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.

翻译：混合整数非线性规划问题广泛存在于能源系统与交通运输等领域，但其求解过程极为困难。机器学习领域的最新进展在优化任务中取得了显著成功，这一方向通常被称为“优化学习”。该方法通过预测模型为具有连续决策变量的优化问题生成解，从而避免了计算代价高昂的优化算法。然而，将学习技术应用于混合整数非线性规划仍面临挑战，主要源于整数决策变量的存在会干扰基于梯度的学习过程。为突破此限制，我们提出了两种可微校正层，在保持梯度信息的同时生成整数输出。结合约束违反的软惩罚机制，我们的框架能够同时处理混合整数非线性规划中的整数性与非线性约束。在包含凸/非凸目标函数/约束条件以及整数/混合整数变量的三类问题上的实验表明，所提出的基于学习的方法能够持续快速地为参数化混合整数非线性规划生成高质量解。随着问题规模增大，传统精确求解器与启发式方法难以找到可行解，而我们的方法仍能持续提供可靠结果。本研究将优化学习的适用范围拓展至混合整数非线性规划领域，为将整数约束整合至深度学习模型开辟了道路。相关代码已发布于 https://github.com/pnnl/L2O-pMINLP。