On the Expressive Power of GNNs to Solve Linear SDPs

2026-04-30

AI Summary

Standard graph neural networks (GNNs) struggle to effectively solve large-scale linear semidefinite programming (SDP) problems. This work systematically analyzes the representational limitations of GNNs in SDP solving and introduces a novel, highly expressive GNN architecture capable of precisely emulating the iterative update dynamics of first-order optimization solvers while explicitly modeling key structural properties of SDPs. Evaluated on both synthetic datasets and the SDPLib benchmark suite, the proposed method significantly reduces prediction error and optimality gap, and achieves up to 80% acceleration in solution time under warm-start settings. These results demonstrate the effectiveness and promise of integrating machine learning with convex optimization for scalable SDP solving.

Abstract

Semidefinite programs (SDPs) are a powerful framework for convex optimization and for constructing strong relaxations of hard combinatorial problems. However, solving large SDPs can be computationally expensive, motivating the use of machine learning models as fast computational surrogates. Graph neural networks (GNNs) are a natural candidate in this setting due to their sparsity-awareness and ability to model variable-constraint interactions. In this work, we study what expressive power is sufficient to recover optimal SDP solutions. We first prove negative results showing that standard GNN architectures fail on recovering linear SDP solutions. We then identify a more expressive architecture that captures the key structure of SDPs and can, in particular, emulate the updates of a standard first-order solver. Empirically, on both synthetic and \textsc{SdpLib} benchmarks of various classes of SDPs, this more expressive architecture achieves consistently lower prediction error and objective gap than theoretically weaker baselines. Finally, using the learned high-quality predictions to warm-start the first-order solver yields practical speedups of up to 80%.

Problem

• Semidefinite Programming
• Graph Neural Networks
• Expressive Power
• Convex Optimization
• Computational Surrogates

Innovation

• Graph Neural Networks
• Semidefinite Programming
• Expressive Power
• First-order Solver
• Warm-start