On Representing Convex Quadratically Constrained Quadratic Programs via Graph Neural Networks
Chenyang Wu, Qian Chen, Akang Wang, Tian Ding, Ruoyu Sun, Wenguo Yang, Qingjiang Shi
TL;DR
This work introduces a tripartite graph representation for convex QCQPs and couples it with tripartite MP-GNNs to represent key QCQP properties. The authors prove a universal approximation result for convex QCQPs, showing MP-GNNs can approximate feasibility, boundedness, optimal value, and the unique minimal-norm optimal solution with arbitrary accuracy. They also show inherent limitations for non-convex QCQPs under this architecture via counterexamples. Empirical experiments on GP, CLS, and TRS datasets demonstrate favorable learning curves, improved generalization with more data and larger models, and substantial runtime speedups over traditional solvers, with validation on real-world-like data. The work advances learning-to-optimize for QCQPs, offering a scalable, structure-aware ML framework and establishing a theoretical connection between QCQPs and GNN representations.
Abstract
Convex quadratically constrained quadratic programs (QCQPs) involve finding a solution within a convex feasible region defined by quadratic constraints while minimizing a convex quadratic objective function. These problems arise in various industrial applications, including power systems and signal processing. Traditional methods for solving convex QCQPs primarily rely on matrix factorization, which quickly becomes computationally prohibitive as the problem size increases. Recently, graph neural networks (GNNs) have gained attention for their potential in representing and solving various optimization problems such as linear programs and linearly constrained quadratic programs. In this work, we investigate the representation power of GNNs in the context of QCQP tasks. Specifically, we propose a new tripartite graph representation for general convex QCQPs and properly associate it with message-passing GNNs. We demonstrate that there exist GNNs capable of reliably representing key properties of convex QCQPs, including feasibility, optimal value, and optimal solution. Our result deepens the understanding of the connection between QCQPs and GNNs, paving the way for future machine learning approaches to efficiently solve QCQPs.