Generalizable Heuristic Generation Through Large Language Models with Meta-Optimization

TL;DR

This paper tackles generalization and exploration limits in LLM-driven heuristic design for combinatorial optimization by introducing MoH, a two-level meta-optimization framework. An outer loop uses an LLM-based meta-optimizer to construct diverse optimizers, while an inner loop employs these optimizers to evolve task-specific heuristics across multiple COP tasks within a multi-task setting. Empirical results on TSP and online BPP show that MoH-minted meta-optimizers yield state-of-the-art performance and strong cross-size generalization, often discovering interpretable optimization strategies (EC, ACO, PSO, SA, Tabu) beyond fixed workflows. The work demonstrates the potential of LLM-driven optimizer design to broaden the search space and reduce reliance on hand-crafted EC pipelines, albeit with higher computational cost and scope for efficiency improvements.

Abstract

Heuristic design with large language models (LLMs) has emerged as a promising approach for tackling combinatorial optimization problems (COPs). However, existing approaches often rely on manually predefined evolutionary computation (EC) optimizers and single-task training schemes, which may constrain the exploration of diverse heuristic algorithms and hinder the generalization of the resulting heuristics. To address these issues, we propose Meta-Optimization of Heuristics (MoH), a novel framework that operates at the optimizer level, discovering effective optimizers through the principle of meta-learning. Specifically, MoH leverages LLMs to iteratively refine a meta-optimizer that autonomously constructs diverse optimizers through (self-)invocation, thereby eliminating the reliance on a predefined EC optimizer. These constructed optimizers subsequently evolve heuristics for downstream tasks, enabling broader heuristic exploration. Moreover, MoH employs a multi-task training scheme to promote its generalization capability. Experiments on classic COPs demonstrate that MoH constructs an effective and interpretable meta-optimizer, achieving state-of-the-art performance across various downstream tasks, particularly in cross-size settings.

Figures (16)

• Figure 1: Generalization performance of the evolved improvement heuristics for TSP.
• Figure 2: Overview of MoH. In iteration $t$, the current meta-optimizer $\mathcal{I}^*_{t-1}$ generates $M$ candidate optimizers in the outer loop. Each candidate optimizer is then evaluated through the inner loop, where it generates $K$ heuristics that are applied to $N$ downstream tasks. For each task, the best heuristic is selected, and its utility contributes to the overall utility of the optimizer. After aggregating utility scores across all tasks, the optimizer with the highest utility is selected as the new meta-optimizer $\mathcal{I}^*_{t}$.
• Figure 3: Optimizer Signature. Its detailed implementation is generated by LLMs, enabling it to iteratively generate or refine novel optimization strategies.
• Figure 4: Training convergence curves under different settings.
• Figure 5: Prompts for generating the meta-optimizer and those embedded within the seed optimizer.