# Learning to Solve Optimization Problems with Hard Linear Constraints

arXiv: https://arxiv.org/abs/2208.10611 · HTML: https://mlsanity.com/paper/2208.10611/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Aug 2022.

## Built on by (most cited first, 23 of 23)

- [End-to-End Feasible Optimization Proxies for Large-Scale Economic Dispatch](https://mlsanity.com/paper/2304.11726/) (2023)
- [Learning to Warm-Start Fixed-Point Optimization Algorithms](https://mlsanity.com/paper/2309.07835/) (2023)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [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)
- [Dual Lagrangian Learning for Conic Optimization](https://mlsanity.com/paper/2402.03086/) (2024)
- [Enforcing Hard Linear Constraints in Deep Learning Models with Decision Rules](https://mlsanity.com/paper/2505.13858/) (2025)
- [Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guarantees](https://mlsanity.com/paper/2410.11061/) (2024)
- [Towards graph neural networks for provably solving convex optimization problems](https://mlsanity.com/paper/2502.02446/) (2025)
- [Deep Learning-based Techniques for Integrated Sensing and Communication Systems: State-of-the-Art, Challenges, and Opportunities](https://mlsanity.com/paper/2509.06968/) (2025)
- [Self-supervised Equality Embedded Deep Lagrange Dual for Approximate Constrained Optimization](https://mlsanity.com/paper/2306.06674/) (2023)
- [Compact Optimality Verification for Optimization Proxies](https://mlsanity.com/paper/2405.21023/) (2024)
- [SnareNet: Flexible Repair Layers for Neural Networks with Hard Constraints](https://mlsanity.com/paper/2602.09317/) (2026)
- [An Efficient Learning-Based Solver for Two-Stage DC Optimal Power Flow with Feasibility Guarantees](https://mlsanity.com/paper/2304.01409/) (2023)
- [HoP: Homeomorphic Polar Learning for Hard Constrained Optimization](https://mlsanity.com/paper/2502.00304/) (2025)
- [Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks](https://mlsanity.com/paper/2605.07113/) (2026)
- [DiffSlack: Learning under Nonlinear Inequality Constraints via Learnable Slack Variables](https://mlsanity.com/paper/2606.05247/) (2026)
- [LOOP-MAC: Learning-Based Multi-Agent Coordination with Guaranteed Local Feasibility in a Virtual Power Plant](https://mlsanity.com/paper/2310.17882/) (2023)
- [Toward Rapid, Optimal, and Feasible Power Dispatch through Generalized Neural Mapping](https://mlsanity.com/paper/2311.04838/) (2023)
- [Impact of Data Poisoning Attacks on Feasibility and Optimality of Neural Power System Optimizers](https://mlsanity.com/paper/2502.05727/) (2025)
- [Navigating Demand Uncertainty in Container Shipping: Deep Reinforcement Learning for Enabling Adaptive and Feasible Master Stowage Planning](https://mlsanity.com/paper/2502.12756/) (2025)
- [WARP: A Benchmark for Primal-Dual Warm-Starting of Interior-Point Solvers](https://mlsanity.com/paper/2605.05728/) (2026)
- [From Accounting to Coordination: A Virtual Water-Aware Electricity-Computation-Water Nexus Framework for Data Center Dispatch](https://mlsanity.com/paper/2605.25854/) (2026)

## Abstract (arXiv)

Constrained optimization problems appear in a wide variety of challenging real-world problems, where constraints often capture the physics of the underlying system. Classic methods for solving these problems rely on iterative algorithms that explore the feasible domain in the search for the best solution. These iterative methods are often the computational bottleneck in decision-making and adversely impact time-sensitive applications. Recently, neural approximators have shown promise as a replacement for the iterative solvers that can output the optimal solution in a single feed-forward providing rapid solutions to optimization problems. However, enforcing constraints through neural networks remains an open challenge. This paper develops a neural approximator that maps the inputs to an optimization problem with hard linear constraints to a feasible solution that is nearly optimal. Our proposed approach consists of four main steps: 1) reducing the original problem to optimization on a set of independent variables, 2) finding a gauge function that maps the infty-norm unit ball to the feasible set of the reduced problem, 3)learning a neural approximator that maps the optimization's inputs to an optimal point in the infty-norm unit ball, and 4) find the values of the dependent variables from the independent variable and recover the solution to the original problem. We can guarantee hard feasibility through this sequence of steps. Unlike the current learning-assisted solutions, our method is free of parameter-tuning and removes iterations altogether. We demonstrate the performance of our proposed method in quadratic programming in the context of the optimal power dispatch (critical to the resiliency of our electric grid) and a constrained non-convex optimization in the context of image registration problems.