# Unrolled Neural Networks for Constrained Optimization

arXiv: https://arxiv.org/abs/2601.17274 · HTML: https://mlsanity.com/paper/2601.17274/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 46 references that are in the map.
- First posted on arXiv 24 Jan 2026.

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Deep Unfolding Network for Image Super-Resolution](https://mlsanity.com/paper/2003.10428/) (2020)
- [Diffusion models as plug-and-play priors](https://mlsanity.com/paper/2206.09012/) (2022)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Deep Generalized Unfolding Networks for Image Restoration](https://mlsanity.com/paper/2204.13348/) (2022)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [Deep Unfolding: Model-Based Inspiration of Novel Deep Architectures](https://mlsanity.com/paper/1409.2574/) (2014)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [Unfolding WMMSE using Graph Neural Networks for Efficient Power Allocation](https://mlsanity.com/paper/2009.10812/) (2020)
- [Self-Supervised Primal-Dual Learning for Constrained Optimization](https://mlsanity.com/paper/2208.09046/) (2022)
- [Iterative Algorithm Induced Deep-Unfolding Neural Networks: Precoding Design for Multiuser MIMO Systems](https://mlsanity.com/paper/2006.08099/) (2020)
- [Constrained Learning with Non-Convex Losses](https://mlsanity.com/paper/2103.05134/) (2021)
- [Deep Unfolding of a Proximal Interior Point Method for Image Restoration](https://mlsanity.com/paper/1812.04276/) (2018)
- [Deep Unfolding with Normalizing Flow Priors for Inverse Problems](https://mlsanity.com/paper/2107.02848/) (2021)
- [InDuDoNet+: A Deep Unfolding Dual Domain Network for Metal Artifact Reduction in CT Images](https://mlsanity.com/paper/2112.12660/) (2021)
- [PDHG-Unrolled Learning-to-Optimize Method for Large-Scale Linear Programming](https://mlsanity.com/paper/2406.01908/) (2024)
- [Expressive Power of Graph Neural Networks for (Mixed-Integer) Quadratic Programs](https://mlsanity.com/paper/2406.05938/) (2024)
- [State-Augmented Learnable Algorithms for Resource Management in Wireless Networks](https://mlsanity.com/paper/2207.02242/) (2022)
- [Graph Signal Restoration Using Nested Deep Algorithm Unrolling](https://mlsanity.com/paper/2106.15910/) (2021)
- [Robust Stochastically-Descending Unrolled Networks](https://mlsanity.com/paper/2312.15788/) (2023)
- [Coordinated Sum-Rate Maximization in Multicell MU-MIMO with Deep Unrolling](https://mlsanity.com/paper/2202.10371/) (2022)
- [Proximal Algorithm Unrolling: Flexible and Efficient Reconstruction Networks for Single-Pixel Imaging](https://mlsanity.com/paper/2505.23180/) (2025)
- [Deep Algorithm Unrolling for Biomedical Imaging](https://mlsanity.com/paper/2108.06637/) (2021)
- [Stochastic Unrolled Federated Learning](https://mlsanity.com/paper/2305.15371/) (2023)
- [Generative Diffusion Models for Resource Allocation in Wireless Networks](https://mlsanity.com/paper/2504.20277/) (2025)

## Abstract (arXiv)

We develop unrolled neural networks to solve constrained optimization problems, offering accelerated, learnable counterparts to dual ascent (DA) algorithms. Our framework, termed constrained dual unrolling (CDU), comprises two coupled neural networks that jointly approximate the saddle point of the Lagrangian. The primal network emulates an iterative optimizer that finds a stationary point of the Lagrangian for a given dual multiplier, sampled from an unknown distribution. The dual network generates trajectories towards the optimal multipliers across its layers while querying the primal network at each layer. Unlike standard unrolling, which maps the updates of a prescribed iterative solver directly to network layers, CDU preserves the underlying optimization dynamics while learning the update mappings. The desired primal-dual dynamics are therefore induced through constrained learning by enforcing primal-descent and dual-ascent constraints. We formulate training the two networks as a nested optimization problem and propose an alternating procedure that updates the primal and dual networks in turn, mitigating uncertainty in the multiplier distribution required for primal network training. We establish theoretical guarantees on the feasibility and optimality of the coupled networks. We numerically evaluate the framework on quadratic programs (QPs) with linear constraints and power allocation in wireless networks.