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

TL;DR

This paper introduces a neural network-based approach to generate instance-specific projections that significantly reduce the complexity of high-dimensional quadratic programming problems, leading to faster solutions with high quality.

Contribution

It presents a novel data-driven framework using graph neural networks to produce tailored projections for QPs, along with an efficient bilevel optimization algorithm for training.

Findings

Outperforms existing methods in solution quality and speed

Provides theoretical analysis of generalization ability

Produces high-quality solutions for unseen QP instances

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.