Data-Driven Optimization with Distributionally Robust Second Order Stochastic Dominance Constraints

Optimization with stochastic dominance constraints has recently received an increasing amount of attention in the quantitative risk management literature. Instead of requiring that the probabilistic description of the uncertain parameters be exactly known, this paper presents a comprehensive study of a data-driven formulation of the distributionally robust second order stochastic dominance constrained problem (DRSSDCP) that hinges on using a type-1 Wasserstein ambiguity set. This formulation allows us to identify solutions with finite sample guarantees and solutions that are asymptotically consistent when observations are independent and identically distributed. It is, furthermore, shown to be axiomatically motivated in an environment with distribution ambiguity. Leveraging recent results in the field of robust optimization, we further formulate the DRSSDCP as a multistage robust optimization problem and further propose a tractable conservative approximation that exploits finite adaptability and a scenario-based lower bounding problem, both of which can reduce to linear programs under mild conditions. We then propose, to the best of our knowledge, the first exact optimization algorithm for this DRSSDCP, the efficiency of which is confirmed by our numerical results. Finally, we illustrate how the data-driven DRSSDCP can be applied in practice on resource-allocation problems with both synthetic and real data. Our empirical results show that, with a proper adjustment of the size of the Wasserstein ball, DRSSDCP can reach “acceptable” out-of-sample feasibility yet still generating strictly better performance than what is achieved by the reference strategy.