From Hand-Crafted to LLM-Based Variation Operators in Metaheuristics: A Tutorial

Large Language Models (LLMs) are increasingly involved in complex mathematical optimization, even if the pragmatic user who triggers them is unaware of it. After all, many real-world problems reduce to the search for better or the best solutions. The field of LLM-as-optimizer has three paradigms: direct optimization, tool-augmented optimization, and tool-creating optimization. Direct optimization uses iterative prompting and heuristic generation to navigate solution spaces. Tool-augmented optimization translates natural language problems into formal specifications and orchestrates external solvers. Tool-creating optimization goes further, using LLMs to discover reusable algorithms or heuristics that can be deployed at zero marginal LLM cost. We describe current performance frontiers based on the benchmarks from the literature. We identify the critical reasoning gap in current architectures and argue for trade-offs between the future potential of direct optimization and the auditability of tool-augmented optimization. Even future, more powerful models might opt for tool-making to improve operational efficiency for repetitive families of problems.

Large language models (LLMs) have made automated heuristic design (AHD) increasingly practical by generating executable heuristic code from task descriptions and evaluator feedback. Yet under a limited query and evaluation budget, search efficiency depends critically on a pre-generation decision. Before each LLM query and black-box evaluation, the system must choose which archived heuristics to reuse as parents and which generation operator should transform them. Existing methods typically choose such actions with predefined rules, leaving the expected outcome of each concrete operator-parent action only indirectly modeled. Therefore, we propose \fullmethod{} (\method{}), a surrogate-guided action-selection module for operator-parent selection in LLM-based AHD. \method{} guides the LLM code-generation process by scoring pre-generation actions with two complementary surrogates. Specifically, a transition surrogate is proposed to predict the latent distribution of the child representation induced by an operator-parent action, while an instance-conditioned utility surrogate is proposed to estimate the expected performance of sampled child latents. Moreover, we propose an uncertainty-aware acquisition rule that combines predicted utility, utility uncertainty, and transition uncertainty to select the next LLM generation action. Across a diverse heuristic-design suite, \method{} is competitive with strong LLM-AHD baselines, and ablation and action-selection analyses suggest that its behavior goes beyond simple archive ranking or fixed operator preferences.

Large Language Model (LLM) based automated heuristic design (AHD) has shown great potential in discovering efficient heuristics. Most existing LLM-AHD frameworks use semantic evolutionary operators that rely entirely on the LLM's pre-trained knowledge. These one-stage methods strictly require the generated code to be valid during the operation and often rely on a ``thought-code'' representation. We argue that this end-to-end generation fundamentally limits the exploration ability within the algorithm search space. In this paper, we propose a two-stage, structure-based evolutionary operator for LLM-AHD. In the first stage, our approach directly performs crossover and mutation on the Abstract Syntax Trees (ASTs) of the heuristic code, intentionally generating diverse but often invalid structural variants. In the second stage, the LLM is employed to repair these invalid heuristics into executable, high-quality code. Depending on the underlying framework, either the raw invalid variants or the repaired heuristics are integrated into the population to preserve potential structural patterns. We demonstrate that the proposed operator can significantly enhance the search ability of state-of-the-art LLM-AHD algorithms, such as EoH-S. Experimental results on the Traveling Salesman Problem (TSP) and the Online Bin Packing Problem (OBP) show that our method effectively improves both optimization performance and convergence speed.