Improving Feasibility via Fast Autoencoder-Based Projections

Precise constraint satisfaction is a prerequisite to deploying learned models in many areas, motivating methods that repair raw neural predictions with a repair procedure. Current methods unroll multiple repair steps in training and softly penalize constraint violations that remain after the unroll. This is compute- and memory-intensive, lacks robustness when the repair fails to converge, and surrenders most of the constraint satisfaction work to the repair. Our central finding is that, contrary to common practice, a single detached projection step suffices in training. We accomplish this with a Projection-Adaptive Loss (PAL), which uses the constraint residual after this single step to adaptively weigh constraint penalties on the raw prediction. In experiments, PAL is the only method that retains virtually perfect feasibility on extremely nonlinear constraints, and matches or outperforms current methods on synthetic and engineering benchmarks. Because it only requires a single detached projection step, PAL trains 2.5x faster than the canonical repair-based method (DC3) on its own ACOPF benchmark. PAL can also be trained when constraints are expensive to evaluate (e.g., via neural surrogates), a setting where current unrolled methods are memory-intractable.

Enforcing nonlinear inequality constraints in neural networks remains challenging, especially when the output is subject to many coupled constraints. Existing hard constraint methods often impose structural restrictions on the constraint set or introduce substantial computational overhead for large-scale nonlinear problems. Here, we propose DiffSlack, a differentiable projection layer for nonlinear inequality-constrained neural prediction. DiffSlack reformulates inequalities as equalities with learnable slack variables, which are predicted as part of the augmented network output and provide a data-driven warm start for damped Gauss-Newton projection. The projection layer maps raw predictions onto the augmented feasible manifold while preserving end-to-end differentiability. A two-stage curriculum further stabilizes training and improves constraint satisfaction. We evaluate DiffSlack on vehicle path planning with 200 nonlinear inequality constraints from collision avoidance, curvature limits, and waypoint spacing. Compared with existing learning-based baselines, DiffSlack achieves a higher planning success rate and stronger geometric constraint satisfaction under a comparable inference budget. Ablation studies further show that the hard projection layer reduces sensitivity to supervision quality. Closed-loop tracking in CARLA and real-world vehicle experiments confirms the executability of the generated trajectories. These results demonstrate that DiffSlack provides a practical and scalable approach to embedding hard inequality constraints into neural networks for engineering applications.

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.