On the Expressive Power of GNNs to Solve Linear SDPs

arXiv cs.LG / 5/1/2026

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Key Points

The paper investigates which graph neural network (GNN) expressive capabilities are sufficient to recover optimal solutions to linear semidefinite programs (SDPs), motivated by the high cost of solving large SDPs.
It proves negative results that common (standard) GNN architectures cannot reliably recover linear SDP solutions.
The authors propose a more expressive GNN architecture designed to capture the core structure of SDPs and to emulate update steps of a standard first-order SDP solver.
Experiments on synthetic data and various 0SdpLib benchmark classes show that the improved architecture achieves lower prediction error and smaller objective gaps than weaker theoretical baselines.
The paper further demonstrates practical gains by using the learned predictions to warm-start the first-order solver, reporting speedups of up to 80%.

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%.