We study contextual stochastic optimization problems. Optimization problems have uncertain parameters stemming from unknown, context-dependent, distributions. Due to the inherent uncertainty in these problems, one is often interested not only in minimizing expected cost, but also to be robust and protect against worst case scenarios. We propose a novel method that combines the learning stage with knowledge of the downstream optimization task. The method prescribes decisions which aim to maximize the likelihood that the cost is below a (user-controlled) threshold. The key idea is (1) to discretize the feasible region into subsets so that the uncertain objective function can be well approximated deterministically within each subset, and (2) devise a secondary optimization problem to prescribe decisions by integrating the individual approximations determined in step (1). We provide theoretical guarantees bounding the underlying regret of decisions proposed by our method. In addition, experimental results demonstrate that our approach is competitive in terms of average regret and yields more robust solutions than other methods proposed in the literature, including up to 20 times lower worst-case cost on a real-world electricity generation problem.

In many real-world applications, predictive methods are used to provide inputs for downstream optimization problems. It has been shown that using the downstream task-based objective to learn the intermediate predictive model is often better than using only intermediate task objectives, such as prediction error. The learning task in the former approach is referred to as end-to-end learning. The difficulty in end-to-end learning lies in differentiating through the optimization problem. Therefore, we propose a neural network architecture that can learn to approximately solve these optimization problems, particularly ensuring its output satisfies the feasibility constraints via alternate projections. We show these projections converge at a geometric rate to the exact projection. Our approach is more computationally efficient than existing methods as we do not need to solve the original optimization problem at each iteration. Furthermore, our approach can be applied to a wider range of optimization problems. We apply this to a shortest path problem for which the first stage forecasting problem is a computer vision task of predicting edge costs from terrain maps, a capacitated multi-product newsvendor problem, and a maximum matching problem. We show that this method out-performs existing approaches in terms of final task-based loss and training time.