Lagrangian relaxation is a versatile mathematical technique employed to relax constraints in an optimization problem, enabling the generation of dual bounds to prove the optimality of feasible solutions and the design of efficient propagators in constraint programming (such as the weighted circuit constraint). However, the conventional process of deriving Lagrangian multipliers (e.g., using subgradient methods) is often computationally intensive, limiting its practicality for large-scale or time-sensitive problems. To address this challenge, we propose an innovative unsupervised learning approach that harnesses the capabilities of graph neural networks to exploit the problem structure, aiming to generate accurate Lagrangian multipliers efficiently. We apply this technique to the well-known Held-Karp Lagrangian relaxation for the travelling salesman problem. The core idea is to predict accurate Lagrangian multipliers and to employ them as a warm start for generating Held-Karp relaxation bounds. These bounds are subsequently utilized to enhance the filtering process carried out by branch-and-bound algorithms. In contrast to much of the existing literature, which primarily focuses on finding feasible solutions, our approach operates on the dual side, demonstrating that learning can also accelerate the proof of optimality. We conduct experiments across various distributions of the metric travelling salesman problem, considering instances with up to 200 cities. The results illustrate that our approach can improve the filtering level of the weighted circuit global constraint, reduce the optimality gap by a factor two for unsolved instances up to a timeout, and reduce the execution time for solved instances by 10%.

翻译：拉格朗日松弛是一种多功能的数学技术，用于松弛优化问题中的约束条件，以生成对偶界来证明可行解的最优性，并设计约束规划中的高效传播算子（例如加权回路约束）。然而，传统推导拉格朗日乘子的过程（例如使用次梯度方法）通常计算成本高昂，限制了其在大规模或时间敏感问题中的实用性。为应对这一挑战，我们提出了一种创新的无监督学习方法，利用图神经网络的能力来挖掘问题结构，旨在高效生成准确的拉格朗日乘子。我们将该技术应用于旅行商问题中著名的Held-Karp拉格朗日松弛。核心思想是预测准确的拉格朗日乘子，并将其作为生成Held-Karp松弛界的启动点。这些界随后用于增强分支定界算法的过滤过程。与现有文献主要集中在寻找可行解不同，我们的方法从对偶侧入手，表明学习也能加速最优性的证明。我们在度量旅行商问题的各种分布上进行了实验，考虑了最多200个城市的实例。结果表明，我们的方法能提高加权回路全局约束的过滤水平，将未解决实例在超时前的最优性差距减少一半，并将已解决实例的执行时间降低10%。