# Optimization with Constraint Learning: A Framework and Survey

arXiv: https://arxiv.org/abs/2110.02121 · HTML: https://mlsanity.com/paper/2110.02121/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 143 references that are in the map.
- First posted on arXiv 5 Oct 2021.

## Builds on (most cited first, 30 of 33)

- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [On Hyperparameter Optimization of Machine Learning Algorithms: Theory and Practice](https://mlsanity.com/paper/2007.15745/) (2020)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [From Predictive to Prescriptive Analytics](https://mlsanity.com/paper/1402.5481/) (2014)
- [Non-convex Optimization for Machine Learning](https://mlsanity.com/paper/1712.07897/) (2017)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [A Survey of Optimization Methods from a Machine Learning Perspective](https://mlsanity.com/paper/1906.06821/) (2019)
- [Efficient softmax approximation for GPUs](https://mlsanity.com/paper/1609.04309/) (2016)
- [CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints](https://mlsanity.com/paper/2105.02343/) (2021)
- [Learning to Solve Large-Scale Security-Constrained Unit Commitment Problems](https://mlsanity.com/paper/1902.01697/) (2019)
- [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)
- [Dynamic Assortment Optimization with Changing Contextual Information](https://mlsanity.com/paper/1810.13069/) (2018)
- [Learning Optimal Power Flow: Worst-Case Guarantees for Neural Networks](https://mlsanity.com/paper/2006.11029/) (2020)
- [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)
- [Verification of Neural Network Behaviour: Formal Guarantees for Power System Applications](https://mlsanity.com/paper/1910.01624/) (2019)
- [Deep Learning Assisted Heuristic Tree Search for the Container Pre-marshalling Problem](https://mlsanity.com/paper/1709.09972/) (2017)
- [A Practical Guide to Robust Optimization](https://mlsanity.com/paper/1501.02634/) (2015)
- [Boosting Combinatorial Problem Modeling with Machine Learning](https://mlsanity.com/paper/1807.05517/) (2018)
- [Optimization Problems for Machine Learning: A Survey](https://mlsanity.com/paper/1901.05331/) (2019)
- [Neural Networks for Encoding Dynamic Security-Constrained Optimal Power Flow](https://mlsanity.com/paper/2003.07939/) (2020)
- [Dynamic optimization with side information](https://mlsanity.com/paper/1907.07307/) (2019)
- [Learning optimization models in the presence of unknown relations](https://mlsanity.com/paper/1401.1061/) (2014)
- [A novel embedded min-max approach for feature selection in nonlinear support vector machine classification](https://mlsanity.com/paper/2004.09863/) (2020)
- [Designing reactive power control rules for smart inverters using support vector machines](https://mlsanity.com/paper/1903.01016/) (2019)
- [Input Convex Neural Networks for Optimal Voltage Regulation](https://mlsanity.com/paper/2002.08684/) (2020)
- [Automating Personnel Rostering by Learning Constraints Using Tensors](https://mlsanity.com/paper/1805.11375/) (2018)

## Built on by (most cited first, 28 of 28)

- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Mixed-Integer Optimization with Constraint Learning](https://mlsanity.com/paper/2111.04469/) (2021)
- [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)
- [Counterfactual Explanations Using Optimization With Constraint Learning](https://mlsanity.com/paper/2209.10997/) (2022)
- [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)
- [Learning Multiple Initial Solutions to Optimization Problems](https://mlsanity.com/paper/2411.02158/) (2024)
- [On data-driven chance constraint learning for mixed-integer optimization problems](https://mlsanity.com/paper/2207.03844/) (2022)
- [Optimization Proxies using Limited Labeled Data and Training Time -- A Semi-Supervised Bayesian Neural Network Approach](https://mlsanity.com/paper/2410.03085/) (2024)
- [Relaxation-Informed Training of Neural Network Surrogate Models](https://mlsanity.com/paper/2604.22746/) (2026)
- [Deep Reinforcement Learning for Traveling Purchaser Problems](https://mlsanity.com/paper/2404.02476/) (2024)
- [Solving 0-1 Integer Programs with Unknown Knapsack Constraints Using Membership Oracles](https://mlsanity.com/paper/2405.14090/) (2024)
- [Conformal Mixed-Integer Constraint Learning with Feasibility Guarantees](https://mlsanity.com/paper/2506.03531/) (2025)
- [Falsification-Based Verification of LLM-Generated Optimization Models: Sound Test Batteries and Their Detection Limits](https://mlsanity.com/paper/2607.16646/) (2026)
- [Output-Constrained Decision Trees](https://mlsanity.com/paper/2405.15314/) (2024)
- [Mathematical Programming Algorithms for Convex Hull Approximation with a Hyperplane Budget](https://mlsanity.com/paper/2407.17341/) (2024)
- [Optimization over Trained (and Sparse) Neural Networks: A Surrogate within a Surrogate](https://mlsanity.com/paper/2505.01985/) (2025)
- [Optimal placement of wind farms via quantile constraint learning](https://mlsanity.com/paper/2510.01093/) (2025)
- [A study on constraint extraction and exception exclusion in care worker scheduling](https://mlsanity.com/paper/2512.24853/) (2025)
- [Efficient reformulations of ReLU deep neural networks for surrogate modelling in power system optimisation](https://mlsanity.com/paper/2601.14673/) (2026)
- [Monotone Optimisation with Learned Projections](https://mlsanity.com/paper/2601.20983/) (2026)
- [Learning efficient representations of complex constraints for scalable optimization](https://mlsanity.com/paper/2603.08283/) (2026)
- [Improving understanding and trust in AI: How users benefit from interval-based counterfactual explanations](https://mlsanity.com/paper/2604.09573/) (2026)
- [Optimization with Dynamic Constraint Learning (DCL)](https://mlsanity.com/paper/2607.25719/) (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)

Many real-life optimization problems frequently contain one or more constraints or objectives for which there are no explicit formulas. If data is however available, these data can be used to learn the constraints. The benefits of this approach are clearly seen, however there is a need for this process to be carried out in a structured manner. This paper therefore provides a framework for Optimization with Constraint Learning (OCL) which we believe will help to formalize and direct the process of learning constraints from data. This framework includes the following steps: (i) setup of the conceptual optimization model, (ii) data gathering and preprocessing, (iii) selection and training of predictive models, (iv) resolution of the optimization model, and (v) verification and improvement of the optimization model. We then review the recent OCL literature in light of this framework, and highlight current trends, as well as areas for future research.