HUANet: Hard-Constrained Unrolled ADMM for Constrained Convex Optimization

Trinh Tran; Binh Nguyen; Truong X. Nghiem

math.OCApr 14, 2026

HUANet: Hard-Constrained Unrolled ADMM for Constrained Convex Optimization

Authors: Trinh Tran, Binh Nguyen, Truong X. Nghiem

Organizations: Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816, USA

Abstract

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.

Figures & tables

Fig. 1 : A neural layer

F_{\theta_{p}}

of HUANet for an ADMM iteration at step

k

Fig. 3 : Run time comparison for the QP problem.

Panel of Figure 3 — Fig. 4 : Average solve time of HUANet and ADMM for the QP and entropy problems.

Fig. 5 : Run time comparison for the entropy problem.

Problem	$n_{x}$	$(n_{\mathrm{eq}},n_{\mathrm{in}})$	Optimality gap (%)	Eq. violation	Ineq. violation	Run time (s)
QP	10	(5, 5)	$4.56\text{\times}{10}^{-3}$ ( $2.42\text{\times}{10}^{-2}$ )	$2.19\text{\times}{10}^{-15}$ ( $3.55\text{\times}{10}^{-15}$ )	$6.99\text{\times}{10}^{-11}$ ( $1.33\text{\times}{10}^{-8}$ )	$8.04\text{\times}{10}^{-5}$ ( $2.28\text{\times}{10}^{-4}$ )
	100	(50, 50)	$5.94\text{\times}{10}^{-2}$ ( $2.62\text{\times}{10}^{-1}$ )	$1.31\text{\times}{10}^{-14}$ ( $2.49\text{\times}{10}^{-14}$ )	$3.55\text{\times}{10}^{-5}$ ( $4.19\text{\times}{10}^{-4}$ )	$1.27\text{\times}{10}^{-4}$ ( $2.97\text{\times}{10}^{-4}$ )
Entropy	10	(1, 5)	$2.83\text{\times}{10}^{-4}$ ( $1.60\text{\times}{10}^{-2}$ )	$7.39\text{\times}{10}^{-17}$ ( $4.44\text{\times}{10}^{-16}$ )	$2.25\text{\times}{10}^{-8}$ ( $3.09\text{\times}{10}^{-6}$ )	$7.34\text{\times}{10}^{-5}$ ( $1.33\text{\times}{10}^{-4}$ )
	100	(1, 50)	$6.18\text{\times}{10}^{-3}$ ( $4.87\text{\times}{10}^{-2}$ )	$9.66\text{\times}{10}^{-17}$ ( $4.45\text{\times}{10}^{-16}$ )	$9.56\text{\times}{10}^{-8}$ ( $2.14\text{\times}{10}^{-6}$ )	$1.15\text{\times}{10}^{-4}$ ( $1.96\text{\times}{10}^{-4}$ )
Energy: HUANet	384	(192, 481)	$5.64\text{\times}{10}^{-2}$ ( $2.74\text{\times}{10}^{-1}$ )	$5.86\text{\times}{10}^{-15}$ ( $7.49\text{\times}{10}^{-15}$ )	$3.81\text{\times}{10}^{-5}$ ( $1.66\text{\times}{10}^{-3}$ )	$1.16\text{\times}{10}^{-3}$ ( $1.33\text{\times}{10}^{-3}$ )
Energy: DC3	384	(192, 481)	$4.05$ ( $5.67$ )	$7.79\text{\times}{10}^{-15}$ ( $1.42\text{\times}{10}^{-14}$ )	$1.54\text{\times}{10}^{-15}$ ( $1.42\text{\times}{10}^{-14}$ )	$1.16\text{\times}{10}^{-4}$ ( $3.40\text{\times}{10}^{-4}$ )

TABLE I : Performance across problem classes. QP and entropy results are for HUANet; energy management results compare HUANet with DC3. Entries report mean (maximum) over

1000

test instances.

Method	Run time (ms)	Speedup*
HUANet	$1.16\pm 0.05$	$12.58\times$
Clarabel	$14.60\pm 0.47$	$1.00\times$
Vanilla ADMM	$3026.93\pm 2669.44$	$0.005\times$

TABLE II : Runtime comparison for the energy problem.

Fig. 6 : Day-ahead microgrid schedule predicted by HUANet over a 24-hour horizon.

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HardNet++: Nonlinear Constraint Enforcement in Neural Networks

Andrea Goertzen, Kaveh Alim, Youngjae Min, Navid Azizan

Massachusetts Institute of Technology

Inequality Constraints Structural Constraints Neural Network Learning-Based Control Hard

Open paper AskPDF arXiv

Enforcing constraint satisfaction in neural network outputs is critical for safety, reliability, and physical fidelity in many control and decision-making applications. While soft-constrained methods penalize constraint violations during training, they do not guarantee constraint adherence during inference. Other approaches guarantee constraint satisfaction via a projection layer, but often rely on the existence of a tractable projection onto the feasible set, limiting their utility in more general problem settings. Many real-world problems of interest are nonlinear and lack the special structure admitting a tractable projection, motivating the development of methods that can enforce general nonlinear constraints. To this end, we introduce HardNet++, a constraint-satisfaction method that enforces linear and nonlinear equality and inequality constraints. Our approach iteratively adjusts the network output via damped local linearizations of the constraints. Each iteration is differentiable, admitting an end-to-end training framework, where the constraint satisfaction layer is active during training. We show that under certain regularity conditions, this procedure enforces nonlinear constraint satisfaction to arbitrary tolerance. Finally, we demonstrate tight constraint adherence without loss of optimality in a learning-for-optimization context, where we apply this method to a nonlinear model predictive control problem.

Abstract

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