We propose a data-driven framework for efficiently solving quadratic programming (QP) problems by reducing the number of variables in high-dimensional QPs using instance-specific projection. A graph neural network-based model is designed to generate projections tailored to each QP instance, enabling us to produce high-quality solutions even for previously unseen problems. The model is trained on heterogeneous QPs to minimize the expected objective value evaluated on the projected solutions. This is formulated as a bilevel optimization problem; the inner optimization solves the QP under a given projection using a QP solver, while the outer optimization updates the model parameters. We develop an efficient algorithm to solve this bilevel optimization problem, which computes parameter gradients without backpropagating through the solver. We provide a theoretical analysis of the generalization ability of solving QPs with projection matrices generated by neural networks. Experimental results demonstrate that our method produces high-quality feasible solutions with reduced computation time, outperforming existing methods.

翻译：我们提出了一种数据驱动的框架，通过利用实例特定的投影来减少高维二次规划问题中的变量数量，从而高效求解二次规划问题。设计了一种基于图神经网络的模型，为每个二次规划实例生成定制化的投影，使我们能够为未见过的二次规划问题生成高质量的解。该模型在异构二次规划问题上进行训练，以最小化投影解上的期望目标值。这被表述为一个双层优化问题：内层优化在给定投影下使用二次规划求解器求解二次规划问题，而外层优化则更新模型参数。我们开发了一种高效算法来求解该双层优化问题，该算法无需通过求解器反向传播即可计算参数梯度。我们对使用神经网络生成的投影矩阵求解二次规划问题的泛化能力进行了理论分析。实验结果表明，我们的方法能够以更短的计算时间生成高质量可行解，性能优于现有方法。