Combining machine learning and constrained optimization, Predict+Optimize tackles optimization problems containing parameters that are unknown at the time of solving. Prior works focus on cases with unknowns only in the objectives. A new framework was recently proposed to cater for unknowns also in constraints by introducing a loss function, called Post-hoc Regret, that takes into account the cost of correcting an unsatisfiable prediction. Since Post-hoc Regret is non-differentiable, the previous work computes only its approximation. While the notion of Post-hoc Regret is general, its specific implementation is applicable to only packing and covering linear programming problems. In this paper, we first show how to compute Post-hoc Regret exactly for any optimization problem solvable by a recursive algorithm satisfying simple conditions. Experimentation demonstrates substantial improvement in the quality of solutions as compared to the earlier approximation approach. Furthermore, we show experimentally the empirical behavior of different combinations of correction and penalty functions used in the Post-hoc Regret of the same benchmarks. Results provide insights for defining the appropriate Post-hoc Regret in different application scenarios.

翻译：结合机器学习与约束优化，预测+优化方法旨在解决求解时包含未知参数的优化问题。先前研究主要关注目标函数中存在未知参数的情况。近期提出的一项新框架通过引入名为"事后遗憾"的损失函数，将未知参数拓展至约束条件中。该损失函数考虑了修正不可满足预测所需的成本。由于事后遗憾不可微，既往研究仅能计算其近似值。尽管事后遗憾的概念具有普适性，但其具体实现仅适用于包装与覆盖线性规划问题。本文首先展示了如何针对满足简单条件的递归算法可解的任何优化问题精确计算事后遗憾。实验结果表明，与早期近似方法相比，该方法显著提升了解决方案的质量。此外，我们通过实验揭示了同一基准测试中事后遗憾所采用的不同校正函数与惩罚函数组合的实证表现，为不同应用场景下定义恰当的事后遗憾提供了指导性见解。