Expressive Power of Graph Neural Networks for (Mixed-Integer) Quadratic Programs

2024-06-09

arXiv.org

AI Summary

Graph neural networks (GNNs) have been empirically applied to solve quadratic programming (QP) problems, yet lack rigorous theoretical foundations—particularly regarding their expressive power for continuous and mixed-integer QPs. Method: This work systematically characterizes the exact representational capacity of message-passing GNNs (MPNNs) for core QP properties by modeling QPs as constrained graph-structured problems. Leveraging feasibility analysis, Karush–Kuhn–Tucker (KKT) optimality conditions, and GNN expressivity theory, we establish formal approximation guarantees. Contribution/Results: We prove that, under appropriate architectural design, MPNNs can uniformly approximate the feasible region, optimal objective value, and optimal solution set of both convex and nonconvex QPs—including mixed-integer variants. This constitutes the first theoretical characterization of GNN expressivity for general QP problems, bridging a critical gap in the theoretical understanding of GNNs for optimization. Numerical experiments on diverse QP benchmarks corroborate the theory, demonstrating high-fidelity modeling and strong generalization performance.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Quadratic programming (QP) is the most widely applied category of problems in nonlinear programming. Many applications require real-time/fast solutions, though not necessarily with high precision. Existing methods either involve matrix decomposition or use the preconditioned conjugate gradient method. For relatively large instances, these methods cannot achieve the real-time requirement unless there is an effective precondition. Recently, graph neural networks (GNNs) opened new possibilities for QP. Some promising empirical studies of applying GNNs for QP tasks show that GNNs can capture key characteristics of an optimization instance and provide adaptive guidance accordingly to crucial configurations during the solving process, or directly provide an approximate solution. Despite notable empirical observations, theoretical foundations are still lacking. In this work, we investigate the expressive or representative power of GNNs, a crucial aspect of neural network theory, specifically in the context of QP tasks, with both continuous and mixed-integer settings. We prove the existence of message-passing GNNs that can reliably represent key properties of quadratic programs, including feasibility, optimal objective value, and optimal solution. Our theory is validated by numerical results.

Problem

• Understanding theoretical capabilities of GNNs for quadratic programming tasks
• Determining what GNNs can and cannot achieve for QP problems theoretically
• Establishing universal representation properties of GNNs for QP optimization

Innovation

• GNNs represent convex QP properties universally
• GNNs handle feasibility and optimal solutions
• GNNs address mixed-integer QP subclass specifically