NeuralQP: A General Hypergraph-based Optimization Framework for Large-scale QCQPs

TL;DR

NeuralQP proposes a general, hypergraph-based optimization framework for large-scale QCQPs that avoids strong problem assumptions and relies on a small-scale solver. It comprises a Hypergraph-based Neural Prediction stage using UniEGNN to learn problem structure and generate a high-quality initial solution, and a Parallel Neighborhood Optimization stage that repairs infeasible candidates and explores multiple neighborhoods in parallel with adaptive partitioning and McCormick-based repairs. The authors prove UniEGNN is equivalent to IPM for quadratic programming, and they empirically demonstrate that NeuralQP delivers higher-quality solutions faster than state-of-the-art solvers on large-scale QCQPs, including QPLIB instances, while maintaining generalization across problem sizes. This work highlights the potential of ML-based optimization to solve nonlinear programs efficiently with limited solver scales, and outlines avenues for expanding datasets and extending to nonlinear and general integer-variable problems.

Abstract

Machine Learning (ML) optimization frameworks have gained attention for their ability to accelerate the optimization of large-scale Quadratically Constrained Quadratic Programs (QCQPs) by learning shared problem structures. However, existing ML frameworks often rely heavily on strong problem assumptions and large-scale solvers. This paper introduces NeuralQP, a general hypergraph-based framework for large-scale QCQPs. NeuralQP features two main components: Hypergraph-based Neural Prediction, which generates embeddings and predicted solutions for QCQPs without problem assumptions, and Parallel Neighborhood Optimization, which employs a McCormick relaxation-based repair strategy to identify and correct illegal variables, iteratively improving the solution with a small-scale solver. We further prove that our framework UniEGNN with our hypergraph representation is equivalent to the Interior-Point Method (IPM) for quadratic programming. Experiments on two benchmark problems and large-scale real-world instances from QPLIB demonstrate that NeuralQP outperforms state-of-the-art solvers (e.g., Gurobi and SCIP) in both solution quality and time efficiency, further validating the efficiency of ML optimization frameworks for QCQPs.

Figures (7)

• Figure 1: Tripartite representation. The green, blue, and red nodes represent the objective, variables, and constraints, and the edge features are associated with coefficients in the original problem.
• Figure 2: Star expansion. A hypergraph is transformed into a bipartite graph, with nodes on the left and hyperedges on the right.
• Figure 3: Variable relational hypergraph representation.
• Figure 4: Half-convolution.
• Figure 5: An overview of the NeuralQP framework. During hypergraph-based Neural Prediction, the QCQP is encoded into a variable relational hypergraph with initial vertex and hyperedge embeddings from the original problem. Then UniEGNN generates neural embeddings for each variable, utilizing both vertex and hyperedge features by converting the hypergraph into a bipartite graph. An MLP layer then predicts the optimal solution based on these embeddings. In Parallel Neighborhood Optimization, predicted solutions are first relaxed and repaired to obtain a feasible solution, followed by neighborhood partition, Parallel Neighborhood Optimization, and McCormick relaxation-based Q-Repair. The neighborhood search solution is again used as a feasible solution if the time limit is not reached; otherwise, the incumbent solution is output as the optimization result.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Lemma 1
• Theorem 2