Data-driven Projection Generation for Efficiently Solving Heterogeneous Quadratic Programming Problems

2025-10-29

AI Summary

To address the low computational efficiency of solving high-dimensional heterogeneous quadratic programs (QPs), this paper proposes a data-driven, instance-adaptive projection-based dimensionality reduction framework. Methodologically, it introduces the first use of graph neural networks (GNNs) to generate instance-specific linear projection matrices that map original high-dimensional decision variables into low-dimensional subspaces. A bilevel optimization algorithm is designed that enables end-to-end differentiable training without requiring backpropagation through embedded QP solvers. Furthermore, we derive a theoretical bound on the projection-induced generalization error. Experiments across diverse heterogeneous QP benchmarks demonstrate that our approach achieves an average 2.1× speedup in solution time while preserving solution feasibility and quality. It consistently outperforms both state-of-the-art learning-augmented methods and conventional heuristics.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Efficiently solving high-dimensional quadratic programming problems
• Generating instance-specific projections via graph neural networks
• Reducing computation time while maintaining solution quality

Innovation

• Data-driven projection reduces QP variables
• Graph neural network generates instance-specific projections
• Bilevel optimization trains model without solver backpropagation