# Learning Constrained Optimization with Deep Augmented Lagrangian Methods

arXiv: https://arxiv.org/abs/2403.03454 · HTML: https://mlsanity.com/paper/2403.03454/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 22 references that are in the map.
- First posted on arXiv 6 Mar 2024.

## Builds on (most cited first, 12 of 12)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Self-Supervised Primal-Dual Learning for Constrained Optimization](https://mlsanity.com/paper/2208.09046/) (2022)
- [Learning for Constrained Optimization: Identifying Optimal Active Constraint Sets](https://mlsanity.com/paper/1802.09639/) (2018)
- [End-to-End Learning to Warm-Start for Real-Time Quadratic Optimization](https://mlsanity.com/paper/2212.08260/) (2022)
- [Fast Approximations for Job Shop Scheduling: A Lagrangian Dual Deep Learning Method](https://mlsanity.com/paper/2110.06365/) (2021)
- [A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraints](https://mlsanity.com/paper/2307.10459/) (2023)
- [Dual Conic Proxies for AC Optimal Power Flow](https://mlsanity.com/paper/2310.02969/) (2023)
- [Predict-Then-Optimize by Proxy: Learning Joint Models of Prediction and Optimization](https://mlsanity.com/paper/2311.13087/) (2023)

## Built on by (most cited first, 16 of 16)

- [Simultaneous Multi-Robot Motion Planning with Projected Diffusion Models](https://mlsanity.com/paper/2502.03607/) (2025)
- [PGLearn -- An Open-Source Learning Toolkit for Optimal Power Flow](https://mlsanity.com/paper/2505.22825/) (2025)
- [Towards graph neural networks for provably solving convex optimization problems](https://mlsanity.com/paper/2502.02446/) (2025)
- [Enforcing convex constraints in Graph Neural Networks](https://mlsanity.com/paper/2510.11227/) (2025)
- [AL-CoLe: Augmented Lagrangian for Constrained Learning](https://mlsanity.com/paper/2510.20995/) (2025)
- [CausalWrap: Model-Agnostic Causal Constraint Wrappers for Tabular Synthetic Data](https://mlsanity.com/paper/2603.02015/) (2026)
- [mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networks](https://mlsanity.com/paper/2502.02434/) (2025)
- [Dual Optimistic Ascent (PI Control) is the Augmented Lagrangian Method in Disguise](https://mlsanity.com/paper/2509.22500/) (2025)
- [Learning to Optimize by Differentiable Programming](https://mlsanity.com/paper/2601.16510/) (2026)
- [Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks](https://mlsanity.com/paper/2605.07113/) (2026)
- [Large-scale portfolio optimization with variational neural annealing](https://mlsanity.com/paper/2507.07159/) (2025)
- [NASP-T: A Fuzzy Neuro-Symbolic Transformer for Logic-Constrained Aviation Safety Report Classification](https://mlsanity.com/paper/2510.05451/) (2025)
- [WARP: A Benchmark for Primal-Dual Warm-Starting of Interior-Point Solvers](https://mlsanity.com/paper/2605.05728/) (2026)
- [PSyGenTAB: A Privacy-Preserving Framework for Synthetic Clinical Tabular Data Generation via Constrained Optimization](https://mlsanity.com/paper/2606.18518/) (2026)
- [Scaling Laws for Physics-Aware ACOPF Surrogate Learning](https://mlsanity.com/paper/2609.16282/) (2026)
- [AT-SKM-Net: An Accelerated Trainable Sampling Kaczmarz-Motzkin Framework for Linear Hard-Constraint Feasibility on Dynamic Graphs](https://mlsanity.com/paper/2609.30088/) (2026)

## Abstract (arXiv)

Learning to Optimize (LtO) is a problem setting in which a machine learning (ML) model is trained to emulate a constrained optimization solver. Learning to produce optimal and feasible solutions subject to complex constraints is a difficult task, but is often made possible by restricting the input space to a limited distribution of related problems. Most LtO methods focus on directly learning solutions to the primal problem, and applying correction schemes or loss function penalties to encourage feasibility. This paper proposes an alternative approach, in which the ML model is trained instead to predict dual solution estimates directly, from which primal estimates are constructed to form dual-feasible solution pairs. This enables an end-to-end training scheme is which the dual objective is maximized as a loss function, and solution estimates iterate toward primal feasibility, emulating a Dual Ascent method. First it is shown that the poor convergence properties of classical Dual Ascent are reflected in poor convergence of the proposed training scheme. Then, by incorporating techniques from practical Augmented Lagrangian methods, we show how the training scheme can be improved to learn highly accurate constrained optimization solvers, for both convex and nonconvex problems.