Generalized Robust Optimization using the Notion of Set-Valued Probability

We propose a novel concept of robustness grounded in the framework of set-valued probabilities, offering a unified and versatile approach to tackling challenges associated with the statistical estimation of uncertain or unknown probabilities. By employing scalarization techniques for set-valued probabilities, we derive optimality conditions. Additionally, we establish generalized convexity properties and stability conditions, which further underpin the robustness of our approach. This comprehensive framework finds significant applications in areas such as financial portfolio management and risk measure theory, where it provides powerful tools for addressing uncertainty, optimizing decision-making, and ensuring resilience against variability in probabilistic models.