Data-Driven Risk-Averse Two-Stage Stochastic Program with ζ-Structure Probability Metrics

The traditional two-stage stochastic programming approach assumes the distribution of the random parameter in a problem is known. In most practices, however, the distribution is actually unknown. Instead, only a series of historic data are available. In this paper, we develop a data-driven stochastic optimization framework to provide a risk-averse decision making under uncertainty. In our approach, starting from a given set of historical data, we first construct a confidence set for the unknown probability distribution utilizing a family of ζ-structure probability metrics. Then, we describe the reference distributions and solution approaches to solving the developed two-stage risk-averse stochastic program, corresponding to the given set of historical data, for the cases in which the true probability distributions are discrete and continuous, respectively. More specifically, for the case in which the true probability distribution is discrete, we reformulate the risk-averse problem to a traditional two-stage robust optimization problem. For the case in which the true probability distribution is continuous, we develop a sampling approach to obtaining the upper and lower bounds of the risk-averse problem, and prove that these two bounds converge to the optimal objective value uniformly at the sample size increases. Furthermore, we prove that, for both cases, the risk-averse problem converges to the risk-neutral one as more data samples are observed, and derive the convergence rate, which indicates the value of data. Finally, the experiment results on newsvendor and facility location problems show how numerically the optimal objective value of the risk-averse stochastic program converges to the risk-neutral one, verifying the effectiveness of our proposed approach.