Learning Through the Hard Problem: Decision-Focused Learning with a Full MILP Combinatorial Layer

While real-world applications frequently require solving problems of mixed-integer linear programs (MILPs) with shared structural properties, commercial solvers process each new instance independently from scratch. Graph neural networks (GNNs) offer a promising alternative by predicting optimal binary decisions to guide solvers through variable fixing or warm-starting. However, hard fixing based on thresholded predictions can render the restricted problem infeasible or substantial suboptimal when the GNN is wrong, and training the predictor with a standard classification loss does not account for how prediction errors propagate into decision quality. We investigate three ways of injecting GNN predictions into a MILP solver: hard fixing, big-$M$ penalization, and a new bounded linear soft-penalization surrogate. We show that the soft surrogate can be trained end-to-end with decision-focused losses (SPO+, Fenchel-Young) without introducing the discontinuities that make hard-thresholded restrictions unstable. We evaluate this framework on an energy-aware lot-sizing problem, both in deterministic and two-stage stochastic settings, and show that the proposed methodology achieves favorable and tunable trade-offs between computational time and solution quality.