Insight-Driven Search: A Framework for Multi-objective Automated Heuristic Design with Large Language Models

Automated Heuristic Design (AHD) with Large Language Models (LLMs) has shown promise in solving optimization tasks. However, existing methods mainly rely on iterative search frameworks and do not systematically structure or reuse task-specific design knowledge elicited from LLM priors and accumulated during online search. Moreover, multi-objective AHD introduces additional challenges, as conflicting objectives such as solution quality and computational efficiency must be balanced during heuristic search. We propose Multi-objective Evolution of Heuristics with Insight-Driven Search (MEoH-IDS), which uses dynamically maintained design insights to guide LLM-based heuristic generation. MEoH-IDS maintains an insight pool by extracting insights from successful heuristics, evaluating their search utility, and filtering less effective ones over time. It selects promising insight combinations using a UCB-inspired policy, with rewards based on non-dominated status in the current Pareto archive. We validate MEoH-IDS on three AHD tasks, including the Traveling Salesman Problem (TSP), Capacitated Vehicle Routing Problem (CVRP), and Vehicle Routing Problem with Time Windows (VRPTW), optimizing both solution quality and computational efficiency. Experimental results show that MEoH-IDS accelerates convergence and discovers heuristics with improved quality--efficiency trade-offs compared with representative LLM-based AHD baselines.