Mixed-Integer Optimization with Constraint Learning

Donato Maragno (), Holly Wiberg (), Dimitris Bertsimas (), Ş. İlker Birbil (), Dick den Hertog () and Adejuyigbe O. Fajemisin ()
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Donato Maragno: Amsterdam Business School, University of Amsterdam, 1018 TV Amsterdam, Netherlands
Holly Wiberg: Heinz College of Information Systems and Public Policy, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Dimitris Bertsimas: Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Ş. İlker Birbil: Amsterdam Business School, University of Amsterdam, 1018 TV Amsterdam, Netherlands
Dick den Hertog: Amsterdam Business School, University of Amsterdam, 1018 TV Amsterdam, Netherlands
Adejuyigbe O. Fajemisin: Amsterdam Business School, University of Amsterdam, 1018 TV Amsterdam, Netherlands

Operations Research, 2025, vol. 73, issue 2, 1011-1028

Abstract: 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 multilayer 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 data sets. 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 and 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.

Keywords: Optimization; mixed-integer optimization; machine learning; constraint learning; prescriptive analytics (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:73:y:2025:i:2:p:1011-1028

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