In this paper, we develop a risk-averse two-stage stochastic program (RTSP) which explicitly incorporates the distributional ambiguity covering both discrete and continuous distributions. Starting from a set of historical data samples, we construct a confidence set for the ambiguous probability distribution through nonparametric statistical estimation of its density function. We then formulate RTSP from the perspective of distributional robustness by hedging against the worst-case distribution within the confidence set and considering the corresponding expected total cost. In particular, we derive an equivalent reformulation for RTSP which explicitly reflects its linkage with a full spectrum of coherent risk measures under various risk-averseness levels. Our reformulation result can also be extended to other interesting stochastic programming models with expectation, conditional value-at-risk, chance, and stochastic dominance constraints. Furthermore, we perform convergence analysis to show that the risk-averseness of RTSP vanishes as the data sample size grows to infinity, in the sense that the optimal objective value and the set of optimal solutions of RTSP converge to those of TSP. Finally, we develop a solution algorithm for the reformulation of RTSP based on the sample average approximation method, and numerical experiments on newsvendor and lot-sizing problems explain and demonstrate the effectiveness of our proposed framework.
This paper was first submitted for publication on June 17th, 2014.