Data-driven inverse optimization for mixed-integer linear programs (MILPs), which seeks to learn an objective function and constraints consistent with observed decisions, is important for building accurate mathematical models in a variety of domains, including power systems and scheduling. However, to the best of our knowledge, existing data-driven inverse optimization methods primarily focus on learning objective functions under known constraints, and learning both objective functions and constraints from data remains largely unexplored. In this paper, we propose a two-stage approach for a class of inverse optimization problems in which the objective is a linear combination of given feature functions and the constraints are parameterized by unknown functions and thresholds. Our method first learns the constraints and then, conditioned on the learned constraints, estimates the objective-function weights. On the theoretical side, we provide finite-sample guarantees for solving the proposed inverse optimization problem. To this end, we develop statistical learning tools for pseudo-metric spaces under sub-Gaussian assumptions and use them to derive a learning-theoretic framework for inverse optimization with both unknown objectives and constraints. On the experimental side, we demonstrate that our method successfully solves inverse optimization problems on scheduling instances formulated as ILPs with up to 100 decision variables.

翻译：针对混合整数线性规划（MILP）的数据驱动逆优化旨在从观测到的决策中学习与之相符的目标函数和约束条件，这对于在电力系统、调度等多个领域建立精确的数学模型具有重要意义。然而，据我们所知，现有的数据驱动逆优化方法主要侧重于在已知约束条件下学习目标函数，而从数据中同时学习目标函数与约束条件的研究仍基本处于空白。本文针对一类逆优化问题提出了一种两阶段方法，其中目标函数为给定特征函数的线性组合，约束条件则由未知函数及阈值参数化。我们的方法首先学习约束条件，然后在已学得的约束条件下估计目标函数的权重。在理论方面，我们为求解所提出的逆优化问题提供了有限样本保证。为此，我们在亚高斯假设下为伪度量空间开发了统计学习工具，并利用这些工具推导出同时包含未知目标与约束的逆优化学习理论框架。在实验方面，我们证明了所提方法能成功求解以整数线性规划（ILP）形式表述、决策变量多达100个的调度实例上的逆优化问题。