ImprovEvolve: Basin-Hopping Meets LLM-Guided Evolutionary Search

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 \textbf{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\%.