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

Precise constraint satisfaction is a prerequisite to deploying learned models in many areas, motivating methods that repair raw neural predictions with a projection procedure. Current repair-based 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. PAL is the only method that retains full 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 $2.5\times$ faster than the canonical repair-based method DC3 on its own AC optimal power flow 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.