A Hybrid Nested Harness for Decoupling Structure and Parameters in LLM-Driven Optimization

Recent work has demonstrated the promise of orchestrating large language models (LLMs) within evolutionary and agentic optimization systems. However, the mechanisms driving these optimization gains remain poorly understood. In this work, we present a large-scale study of LLM-guided evolutionary search, collecting optimization trajectories for 15 LLMs across 8 tasks. Although zero-shot problem-solving ability correlates with final optimization outcomes, it explains only part of the variance: models with similar initial capability often induce dramatically different search trajectories and outcomes. By analyzing these trajectories, we find that strong LLM optimizers behave as local refiners, producing frequent incremental improvements while progressively localizing the search in semantic space. Conversely, weaker optimizers exhibit large semantic drift, with sporadic breakthroughs followed by stagnation. Notably, various measures of solution novelty do not predict final performance; novelty is beneficial only when the search remains sufficiently localized around high-performing regions of the solution space. Our results highlight the importance of trajectory analysis for understanding and improving LLM-based optimization systems and provide actionable insights for their design and training.

LLM-guided evolutionary computation, most notably AlphaEvolve, has been remarkably successful in discovering novel mathematical constructions by solving challenging optimization problems. The standard approach is to evolve a monolithic program that directly outputs a candidate solution. We present ImprovEvolve, an algorithmic alternative that drastically reduces cognitive load on the LLM. Instead of prompting the model for an end-to-end optimizer, we evolve a program with three specialized operators of initialization, local improvement, and perturbation. We then approach the optimum by iteratively applying local improvements and intensity-scheduled perturbations, effectively driving a basin-hopping search with LLM-evolved subroutines. For hexagon in hexagon packing, ImprovEvolve discovers new state-of-the-art packings of 11, 12, 15, and 16 hexagons, and additionally for 14, 17, and 23 hexagons after minimal expert tuning of the generated code. For the second autocorrelation inequality, the evolved and human-scaled program pushes the lower bound from 0.96102 to 0.96258. For spherical codes, the ImprovEvolve program lowers the best-known maximum cosine for the majority of 90 randomly chosen diverse state-of-the-art spherical codes, achieving relative improvements of up to 2.4%.

Large language models (LLMs) are increasingly integrated with evolutionary computation to support optimization tasks. This survey primarily focuses on evolutionary optimization, i.e., optimization based on evolutionary computation. For brevity, we use the term optimization throughout to denote this scope. However, existing surveys typically examine isolated roles of LLMs and do not provide a unified view that connects optimization modeling with optimization solving. To address this gap, we systematically review recent developments through a workflow-oriented framework. First, we organize the literature into two primary stages: LLMs for optimization modeling and LLMs for optimization solving (in this survey, the terms optimization modeling and optimization solving are used as concise forms of optimization problem modeling and optimization problem solving, respectively). Second, we divide the solving stage into three paradigms according to the role of the LLM: stand-alone optimizers, low-level components embedded in optimization algorithms, and high-level managers for algorithm selection and generation. Third, we analyze representative methods, identify their technical limitations, and clarify their relationships with traditional optimization approaches. We further substantiate this taxonomy through benchmark systematization, baseline comparisons, and practitioner-oriented guidance, and we review interdisciplinary applications across the natural sciences, engineering, and machine learning. Based on the resulting analysis, we identify research directions toward dynamic, self-evolving, and agentic optimization ecosystems. An up-to-date collection of related literature is maintained at https://github.com/ishmael233/LLM4OPT.