Only Project Once: Projection-Adaptive Loss for Exact Constraint Satisfaction

2026-10-03

AI Summary

This study addresses the high computational overhead, excessive memory consumption, and limited robustness of existing constraint satisfaction methods by proposing the Projection-Adaptive Loss (PAL) framework. Departing from conventional multi-step unfolding paradigms, PAL integrates deep learning with a projected gradient method, enforcing nonlinear constraints during training through a single decoupled projection step coupled with an adaptive weighting mechanism. Experimental results demonstrate that PAL accelerates training by 2.5× while maintaining near-perfect constraint feasibility under extreme nonlinearity. By significantly outperforming established baselines, this work achieves an effective unification of resource efficiency and solution accuracy.

Abstract

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.

Problem

• constraint satisfaction
• neural network repair
• projection
• computational efficiency
• memory intractability

Innovation

• Projection-Adaptive Loss
• Exact Constraint Satisfaction
• Detached Projection
• Adaptive Penalty
• Neural Surrogates