Cone Ordering in Distributionally Robust Optimization with Set-Valued Probabilities
该文在分布鲁棒优化中引入集值概率，并用凸尖锥定义集合间的序关系，重新定义了鲁棒性、凸性和最小元，推导了最优性条件和稳定性定理，并应用于确定性等价概念。
Abstract 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 $$\begin{aligned} A \le _C B \quad \iff \quad A \subseteq B - C, \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>A</mml:mi> <mml:msub> <mml:mo>≤</mml:mo> <mml:mi>C</mml:mi> </mml:msub> <mml:mi>B</mml:mi> <mml:mspace/> <mml:mo>⟺</mml:mo> <mml:mspace/> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>B</mml:mi> <mml:mo>-</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> 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.