Data-driven chance constrained stochastic program

Chance constrained programming is an effective and convenient approach to control riskin decision making under uncertainty. However, due to unknown probability distributions ofrandom parameters, the solution obtained from a chance constrained optimization problem canbe biased. In addition, instead of knowing the true distributions of random parameters, inpractice, only a series of historical data, which can be considered as samples taken from thetrue (while ambiguous) distribution, can be observed and stored. In this paper, we derivestochastic programs with data-driven chance constraints (DCCs) to tackle these problems anddevelop equivalent reformulations. For a given historical data set, we construct two types ofconfidence sets for the ambiguous distribution through nonparametric statistical estimation of itsmoments and density functions, depending on the amount of available data. We then formulateDCCs from the perspective of robust feasibility, by allowing the ambiguous distribution to runadversely within its confidence set. After deriving equivalent reformulations, we provide exactand approximate solution approaches for stochastic programs with DCCs under both moment-based and density-based confidence sets. In addition, we derive the relationship between theconservatism of DCCs and the sample size of historical data, which shows quantitatively whatwe call the value of data.