Learning to Optimize for Mixed-Integer Non-linear Programming

Mixed-integer non-linear programs (MINLPs) arise in various domains, such as energy systems and transportation, but are notoriously difficult to solve. Recent advances in machine learning have led to remarkable successes in optimization tasks, an area broadly known as learning to optimize. This approach includes using predictive models to generate solutions for optimization problems with continuous decision variables, thereby avoiding the need for computationally expensive optimization algorithms. However, applying learning to MINLPs remains challenging primarily due to the presence of integer decision variables, which complicate gradient-based learning. To address this limitation, we propose two differentiable correction layers that generate integer outputs while preserving gradient information. Combined with a soft penalty for constraint violation, our framework can tackle both the integrality and non-linear constraints in a MINLP. Experiments on three problem classes with convex/non-convex objective/constraints and integer/mixed-integer variables show that the proposed learning-based approach consistently produces high-quality solutions for parametric MINLPs extremely quickly. As problem size increases, traditional exact solvers and heuristic methods struggle to find feasible solutions, whereas our approach continues to deliver reliable results. Our work extends the scope of learning-to-optimize to MINLP, paving the way for integrating integer constraints into deep learning models. Our code is available at https://github.com/pnnl/L2O-pMINLP.