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 internal 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：我们考虑求解一类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。我们引入割平面层（CPL）的概念，即一个可微的割平面生成器，它将问题数据及先前迭代结果映射为割平面。我们提出一种CPL实现方法以生成分割割，并通过组合多个CPL设计了一种可微割平面算法，该算法利用了参数化实例的重复特性。在离线阶段，我们通过更新控制CPL的内部参数来训练算法，从而改变割平面生成方式。训练完成后，我们的算法能以可预测的执行时间和固定数量的割平面，计算出具有低整数间隙的解。初步计算测试表明，该算法能够泛化到未见过的实例，并捕捉潜在的参数化结构。