Global Optimization for Combinatorial Geometry Problems Revisited in the Era of LLMs

TL;DR

This paper demonstrates that modern nonlinear optimization solvers can match or surpass recent LLM-driven solutions for complex geometric and combinatorial problems, highlighting their maturity and potential in algorithm discovery.

Contribution

It shows that off-the-shelf NLP solvers can effectively solve problems previously tackled by LLM-driven methods, without modifications, achieving competitive or better results.

Findings

Both solvers reproduce and improve upon previous solutions.

NLP technology now effectively handles complex geometric problems.

Global NLP solvers are promising tools for future algorithm discovery.

Abstract

Recent progress in LLM-driven algorithm discovery, exemplified by DeepMind's AlphaEvolve, has produced new best-known solutions for a range of hard geometric and combinatorial problems. This raises a natural question: to what extent can modern off-the-shelf global optimization solvers match such results when the problems are formulated directly as nonlinear optimization problems (NLPs)? We revisit a subset of problems from the AlphaEvolve benchmark suite and evaluate straightforward NLP formulations with two state-of-the-art solvers, the commercial FICO Xpress and the open-source SCIP. Without any solver modifications, both solvers reproduce, and in several cases improve upon, the best solutions previously reported in the literature, including the recent LLM-driven discoveries. Our results not only highlight the maturity of generic NLP technology and its ability to tackle nonlinear mathematical problems that were out of reach for general-purpose solvers only a decade ago, but also position global NLP solvers as powerful tools that may be exploited within LLM-driven algorithm discovery.