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 (Formula presented.) 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.