We consider the problem of solving a family of parametric mixed-integer linear optimization problems where some entries in the input data change. We introduce the concept of $cutting-plane$ $layer$ (CPL), $i.e.$, a differentiable cutting-plane generator mapping the problem data and previous iterates to cutting planes. We propose a CPL implementation to generate split cuts, and by combining several CPLs, we devise a differentiable cutting-plane algorithm that exploits the repeated nature of parametric instances. In an offline phase, we train our algorithm by updating the parameters controlling the CPLs, thus altering cut generation. Once trained, our algorithm computes, with predictable execution times and a fixed number of cuts, solutions with low integrality gaps. Preliminary computational tests show that our algorithm generalizes on unseen instances and captures underlying parametric structures.

翻译：我们考虑解决一族参数化混合整数线性优化问题，其中输入数据的某些条目发生变化。我们引入了$切割平面层$（CPL）的概念，即一个可微的切割平面生成器，将问题数据和先前迭代映射到切割平面。我们提出了一种用于生成分割切割的CPL实现，并通过组合多个CPL，设计了一种利用参数化实例重复特性的可微切割平面算法。在离线阶段，我们通过更新控制CPL的参数来训练算法，从而改变切割生成过程。训练完成后，我们的算法能够以可预测的执行时间和固定数量的切割，计算出具有低积分间隙的解。初步计算测试表明，我们的算法能够泛化到未见过的实例，并捕捉潜在的参数化结构。