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arXiv 2111.04469math.OCcs.LGstat.ML
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基于约束学习的混合整数优化

Mixed-Integer Optimization with Constraint Learning

Amsterdam Business School, University of Amsterdam（阿姆斯特丹大学阿姆斯特丹商学院）；Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运筹学中心）；Sloan School of Management, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）

Donato Maragno, Holly Wiberg, Dimitris Bertsimas, S. Ilker Birbil, Dick den Hertog, Adejuyigbe Fajemisin

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AI总结：

本文提出一种基于约束学习的混合整数优化框架，将机器学习模型嵌入优化问题，并通过信任区域和集成学习处理数据不确定性，在粮食计划署规划和化疗优化中验证了其有效性。

摘要：

我们为带有学习约束的混合整数优化建立了广泛的方法论基础。我们提出了一种端到端的数据驱动决策流程，其中约束和目标函数通过机器学习直接从数据中学习，并将训练好的模型嵌入到优化公式中。我们利用许多机器学习方法的混合整数优化可表示性，包括线性模型、决策树、集成模型和多层感知器，这使我们能够捕捉决策、上下文变量和结果之间的各种潜在关系。我们还引入了两种方法来处理从数据中学习所固有的不确定性。首先，我们利用观测值的凸包来刻画决策信任区域，以确保推荐结果可信并避免外推。我们使用列生成高效地纳入这一表示，并提出了一种更灵活的公式来处理低密度区域和高维数据集。然后，我们提出了一种集成学习方法，在多个自助抽样估计器或多个算法上强制约束满足。结合领域驱动的组成部分，嵌入模型和信任区域定义了一个用于处方生成的混合整数优化问题。我们将该框架实现为一个面向实践者的Python包（OptiCL）。我们在世界粮食计划署规划和化疗优化两个案例中展示了该方法。案例研究表明，该框架能够生成高质量的处方，并体现了信任区域带来的附加价值、使用集成方法控制模型鲁棒性的作用、对多种机器学习方法的考虑以及纳入多个学习约束的意义。

英文摘要：

We establish a broad methodological foundation for mixed-integer optimization with learned constraints. We propose an end-to-end pipeline for data-driven decision making in which constraints and objectives are directly learned from data using machine learning, and the trained models are embedded in an optimization formulation. We exploit the mixed-integer optimization-representability of many machine learning methods, including linear models, decision trees, ensembles, and multi-layer perceptrons, which allows us to capture various underlying relationships between decisions, contextual variables, and outcomes. We also introduce two approaches for handling the inherent uncertainty of learning from data. First, we characterize a decision trust region using the convex hull of the observations, to ensure credible recommendations and avoid extrapolation. We efficiently incorporate this representation using column generation and propose a more flexible formulation to deal with low-density regions and high-dimensional datasets. Then, we propose an ensemble learning approach that enforces constraint satisfaction over multiple bootstrapped estimators or multiple algorithms. In combination with domain-driven components, the embedded models and trust region define a mixed-integer optimization problem for prescription generation. We implement this framework as a Python package (OptiCL) for practitioners. We demonstrate the method in both World Food Programme planning and chemotherapy optimization. The case studies illustrate the framework's ability to generate high-quality prescriptions as well as the value added by the trust region, the use of ensembles to control model robustness, the consideration of multiple machine learning methods, and the inclusion of multiple learned constraints.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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