. The framework offers a unifying perspective linking robust optimization, set-valued analysis, and cone ordering preferences. An application to the notion of Certainty Equivalent is provided at the end.