# CG4AI: A Column Generation Framework for Training AI Models Under Constraints

arXiv: https://arxiv.org/abs/2608.26375 · HTML: https://mlsanity.com/paper/2608.26375/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [Probably Approximately Correct Constrained Learning](https://mlsanity.com/paper/2006.05487/) (2020)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks](https://mlsanity.com/paper/2211.01340/) (2022)
- [Enforcing Hard Linear Constraints in Deep Learning Models with Decision Rules](https://mlsanity.com/paper/2505.13858/) (2025)
- [Near-Optimal Solutions of Constrained Learning Problems](https://mlsanity.com/paper/2403.11844/) (2024)
- [Feed-Forward Neural Networks as a Mixed-Integer Program](https://mlsanity.com/paper/2402.06697/) (2024)

## Abstract (arXiv)

Standard machine-learning training minimizes a loss function over a dataset, but does not guarantee that the resulting model will satisfy predefined rules or constraints on its outputs. In many real-world applications, ranging from autonomous systems to network routing, such guarantees are essential. We propose CG4AI, a framework that builds a convex combination of AI models while enforcing linear constraints on the combined output. A master linear program (LP) determines the optimal mixture weights, while a pricing subproblem generates new models guided by LP dual variables, focusing attention on the most violated constraints. A cutting-plane procedure extends feasibility guarantees beyond the training set. We apply CG4AI to two problems: (i) digit classification on MNIST, where we demonstrate four distinct uses of constraints, learning from constraints alone, improving adversarial robustness, correcting misclassified examples, and enforcing output relabeling; and (ii) the multi-commodity flow problem, where link capacity constraints are enforced on neural-network routing predictors. Experiments on MNIST and standard SNDLIB benchmark networks show that CG4AI reliably produces feasible predictors while achieving better accuracy than single-model baselines.