Data-Driven Distributionally Robust Risk-Averse Two-Stage Stochastic Linear Programming over Wasserstein Ball

In this paper, we consider a data-driven distributionally robust two-stage stochastic linear optimization problem over 1-Wasserstein ball centered at a discrete empirical distribution. Differently from the traditional two-stage stochastic programming which involves the expected recourse function as the preference criterion and hence is risk-neutral, we take the conditional value-at-risk (CVaR) as the risk measure in order to model its effects on decision making problems. We mainly explore tractable reformulations for the proposed robust two-stage stochastic programming with mean-CVaR criterion by analyzing the first case where uncertainties are only in the objective function and then the second case where uncertainties are only in the constraints. We demonstrate that the first model can be exactly reformulated as a deterministic convex programming. Furthermore, it is shown that under several different support sets, the resulting convex optimization problems can be converted into computationally tractable conic programmings. Besides, the second model is generally NP-hard since checking constraint feasibility can be reduced to a norm maximization problem over a polytope. However, even with the case of uncertainty in constraints, tractable conic reformulations can be established when the extreme points of the polytope are known. Finally, we present numerical results to discuss how to control the risk for the best decisions and illustrate the computational effectiveness and superiority of the proposed models.