# Mixed-Integer Optimization with Constraint Learning

arXiv: https://arxiv.org/abs/2111.04469 · HTML: https://mlsanity.com/paper/2111.04469/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 58 references that are in the map.
- First posted on arXiv 4 Nov 2021.

## Builds on (most cited first, 15 of 15)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Recursive Partitioning for Heterogeneous Causal Effects](https://mlsanity.com/paper/1504.01132/) (2015)
- [From Predictive to Prescriptive Analytics](https://mlsanity.com/paper/1402.5481/) (2014)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [Learning in High Dimension Always Amounts to Extrapolation](https://mlsanity.com/paper/2110.09485/) (2021)
- [Global Deterministic Optimization with Artificial Neural Networks Embedded](https://mlsanity.com/paper/1801.07114/) (2018)
- [ReLU Networks as Surrogate Models in Mixed-Integer Linear Programs](https://mlsanity.com/paper/1907.03140/) (2019)
- [Optimization with Constraint Learning: A Framework and Survey](https://mlsanity.com/paper/2110.02121/) (2021)
- [JANOS: An Integrated Predictive and Prescriptive Modeling Framework](https://mlsanity.com/paper/1911.09461/) (2019)
- [Optimization of Tree Ensembles](https://mlsanity.com/paper/1705.10883/) (2017)
- [Neural Networks for Encoding Dynamic Security-Constrained Optimal Power Flow](https://mlsanity.com/paper/2003.07939/) (2020)
- [Learning optimization models in the presence of unknown relations](https://mlsanity.com/paper/1401.1061/) (2014)
- [Input Convex Neural Networks for Optimal Voltage Regulation](https://mlsanity.com/paper/2002.08684/) (2020)

## Built on by (most cited first, 30 of 31)

- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Finding Regions of Counterfactual Explanations via Robust Optimization](https://mlsanity.com/paper/2301.11113/) (2023)
- [Artificial Intelligence for Operations Research: Revolutionizing the Operations Research Process](https://mlsanity.com/paper/2401.03244/) (2024)
- [Optimization Over Trained Neural Networks: Taking a Relaxing Walk](https://mlsanity.com/paper/2401.03451/) (2024)
- [ORGEval: Graph-Theoretic Evaluation of LLMs in Optimization Modeling](https://mlsanity.com/paper/2510.27610/) (2025)
- [Counterfactual Explanations Using Optimization With Constraint Learning](https://mlsanity.com/paper/2209.10997/) (2022)
- [PySCIPOpt-ML: Embedding Trained Machine Learning Models into Mixed-Integer Programs](https://mlsanity.com/paper/2312.08074/) (2023)
- [Computational Tradeoffs of Optimization-Based Bound Tightening in ReLU Networks](https://mlsanity.com/paper/2312.16699/) (2023)
- [Global Optimization: A Machine Learning Approach](https://mlsanity.com/paper/2311.01742/) (2023)
- [Neur2BiLO: Neural Bilevel Optimization](https://mlsanity.com/paper/2402.02552/) (2024)
- [On data-driven chance constraint learning for mixed-integer optimization problems](https://mlsanity.com/paper/2207.03844/) (2022)
- [Metaheuristics for (Variable-Size) Mixed Optimization Problems: A Unified Taxonomy and Survey](https://mlsanity.com/paper/2401.03880/) (2024)
- [Counterfactual Explanations for Linear Optimization](https://mlsanity.com/paper/2405.15431/) (2024)
- [Relaxation-Informed Training of Neural Network Surrogate Models](https://mlsanity.com/paper/2604.22746/) (2026)
- [Optimizing over an ensemble of neural networks](https://mlsanity.com/paper/2112.07007/) (2021)
- [Constraint Learning to Define Trust Regions in Predictive-Model Embedded Optimization](https://mlsanity.com/paper/2201.04429/) (2022)
- [Tight Mixed-Integer Optimization Formulations for Prescriptive Trees](https://mlsanity.com/paper/2302.14744/) (2023)
- [Learning Fair Policies for Multi-stage Selection Problems from Observational Data](https://mlsanity.com/paper/2312.13173/) (2023)
- [Conformal Mixed-Integer Constraint Learning with Feasibility Guarantees](https://mlsanity.com/paper/2506.03531/) (2025)
- [A General Neural Backbone for Mixed-Integer Linear Optimization via Dual Attention](https://mlsanity.com/paper/2601.04509/) (2026)
- [Deep Neural Network for Constraint Acquisition through Tailored Loss Function](https://mlsanity.com/paper/2403.02042/) (2024)
- [Optimization over Trained (and Sparse) Neural Networks: A Surrogate within a Surrogate](https://mlsanity.com/paper/2505.01985/) (2025)
- [Learning-Augmented Power System Operations: A Unified Optimization View](https://mlsanity.com/paper/2505.05203/) (2025)
- [Optimal placement of wind farms via quantile constraint learning](https://mlsanity.com/paper/2510.01093/) (2025)
- [Learning efficient representations of complex constraints for scalable optimization](https://mlsanity.com/paper/2603.08283/) (2026)
- [Fusing Backdoors, Machine Learning, and Optimization for Large-Scale Parametric Mixed-Integer Programs](https://mlsanity.com/paper/2606.21440/) (2026)
- [Optimization with Dynamic Constraint Learning (DCL)](https://mlsanity.com/paper/2607.25719/) (2026)
- [Input convex neural networks as surrogates in mathematical optimisation](https://mlsanity.com/paper/2608.09707/) (2026)
- [DCEmbed: Scalable Optimization over Neural Surrogates](https://mlsanity.com/paper/2609.33879/) (2026)
- [TACIT: Optimization Models that Learn from Their Mistakes](https://mlsanity.com/paper/2609.38434/) (2026)

## Abstract (arXiv)

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.