EG-Opt: Efficient Learning for Nonlinear Programming via Explicit Gradients from Sample-Calibrated Linear Surrogates

Large-scale nonlinear programs remain challenging, especially in repeated applications. Learning-based methods enable fast inference. However, feasibility-preserving training often requires repeated nonlinear solutions or iterative projections, while implicit differentiation or unrolling adds substantial computational overhead. We propose EG-Opt, an efficient framework based on explicit gradients from sample-calibrated linear surrogates. During training, EG-Opt replaces repeated nonlinear completion with a reusable linear surrogate enabling explicit gradient propagation and efficient batching, caching, and factorization reuse. In addition, we derive local error bounds for state and gradient, and provide sufficient conditions for surrogate gradients to be descent directions for the exact completion loss. At inference, the linear surrogate warm-starts the original nonlinear completion solver, reducing iterations and helping recover the reference solution branch associated with the labels when nonlinear completion admits multiple roots, while ensuring equality feasibility. Across ten benchmarks spanning alternate-current optimal power flow, water distribution, gas flow optimization, and a controlled nonlinear problem, EG-Opt achieves full equality and inequality feasibility with high solution quality. It speeds up end-to-end training by up to 128× versus implicit differentiation and 2,160× versus two-point zeroth-order training while accelerating inference by approximately 1.5× over nonlinear-completion baselines.