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On the Universality and Complexity of GNN for Solving Second-order Cone Programs

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
graph neural networksecond order cone programminglearning to optimizeWeisfeiler-Lehman testsample complexity
Abstract:

Graph Neural Networks (GNNs) have demonstrated both empirical efficiency and universal expressivity for solving constrained optimization problems such as linear and quadratic programming. However, extending this paradigm to more general convex problems with universality guarantees, particularly Second-Order Cone Programs (SOCPs), remains largely unexplored. We address this challenge by proposing a novel graph representation that captures the inherent structure of conic constraints. We then establish a key universality theorem: there exist GNNs that can provably approximate essential SOCP properties, including instance feasibility and optimal solutions. We further derive the sample complexity for GNN generalization based on Rademacher complexity, filling an important gap for Weisfeiler-Lehman-based GNNs in learning-to-optimize paradigms. Our results provide a rigorous foundation linking GNN expressivity and generalization power to conic optimization structure, opening new avenues for scalable, data-driven SOCP solvers. The approach extends naturally to pp-order cone programming for any p≥1p \geq 1 while preserving universal expressivity and requiring no structural modifications to the GNN architecture. Numerical experiments on randomly generated SOCPs and real-world power grid problems demonstrate the effectiveness of our approach, achieving superior prediction accuracy with significantly fewer parameters than fully connected neural networks.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper establishes universal approximation theorems for GNNs solving second-order cone programs and derives sample complexity bounds via Rademacher complexity. It resides in the 'Universal Approximation and Complexity Analysis' leaf under 'Theoretical Foundations and Universal Expressivity', which contains only two papers total (including this work). This sparse population suggests the theoretical analysis of GNN expressivity for SOCP is an emerging research direction, contrasting sharply with the more crowded application branches (e.g., four papers in distribution network reconfiguration, three in cell-free MIMO power control).

The taxonomy reveals that most neighboring work focuses on domain-specific implementations rather than foundational theory. The sibling branches under 'Power System Optimization Applications' and 'Wireless Communication and Beamforming' contain seventeen application-oriented papers that leverage SOCP formulations but do not establish universal expressivity guarantees. The 'Signal Processing and Filter Design' branch addresses graph filter optimization using SOCP but remains distinct from learning-to-optimize paradigms. This structural isolation underscores that rigorous approximation theory for GNN-based SOCP solvers occupies a relatively unexplored niche within the broader field.

Among nine candidates examined, the universal expressivity contribution shows one refutable candidate from three examined, while the novel graph representation shows zero refutations across six candidates. The sample complexity analysis was not tested against prior work (zero candidates examined). The limited search scope means these statistics reflect top-K semantic matches rather than exhaustive coverage. The graph representation appears more distinctive within this sample, whereas the expressivity guarantees encounter at least one overlapping prior result among the small candidate pool examined.

Given the sparse theoretical branch and limited search scope (nine candidates total), the work appears to address a gap in foundational SOCP-GNN theory. However, the analysis cannot confirm whether exhaustive literature review would reveal additional overlapping results, particularly for the expressivity contribution where one refutation was found. The taxonomy structure suggests the field prioritizes applications over theory, positioning this work as foundational infrastructure for the seventeen application-focused papers.

Taxonomy

Core-task Taxonomy Papers
22
3
Claimed Contributions
9
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: solving second-order cone programs using graph neural networks. The field structure reflects a dual emphasis on theoretical rigor and diverse application domains. At the highest level, one branch explores Theoretical Foundations and Universal Expressivity, examining the representational power and complexity guarantees of GNN-based SOCP solvers, as exemplified by GNN SOCP Universality[0] and GNN SOCP Expressivity[19]. In parallel, several application-oriented branches have emerged: Power System Optimization Applications leverage GNN architectures for tasks such as voltage regulation and optimal power flow (e.g., GCN Voltage Regulation[3], Physics Informed ACOPF[2]), while Wireless Communication and Beamforming addresses resource allocation and precoding in multi-user networks (e.g., GNN Power Control[1], GNN Cell Free[5]). Additional branches cover Signal Processing and Filter Design, which adapts graph filters to structured optimization (ARMA Graph Filters[9]), and Mathematical Structures and Graph Theory, which investigates the algebraic underpinnings of graph-based convex solvers. Recent work reveals a tension between universal approximation guarantees and computational scalability in real-time settings. Many studies in power systems and wireless domains prioritize fast, decentralized inference over strict optimality certificates, whereas the theoretical branch seeks to characterize when and how GNNs can provably represent SOCP solutions. GNN SOCP Universality[0] sits squarely within the Theoretical Foundations cluster, focusing on universal approximation and complexity analysis alongside GNN SOCP Expressivity[19]. Compared to application-driven papers like GNN Power Control[1] or Physics Informed ACOPF[2], which embed domain physics into network design, GNN SOCP Universality[0] emphasizes foundational expressivity results that underpin these practical deployments. This positioning highlights an open question: bridging rigorous approximation theory with the sample efficiency and generalization demands of large-scale engineering applications.

Claimed Contributions

Novel graph representation for second-order cone programs

The authors introduce a specialized graph structure that decomposes second-order cone constraints into four types of nodes (decision variables, polyhedron constraints, minor conic constraints, and major conic constraints) with corresponding edges, enabling efficient representation of the hybrid linear and non-linear structure inherent in SOCPs.

6 retrieved papers
Universal expressivity guarantees for SOCP-GNNs

The authors establish a universality theorem proving that their proposed GNN architecture can provably approximate essential SOCP properties such as feasibility and optimal solutions. This theoretical guarantee extends to p-order cone programming for any p ≥ 1 without requiring structural modifications to the GNN.

3 retrieved papers
Can Refute
Sample complexity analysis for GNN generalization

The authors provide the first generalization analysis for Weisfeiler-Lehman-based GNNs in the learning-to-optimize paradigm, deriving sample complexity bounds based on Rademacher complexity. This framework fills an important theoretical gap and extends to other WL-based GNN approaches for optimization problems.

0 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Novel graph representation for second-order cone programs

The authors introduce a specialized graph structure that decomposes second-order cone constraints into four types of nodes (decision variables, polyhedron constraints, minor conic constraints, and major conic constraints) with corresponding edges, enabling efficient representation of the hybrid linear and non-linear structure inherent in SOCPs.

Contribution

Universal expressivity guarantees for SOCP-GNNs

The authors establish a universality theorem proving that their proposed GNN architecture can provably approximate essential SOCP properties such as feasibility and optimal solutions. This theoretical guarantee extends to p-order cone programming for any p ≥ 1 without requiring structural modifications to the GNN.

Contribution

Sample complexity analysis for GNN generalization

The authors provide the first generalization analysis for Weisfeiler-Lehman-based GNNs in the learning-to-optimize paradigm, deriving sample complexity bounds based on Rademacher complexity. This framework fills an important theoretical gap and extends to other WL-based GNN approaches for optimization problems.

Table of Contents