Forecasting Outside the Box: Application-Driven Optimal Pointwise Forecasts for Stochastic Optimization

We study a class of two-stage stochastic programs, namely, those with fixed recourse matrix and fixed costs, and linear second stage. We show that, under mild assumptions, the problem can be solved with just one scenario, which we call an ``optimal scenario.'' Such a scenario does not have to be unique and may fall outside the support of the underlying distribution. Although finding an optimal scenario in general might be hard, we show that the result can be particularly useful in the case of stochastic optimization problems with contextual information, where the goal is to optimize the expected value of a certain function given some contextual information (e.g., previous demand, customer type, etc.) that accompany the main data of interest. The contextual information allows for a better estimation of the quantity of interest via machine learning methods. We focus on a class of learning methods -- sometimes called in the literature decision-focused learning -- that integrate the learning and optimization procedures by means of a bilevel optimization formulation, which determines the parameters for pointwise forecasts. By using the optimal scenario result, we prove that when such models are applied to the class of contextual two-stage problems considered in this paper, the pointwise forecasts computed from the bilevel optimization formulation actually yield asymptotically the best approximation of an optimal scenario within the modeler's pre-specified set of parameterized forecast functions. Numerical results conducted with inventory problems from the literature (with synthetic data) as well as a bike-sharing problem with real data demonstrate that the proposed approach performs well when compared to benchmark methods from the literature.