Learning Initial Basis Selection for Linear Programming via Duality-Inspired Tripartite Graph Representation and Comprehensive Supervision

For the fundamental linear programming (LP) problems, the simplex method remains popular, which usually requires an appropriate initial basis as a warm start to accelerate the solving process. Predicting an initial basis close to an optimal one can often accelerate the solver, but a closer initial basis does not always result in greater acceleration. To achieve better acceleration, we propose a GNN model based on a tripartite graph representation inspired by LP duality. This approach enables more effective feature extraction for general LP problems and enhances the expressiveness of GNNs. Additionally, we introduce novel loss functions targeting basic variable selection and basis feasibility, along with data preprocessing schemes, to further improve learning capability. In addition to achieving high prediction accuracy, we enhance the quality of the initial basis for practical use. Experimental results show that our approach greatly surpasses the state-of-the-art method in predicting initial basis with greater accuracy and in reducing the number of iterations and solving time of the LP solver.

Linear programming is a key mathematical tool used to solve many real-world optimization problems, like scheduling or resource allocation. One popular method to solve these problems, called the simplex method, works faster if it starts from a good initial guess, but finding that guess is tricky. Simply having a guess closer to the best solution doesn’t always speed things up.We developed a new machine learning model using graph neural networks (GNNs) that better understands the structure of these problems by representing them in a novel way inspired by mathematical theory. Our model learns to predict better starting points for the simplex method, improving not only the accuracy of the guess but also making it more useful in practice.In tests, our approach outperformed existing methods by predicting initial solutions more precisely, which led to faster problem-solving and reduced computing time. This advancement can help speed up many optimization tasks in industries relying on linear programming.