HUANet: Hard-Constrained Unrolled ADMM for Constrained Convex Optimization

HUANet presents a deep neural network architecture that unrolls Alternating Direction Method of Multipliers (ADMM) iterations to solve constrained convex optimization problems efficiently. This framework addresses limitations in existing end-to-end learning methods by explicitly enforcing equality constraints via a differentiable correction stage and promoting convergence through the integration of first-order optimality conditions into the training loss.

Problem Formulation and Motivation

Constrained convex optimization is crucial across many disciplines, but traditional solvers face challenges related to computational efficiency, feasibility guarantees, and solution optimality for large-scale problems. The Alternating Direction Method of Multipliers (ADMM) is a widely adopted first-order method due to its decomposability; however, it often requires a large number of iterations and suffers from slow tail convergence. Learning-to-optimize (L2O) methods have emerged, but end-to-end surrogate models struggle with constraint satisfaction. Algorithm unrolling methods embed domain knowledge into neural network architectures by representing each classical algorithm iteration as a layer, offering better data efficiency and interpretability than purely black-box mappings.

HUANet Architecture and Operation

HUANet is a deep unrolled ADMM framework where each ADMM iteration (3) is mapped to a sequential neural layer, denoted as Fθp. The core of the architecture is the Hard-Constrained Neural Network (HNNθp), which consists of a Multi-Layer Perceptron (MLP) that generates an unconstrained prior estimate of the primal–slack pair, followed by a correction stage. This correction stage acts as an equality feasibility corrector, mapping the prior estimate onto the affine subspace defined by the equality constraints. The output is a posterior estimation (ˆx k+1, s k+1) that satisfies both equality constraints exactly.

Optimality-Driven Convergence Training

The training of HUANet is guided by a self-supervised loss function designed to promote solution quality and convergence. This loss function combines three components: the terminal objective value promoting solution quality, an inequality-constraint violation penalty at the final layer enforcing feasibility, and KKT residual terms facilitating convergence. Specifically, the KKT residual (r k+1) is computed using first-order optimality conditions (10) and (11), which are then incorporated into the training loss to drive the network toward true optimal solutions. This approach ensures that when r k+1 = 0, the layer mimics an ADMM iteration, promoting convergence.

Empirical Validation and Performance

HUANet was validated on three problem types: LASSO, Quadratic Programming (QP), and Entropy Maximization. Experimental results show that HUANet achieves small optimality gaps (averaging less than 0.6% across both dimensional settings) and satisfies equality constraints to machine precision. Crucially, in high-dimensional scenarios (e.g., nx = 100), HUANet demonstrates significant speedup over classical solvers like Clarabel and OSQP, running orders of magnitude faster (e.g., 8× faster than Clarabel for LASSO at nx=100). The scalability advantage is highlighted by the comparison against vanilla ADMM, where HUANet achieves approximately 4700× speedup at nx = 100 compared to ADMM's 1240× increase.

Conclusion and Future Directions

HUANet successfully provides a scalable framework for parametric constrained convex optimization by accelerating ADMM through unrolled neural networks and guaranteeing equality constraint satisfaction via a correction stage. The integration of KKT optimality conditions into the training process facilitates convergence to optimal solutions without requiring ground-truth solutions. Future work will focus on embedding both equality and inequality constraints directly into the network architecture, eliminating the need for explicit correction stages, and extending the framework to nonconvex problems.

The gist

HUANet is a deep unrolled ADMM framework that enforces equality constraint satisfaction at every unrolled iteration via a correction stage in each neural layer while integrating first-order optimality conditions into the training loss to drive convergence toward optimal solutions.

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How it works

HUANet unfolds the ADMM iterations (3) into N sequential neural layers, where each layer executes an identical neural mapping Fθp. The architecture of each layer is defined by a hard-constrained neural network (HNNθp), which consists of an MLP generating a prior estimate without enforcing equality constraints, followed by a correction stage. This stage acts as an equality feasibility corrector, mapping the prior estimate onto the affine subspace defined by the equality constraints to produce a posterior estimation (ˆx k+1, s k+1). The MLP learns the mapping from inputs (q k, λ) to this unconstrained estimate.

Based on the provided scientific paper, HUANet: Hard-Constrained Unrolled ADMM for Constrained Convex Optimization, here are specific improvements that can be made to existing AI systems, leveraging the capabilities described in HUANet:

The core improvement is shifting from black-box solution mapping to a structured, iterative optimization framework that explicitly enforces constraints and drives convergence towards optimality.

Here are the specific improvements and what the improved AI system can do:

1. Enhance solving of high-dimensional, constrained convex problems (e.g., in robotics path planning, power grid distribution, portfolio optimization).

2. Achieve significant computational speedups over classical iterative solvers (like vanilla ADMM) and traditional solvers (like OSQP or Clarabel), especially as problem dimensions scale from low to high dimensions.

3. Guarantee strict feasibility for equality constraints at every iteration of the optimization process, achieving machine precision satisfaction where previous end-to-end methods failed.

4. Accelerate the convergence rate of iterative optimization algorithms by incorporating first-order optimality conditions directly into the training loss function, ensuring iterates move systematically toward a true optimal solution rather than just satisfying a surrogate objective.

Specific capabilities of the improved AI system:

1. A deep learning model capable of solving complex, large-scale constrained convex optimization problems (like LASSO regression or Quadratic Programming) in real-time or near real-time scenarios, overcoming the computational bottlenecks inherent in classical solvers for high dimensions.

2. A guaranteed feasible solution generator for constrained systems where strict adherence to equality constraints is paramount (e.g., ensuring physical system constraints are never violated).

3. An accelerated solver that can process new problem instances quickly by leveraging its learned mapping from problem parameters to solutions, reducing the need to re-execute computationally expensive classical optimization routines for every parameter change.

4. A training paradigm that self-supervises the learning process using KKT residuals, allowing the AI system to learn the underlying mathematical structure of ADMM convergence rather than just approximating a final output.

This paper presents HUANet, a constrained deep neural network architecture that unrolls the Alternating Direction Method of Multipliers (ADMM) into a trainable neural network for accelerating parametric constrained convex optimization. Existing end-to-end learning methods operate as black-box mappings from parameters to solutions, often without explicitly incorporating optimality principles or guaranteeing constraint satisfaction. To address these limitations, HUANet embeds a hard-constrained neural network within each unrolled ADMM iteration, where a differentiable correction stage enforces the affine equalities of the primal subproblem. Furthermore, we incorporate first-order optimality conditions into a self-supervised training loss to promote the convergence of the proposed unrolled algorithm. Extensive numerical experiments for benchmark optimization problems and a control application demonstrate and validate the effectiveness of HUANet in accelerating constrained convex optimization solving.