BEAM: Bi-level Memory-adaptive Algorithmic Evolution for LLM-Powered Heuristic Design

Large Language Models (LLMs) have advanced Automatic Heuristic Design (AHD) by enabling heuristic generation through reasoning and code synthesis. In LLM-based AHD, the LLM reasons about algorithm design and generates executable heuristic code. Existing architectures adopt two main paradigms: Natural Evolution applies crossover and mutation to this code to explore diverse strategies, but discards the reasoning traces behind the design decisions, weakening knowledge retention; Metacognitive Evolution retains these reasoning traces and refines them through reflection, but lacks population-level recombination, limiting exploration. These limitations reduce search efficiency, stability, and solution quality on complex problems. To address this gap, we propose MeEvo, an AHD framework that cyclically couples Natural Evolution and Metacognitive Evolution with operator balance that shifts from exploration to exploitation. Natural Evolution explores heuristic code while recording LLM-generated reasoning traces, fitness values, errors and best heuristic into a shared history; Metacognitive Evolution then reflects on this history to generate improved heuristics that feed into the next Natural Evolution cycle. This design enables population-driven exploration and reflection-driven refinement to reinforce each other. Experiments on five optimization problems show that MeEvo achieves stronger performance and lower variance than tested LLM-based AHD architectures, especially on complex constrained tasks.

Designing heuristics for combinatorial optimization problems (COPs) is a fundamental yet challenging task that traditionally requires extensive domain expertise. Recently, Large Language Model (LLM)-based Automated Heuristic Design (AHD) has shown promise in autonomously generating heuristic components with minimal human intervention. However, most existing LLM-based AHD methods enforce fixed algorithmic templates to ensure executability, which confines the search to component-level tuning and limits system-level algorithmic expressiveness. To enable open-ended solver synthesis beyond rigid templates, we propose Automated Algorithm Design via Evolutionary Program Trees (A2DEPT), which treats LLMs as system-level algorithm architects. A2DEPT explores the vast program space via a tree-structured evolutionary search with hybrid selection and hierarchical operators, enabling iterative refinement of complete algorithms. To make open-ended generation practical, we enforce executability with a lightweight program-maintenance loop that performs feedback-driven repair. In experiments, A2DEPT consistently outperforms representative LLM-based baselines on both standard and highly constrained benchmarks. On the standard benchmarks, it reduces the mean normalized optimality gap by 9.8% relative to the strongest competing AHD baseline.

In this paper, we introduce ES-AHD, a novel framework that fundamentally integrates Evolution Strategy (ES) into Large Language Model (LLM)-driven Automatic Heuristic Design (AHD). Existing evolutionary approaches predominantly rely on random, individual-level mutation, leading to blind search and an imbalance between exploration and exploitation. To address these issues, ES-AHD introduces two core mechanisms. First, Semantic Recombination via LLMs discards traditional point-to-point reproduction. By leveraging the LLM's contextual reasoning to explicitly extract core insights from top-performing individuals, the algorithm establishes a promising semantic search direction. This transforms random code mutation into targeted, center-guided sampling inspired by ES. Second, Stochastic Covariance Adaptation via Temperature Sampling dynamically addresses the exploration-exploitation dilemma. By mapping the covariance matrix in ES to the LLM's sampling temperature, the framework employs a stochastic random walk mechanism with momentum. This approach primarily shrinks the search radius for micro-level code refinement, while retaining the critical ability to occasionally sample higher temperatures to escape semantic local optima. Ultimately, ES-AHD provides a highly directional, robust, and efficient search paradigm, significantly accelerating the generation of high-quality heuristic algorithms. The source code is available at: https://github.com/Mriya0306/ES-AHD.