Robust optimization provides a mathematical framework for modeling and solving decision-making problems under worst-case uncertainty. This work addresses two-stage robust optimization (2RO) problems (also called adjustable robust optimization), wherein first-stage and second-stage decisions are made before and after uncertainty is realized, respectively. This results in a nested min-max-min optimization problem which is extremely challenging computationally, especially when the decisions are discrete. We propose Neur2RO, an efficient machine learning-driven instantiation of column-and-constraint generation (CCG), a classical iterative algorithm for 2RO. Specifically, we learn to estimate the value function of the second-stage problem via a novel neural network architecture that is easy to optimize over by design. Embedding our neural network into CCG yields high-quality solutions quickly as evidenced by experiments on two 2RO benchmarks, knapsack and capital budgeting. For knapsack, Neur2RO finds solutions that are within roughly $2\%$ of the best-known values in a few seconds compared to the three hours of the state-of-the-art exact branch-and-price algorithm; for larger and more complex instances, Neur2RO finds even better solutions. For capital budgeting, Neur2RO outperforms three variants of the $k$-adaptability algorithm, particularly on the largest instances, with a 5 to 10-fold reduction in solution time. Our code and data are available at https://github.com/khalil-research/Neur2RO.

翻译：鲁棒优化为在最坏情况不确定性下建模和求解决策问题提供了数学框架。本文研究两阶段鲁棒优化（2RO）问题（亦称可调整鲁棒优化），其中第一阶段和第二阶段的决策分别在不确定性实现前后作出。这导致了一个嵌套的极小-极大-极小优化问题，计算上极具挑战性，尤其是当决策为离散时。我们提出Neur2RO，一种高效的机器学习驱动的列与约束生成（CCG）实例化方法，CCG是求解2RO的经典迭代算法。具体而言，我们通过一种新颖的神经网络架构来学习估计第二阶段问题的价值函数，该架构在设计上易于优化。将我们的神经网络嵌入CCG中，能够快速获得高质量解，这一点通过两个2RO基准测试（背包问题和资本预算问题）的实验得到了验证。对于背包问题，Neur2RO在数秒内找到的解与已知最优值的差距约在2%以内，而最先进的精确分支定价算法需耗时三小时；对于更大、更复杂的实例，Neur2RO能找到更优的解。对于资本预算问题，Neur2RO优于k-适应性算法的三种变体，尤其在最大规模实例上，求解时间减少了5到10倍。我们的代码和数据可在https://github.com/khalil-research/Neur2RO获取。