Directing to Diversify: Multi-Directional Search for Automatic Heuristic Design

Large Language Models (LLMs) have advanced Automated Heuristic Design (AHD) in combinatorial optimization (CO) in the past few years. However, existing discovery pipelines often require extensive manual trial-and-error or reliance on domain expertise to adapt to new or complex problems. This stems from tightly coupled internal mechanisms that limit systematic improvement of the LLM-driven design process. To address this challenge, we propose a structured framework for LLM-driven AHD that explicitly decomposes the heuristic discovery process into modular stages: a forward pass for evaluation, a backward pass for analytical feedback, and an update step for program refinement. This separation provides a clear abstraction for iterative refinement and enables principled improvements of individual components. We validate our framework across four diverse real-world CO domains, where it consistently outperforms baselines, achieving up to $0.17$ improvement in QYI on unseen test sets. Finally, we show that several popular AHD methods are restricted instantiations of our framework. By integrating them in our structured pipeline, we can upgrade the components modularly and significantly improve their performance.

The integration of Large Language Models (LLMs) with evolutionary computation has emerged as a powerful paradigm for automated heuristic design in combinatorial optimization. However, existing approaches suffer from mode collapse, converging to homogeneous populations that lack semantic diversity and fail to explore the full algorithmic space. We propose Quality-Diversity Evolution (QDEvo), a multi-objective framework that integrates Quality-Diversity optimization with LLM-driven heuristic search, maintaining an unbounded archive of semantically diverse algorithms using pre-trained code embeddings and incorporating hierarchical self-reflection to guide the evolutionary process. Extensive experiments across standard benchmarks and real-world industrial applications demonstrate that QDEvo significantly outperforms state-of-the-art methods in both Hypervolume and Inverted Generational Distance metrics. Our framework enables the discovery of heuristics that are simultaneously high-performing, computationally efficient, and semantically diverse, providing practitioners with a rich portfolio of solutions for complex optimization problems.

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.