Data-Driven Chance Constrained Stochastic Program

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



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