Towards graph neural networks for provably solving convex optimization problems

TL;DR

This paper introduces a graph neural network framework with provable guarantees for solving convex optimization problems, demonstrating improved solution quality, feasibility, and efficiency over existing neural and traditional methods.

Contribution

The authors develop an iterative MPNN approach that can simulate interior-point methods with provable feasibility guarantees for convex optimization problems.

Findings

Outperforms existing neural baselines in solution quality and feasibility

Generalizes well to unseen problem sizes

Achieves faster solutions than some state-of-the-art solvers like Gurobi

Abstract

Recently, message-passing graph neural networks (MPNNs) have shown potential for solving combinatorial and continuous optimization problems due to their ability to capture variable-constraint interactions. While existing approaches leverage MPNNs to approximate solutions or warm-start traditional solvers, they often lack guarantees for feasibility, particularly in convex optimization settings. Here, we propose an iterative MPNN framework to solve convex optimization problems with provable feasibility guarantees. First, we demonstrate that MPNNs can provably simulate standard interior-point methods for solving quadratic problems with linear constraints, covering relevant problems such as SVMs. Secondly, to ensure feasibility, we introduce a variant that starts from a feasible point and iteratively restricts the search within the feasible region. Experimental results show that our approach outperforms existing neural baselines in solution quality and feasibility, generalizes well to unseen problem sizes, and, in some cases, achieves faster solution times than state-of-the-art solvers such as Gurobi.