Risk-Averse Two-Stage Stochastic Program with Distributional Ambiguity

In this paper, we develop a risk-averse two-stage stochastic program (RTSP) that explicitly incorporates the distributional ambiguity covering both discrete and continuous distributions. We formulate RTSP from the perspective of distributional robustness by hedging against the worst-case distribution within an ambiguity set and considering the corresponding expected total cost. In particular, we derive an equivalent reformulation for RTSP that indicates that each worst-case expectation over an L-1-norm-based ambiguity set reduces to a convex combination of a conditional value-at-risk and an essential supremum. Our reformulation explicitly shows how additional data can help reduce the conservatism of the problem from the traditional two-stage robust optimization to the traditional risk-neutral two-stage stochastic program (TSP). Accordingly, we develop solution algorithms for the reformulations of RTSP based on the sample average approximation method. We also extend the studies to ambiguity sets based on L-infinity-norm, joint L-1- and L-infinity-norm, and joint L-1-norm and the first two moments, respectively. Furthermore, starting from a set of historical data samples, we utilize nonparametric statistics to construct these ambiguity sets. We perform convergence analysis to show that the ambiguity-aversion of RTSP vanishes as the data size increases to infinity, in the sense that the optimal objective value and the set of optimal solutions of RTSP converge to those of risk-neutral TSP. Finally, numerical experiments on newsvendor and lot-sizing problems explain and demonstrate the effectiveness of our proposed method.