# Constrained Graph Diffusion for Mixed Integer Optimization

arXiv: https://arxiv.org/abs/2608.13079 · HTML: https://mlsanity.com/paper/2608.13079/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 37 references that are in the map.
- First posted on arXiv 13 Aug 2026.

## Builds on (most cited first, 26 of 26)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Learning Combinatorial Optimization Algorithms over Graphs](https://mlsanity.com/paper/1704.01665/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Solving Mixed Integer Programs Using Neural Networks](https://mlsanity.com/paper/2012.13349/) (2020)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Reinforcement Learning for Integer Programming: Learning to Cut](https://mlsanity.com/paper/1906.04859/) (2019)
- [Hybrid Models for Learning to Branch](https://mlsanity.com/paper/2006.15212/) (2020)
- [MIPaaL: Mixed Integer Program as a Layer](https://mlsanity.com/paper/1907.05912/) (2019)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms](https://mlsanity.com/paper/1908.02788/) (2019)
- [Learning To Cut By Looking Ahead: Cutting Plane Selection via Imitation Learning](https://mlsanity.com/paper/2206.13414/) (2022)
- [Self-Supervised Primal-Dual Learning for Constrained Optimization](https://mlsanity.com/paper/2208.09046/) (2022)
- [A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimization](https://mlsanity.com/paper/2406.01661/) (2024)
- [End to end learning and optimization on graphs](https://mlsanity.com/paper/1905.13732/) (2019)
- [Online Mixed-Integer Optimization in Milliseconds](https://mlsanity.com/paper/1907.02206/) (2019)
- [End-to-End Learning to Warm-Start for Real-Time Quadratic Optimization](https://mlsanity.com/paper/2212.08260/) (2022)
- [Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guarantees](https://mlsanity.com/paper/2410.11061/) (2024)
- [Unsupervised Training of Diffusion Models for Feasible Solution Generation in Neural Combinatorial Optimization](https://mlsanity.com/paper/2411.00003/) (2024)
- [Unsupervised Diffusion Solver for Combinatorial Optimization via Combinatorial Adjoint Matching](https://mlsanity.com/paper/2605.30920/) (2026)
- [Boosting Cross-problem Generalization in Diffusion-Based Neural Combinatorial Solver via Inference Time Adaptation](https://mlsanity.com/paper/2502.12188/) (2025)

## Abstract (arXiv)

This paper proposes a novel learning-based approach to approximately solve instances of mixed-integer optimization problems. These problems are computationally challenging, as they require jointly determining discrete and continuous decisions while satisfying complex combinatorial constraints. problem-agnostic and can accommodate a broad class of mixed-integer optimization problems through suitable projection operators. We introduce Constrained Graph Diffusion (CGD), a learning-based framework that approximately solves recurring instances of such problems by learning a conditional distribution over their discrete decisions. CGD uses a graph-based diffusion model and incorporates constraint information directly into the reverse diffusion process, steering intermediate predictions toward the feasible region throughout generation. By operating on continuous relaxations of the discrete variables, CGD defines a differentiable constrained generation pathway up to terminal discrete recovery. Once the discrete decision is recovered and fixed, a numerical optimizer solves the remaining continuous problem, avoiding online combinatorial search over the binary variables while retaining numerical optimization for continuous completion. We evaluate CGD on AC-OPF with branch switching and discrete portfolio optimization, demonstrating substantial improvements in feasibility and solution quality over learning-based baselines while achieving speedups of up to 543× over state-of-the-art MIP solvers on large instances.