Problem-driven scenario clustering in stochastic optimization

In stochastic optimisation, the large number of scenarios required to faithfully represent the underlying uncertainty is often a barrier to finding efficient numerical solutions. This motivates the scenario reduction problem: by finding a smaller subset of scenarios, reduce the numerical complexity while keeping the error at an acceptable level. In this paper we propose a novel and computationally efficient methodology to tackle the scenario reduction problem for two-stage problems when the error to be minimised is the implementation error, i.e. the error incurred by implementing the solution of the reduced problem in the original problem. Specifically, we develop a problem-driven scenario clustering method that produces a partition of the scenario set. Each cluster contains a representative scenario that best reflects the optimal value of the objective function in each cluster of the partition to be identified. We demonstrate the efficiency of our method by applying it to two challenging two-stage stochastic combinatorial optimization problems: the two-stage stochastic network design problem and the two-stage facility location problem. When compared to alternative clustering methods and Monte Carlo sampling, our method is shown to clearly outperform all other methods.