Dealing with uncertainty in optimization parameters is an important and longstanding challenge. Typically, uncertain parameters are predicted accurately, and then a deterministic optimization problem is solved. However, the decisions produced by this so-called \emph{predict-then-optimize} procedure can be highly sensitive to uncertain parameters. In this work, we contribute to recent efforts in producing \emph{decision-focused} predictions, i.e., to build predictive models that are constructed with the goal of minimizing a \emph{regret} measure on the decisions taken with them. We begin by formulating the exact expected regret minimization as a pessimistic bilevel optimization model. Then, we establish NP-completeness of this problem, even in a heavily restricted case. Using duality arguments, we reformulate it as a non-convex quadratic optimization problem. Finally, we show various computational techniques to achieve tractability. We report extensive computational results on shortest-path instances with uncertain cost vectors. Our results indicate that our approach can improve training performance over the approach of Elmachtoub and Grigas (2022), a state-of-the-art method for decision-focused learning.

翻译：处理优化参数中的不确定性是一项重要且长期存在的挑战。通常，人们会先对不确定参数进行精确预测，然后求解确定性优化问题。然而，这种所谓“先预测后优化”流程所产生的决策可能对不确定参数高度敏感。在本研究中，我们致力于近期旨在生成“决策导向”预测的工作，即构建以最小化基于这些预测所做决策的“遗憾”度量为目标的预测模型。我们首先将精确的期望遗憾最小化问题表述为一个悲观双层优化模型。随后，我们证明了该问题即使在高度受限的情况下也是NP完全问题。通过引入对偶论证，我们将其重构为一个非凸二次优化问题。最后，我们展示了多种实现计算可行性的技术。我们在具有不确定成本向量的最短路径实例上进行了广泛的计算实验。结果表明，相较于Elmachtoub和Grigas（2022）提出的决策导向学习先进方法，我们的方法能够提升训练性能。