Learn to Relax with Large Language Models: Solving Nonlinear Combinatorial Optimization Problems via Bidirectional Coevolution

Nonlinear Combinatorial Optimization Problems (NCOPs) present a formidable computational hurdle in practice, as their nonconvex nature gives rise to multi-modal solution spaces that defy efficient optimization. Traditional constraint relaxation approaches rely heavily on expert-driven, iterative design processes that lack systematic automation and scalable adaptability. While recent Large Language Model (LLM)-based optimization methods show promise for autonomous problem-solving, they predominantly function as passive constraint validators rather than proactive strategy architects, failing to handle the sophisticated constraint interactions inherent to NCOPs.To address these limitations, we introduce the first end-to-end \textbf{Auto}mated \textbf{C}onstraint \textbf{O}ptimization (AutoCO) method, which revolutionizes NCOPs resolution through learning to relax with LLMs.Specifically, we leverage structured LLM reasoning to generate constraint relaxation strategies, which are dynamically evolving with algorithmic principles and executable code through a unified triple-representation scheme. We further establish a novel bidirectional (global-local) coevolution mechanism that synergistically integrates Evolutionary Algorithms for intensive local refinement with Monte Carlo Tree Search for systematic global strategy space exploration, ensuring optimal balance between intensification and diversification in fragmented solution spaces. Finally, comprehensive experiments on three challenging NCOP benchmarks validate AutoCO's consistent effectiveness and superior performance over the baselines.