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Warm-starting active-set solvers using graph neural networks

Ella J. Schmidtobreick, Daniel Arnström, Paul Häusner, Jens Sjölund

TL;DR

The paper addresses speeding up quadratic programs by warm-starting a dual active-set solver with a structure-aware graph neural network that predicts the active constraint set $\mathcal{A}^*$. It represents QPs as bipartite graphs and uses a LEConv-based GNN to map problem structure to constraint activity, enabling fast, accurate predictions that feed into the DAQP solver. Empirical results on synthetic data and MPC-like tasks show consistent reductions in solver iterations and solve time, with strong generalization to larger problem sizes and performance comparable to an MLP baseline. The approach offers practical gains for real-time optimization in sequential settings such as model predictive control, while highlighting avenues for predicting the number of active constraints and extending to other problem classes.

Abstract

Quadratic programming (QP) solvers are widely used in real-time control and optimization, but their computational cost often limits applicability in time-critical settings. We propose a learning-to-optimize approach using graph neural networks (GNNs) to predict active sets in the dual active-set solver DAQP. The method exploits the structural properties of QPs by representing them as bipartite graphs and learning to identify the optimal active set for efficiently warm-starting the solver. Across varying problem sizes, the GNN consistently reduces the number of solver iterations compared to cold-starting, while performance is comparable to a multilayer perceptron (MLP) baseline. Furthermore, a GNN trained on varying problem sizes generalizes effectively to unseen dimensions, demonstrating flexibility and scalability. These results highlight the potential of structure-aware learning to accelerate optimization in real-time applications such as model predictive control.

Warm-starting active-set solvers using graph neural networks

TL;DR

The paper addresses speeding up quadratic programs by warm-starting a dual active-set solver with a structure-aware graph neural network that predicts the active constraint set . It represents QPs as bipartite graphs and uses a LEConv-based GNN to map problem structure to constraint activity, enabling fast, accurate predictions that feed into the DAQP solver. Empirical results on synthetic data and MPC-like tasks show consistent reductions in solver iterations and solve time, with strong generalization to larger problem sizes and performance comparable to an MLP baseline. The approach offers practical gains for real-time optimization in sequential settings such as model predictive control, while highlighting avenues for predicting the number of active constraints and extending to other problem classes.

Abstract

Quadratic programming (QP) solvers are widely used in real-time control and optimization, but their computational cost often limits applicability in time-critical settings. We propose a learning-to-optimize approach using graph neural networks (GNNs) to predict active sets in the dual active-set solver DAQP. The method exploits the structural properties of QPs by representing them as bipartite graphs and learning to identify the optimal active set for efficiently warm-starting the solver. Across varying problem sizes, the GNN consistently reduces the number of solver iterations compared to cold-starting, while performance is comparable to a multilayer perceptron (MLP) baseline. Furthermore, a GNN trained on varying problem sizes generalizes effectively to unseen dimensions, demonstrating flexibility and scalability. These results highlight the potential of structure-aware learning to accelerate optimization in real-time applications such as model predictive control.

Paper Structure

This paper contains 18 sections, 11 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Active-set method
  4. Dual active-set solver
  5. Graph Neural Networks
  6. Method
  7. Graph representation
  8. Learning problem
  9. Model architecture
  10. Experiments
  11. Synthetic data generation
  12. Active-set prediction using GNN
  13. Impact of problem structure on predictive performance
  14. Generalization across variable problem sizes
  15. Effect of predictive model on solver efficiency
  16. ...and 3 more sections

Figures (8)

  • Figure 1: Method overview: The problem is formulated as a QP and represented as a graph, which serves as input to the Graph Neural Network (GNN) for node-level prediction. The resulting prediction is then used to warm-start the DAQP solver.
  • Figure 2: A quadratic program can be represented as a bipartite graph having a set of variable nodes and a set of constraint nodes, with edge features describing relationships between variables.
  • Figure 3: Comparison of iterations when warm-starting the DAQP solver with predictions from our graph neural network, a standard multilayer perceptron and cold-starting the solver without any learned prediction.
  • Figure 4: Comparison of solve time and prediction time for (a) our graph neural network, (b) a standard multilayer perceptron, and (c) the cold-started active set method.
  • Figure 5: Comparison of iterations (left) and solve time (right) of cold-starting (purple) and warm-starting the DAQP solver using the GNN (green) on problem instances with $100$ variables and $200$ constraints. The dashed line represents the mean.
  • ...and 3 more figures

Theorems & Definitions (1)

  • definition 1