# Diffusion-based learning framework for Constrained Nonconvex Optimization with Weighted Bootstrapped Refinement

arXiv: https://arxiv.org/abs/2502.10330 · HTML: https://mlsanity.com/paper/2502.10330/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 40 of its 56 references that are in the map.
- First posted on arXiv 14 Feb 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 40)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Learning to Optimize](https://mlsanity.com/paper/1606.01885/) (2016)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Learning a Diffusion Model Policy from Rewards via Q-Score Matching](https://mlsanity.com/paper/2312.11752/) (2023)
- [Learning Human-to-Humanoid Real-Time Whole-Body Teleoperation](https://mlsanity.com/paper/2403.04436/) (2024)
- [Erdos Goes Neural: an Unsupervised Learning Framework for Combinatorial Optimization on Graphs](https://mlsanity.com/paper/2006.10643/) (2020)
- [Policy Representation via Diffusion Probability Model for Reinforcement Learning](https://mlsanity.com/paper/2305.13122/) (2023)
- [Reinforcement Learning for Integer Programming: Learning to Cut](https://mlsanity.com/paper/1906.04859/) (2019)
- [Diffusion-based Reinforcement Learning via Q-weighted Variational Policy Optimization](https://mlsanity.com/paper/2405.16173/) (2024)
- [Learning Combinatorial Embedding Networks for Deep Graph Matching](https://mlsanity.com/paper/1904.00597/) (2019)
- [Predicting AC Optimal Power Flows: Combining Deep Learning and Lagrangian Dual Methods](https://mlsanity.com/paper/1909.10461/) (2019)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Deep Graph Matching Consensus](https://mlsanity.com/paper/2001.09621/) (2020)
- [Learning Optimal Solutions for Extremely Fast AC Optimal Power Flow](https://mlsanity.com/paper/1910.01213/) (2019)
- [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)
- [Enhancing Deep Reinforcement Learning: A Tutorial on Generative Diffusion Models in Network Optimization](https://mlsanity.com/paper/2308.05384/) (2023)
- [DeepOPF: A Deep Neural Network Approach for Security-Constrained DC Optimal Power Flow](https://mlsanity.com/paper/1910.14448/) (2019)
- [Fast T2T: Optimization Consistency Speeds Up Diffusion-Based Training-to-Testing Solving for Combinatorial Optimization](https://mlsanity.com/paper/2502.02941/) (2025)
- [Can Push-forward Generative Models Fit Multimodal Distributions?](https://mlsanity.com/paper/2206.14476/) (2022)
- [Diffusion Models as Constrained Samplers for Optimization with Unknown Constraints](https://mlsanity.com/paper/2402.18012/) (2024)
- [Learning Warm-Start Points for AC Optimal Power Flow](https://mlsanity.com/paper/1905.08860/) (2019)
- [Model-Based Diffusion for Trajectory Optimization](https://mlsanity.com/paper/2407.01573/) (2024)
- [Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics](https://mlsanity.com/paper/2502.08696/) (2025)
- [CGD: Constraint-Guided Diffusion Policies for UAV Trajectory Planning](https://mlsanity.com/paper/2405.01758/) (2024)
- [DiffuSolve: Diffusion-based Solver for Non-convex Trajectory Optimization](https://mlsanity.com/paper/2403.05571/) (2024)
- [Smart-PGSim: Using Neural Network to Accelerate AC-OPF Power Grid Simulation](https://mlsanity.com/paper/2008.11827/) (2020)

## Built on by (most cited first, 1 of 1)

- [Sample from What You See: Visuomotor Policy Learning via Diffusion Bridge with Observation-Embedded Stochastic Differential Equation](https://mlsanity.com/paper/2512.07212/) (2025)

## Abstract (arXiv)

Recent advances in diffusion models show promising potential to accelerate nonconvex problem solving by leveraging their multimodality. However, most existing diffusion-based optimization approaches rely on supervised learning and lack a mechanism to enforce constraint satisfaction, which is required in real-world applications. In that case, we investigate and theoretically analyze the inherent problem of supervised diffusion solvers and identify the distributional misalignment problem, i.e., the generated solution distribution often exhibits low probability mass on the feasible region. To resolve this issue, we propose DiOpt, a new diffusion-based learning framework for constrained nonconvex optimization, which effectively learns the mapping from noise to the constraint region. Specifically, this framework operates in two distinct phases: an initial warm-start phase, implemented via supervised learning, followed by a bootstrapping training phase. This dual-phase architecture is designed to iteratively refine solutions, thereby improving the objective function with high constraint satisfaction. Finally, we also employ a solution selection technique in inference for better optimality. Notably, DiOpt is the first successful integration of the diffusion solver in constrained nonconvex optimization. Evaluations on diverse nonconvex tasks demonstrate the superiority of DiOpt in both optimality and constraint satisfaction. Our official page is released at https://dingsht.tech/diopt-webpage.