On the Expressive Power of GNNs to Solve Linear SDPs

TL;DR

This paper investigates the expressive power of GNNs for solving linear SDPs, showing limitations of standard GNNs, proposing a more expressive architecture, and demonstrating practical speedups in solving SDPs.

Contribution

It identifies the limitations of standard GNNs in solving SDPs, proposes a more expressive GNN architecture, and empirically demonstrates improved accuracy and speedups.

Findings

Standard GNNs fail to recover linear SDP solutions.

A more expressive GNN architecture captures SDP structure effectively.

Using learned predictions to warm-start solvers yields up to 80% speedup.

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%.