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Global Optimization for Combinatorial Geometry Problems

Timo Berthold, Dominik Kamp, Gioni Mexi, Sebastian Pokutta, Imre Pólik

TL;DR

The paper investigates whether standard global NLP solvers can match or exceed results from LLM-driven algorithm discovery on challenging geometric/combinatorial problems by reformulating them as nonlinear programs. It solves three AlphaEvolve benchmark problems—the min-max distance ratio, circle packing with variable radii, and hexagon packing inside a hexagon—using off-the-shelf solvers (SCIP and FICO Xpress) without solver modifications, achieving matching or improved best-known solutions. Key modeling choices include using squared distances $d_{ij}^2$ to avoid square roots, exploiting scale-invariance by fixing $t_{ ext{min}}$ or $t_{ ext{max}}$, and employing Farkas-based separation for hexagon packing, all within nonconvex quadratically constrained programs. The results demonstrate that modern NLP solvers provide a robust, industry-ready baseline that can complement LLM-driven discovery, enabling rapid, provably high-quality solutions and suggesting a productive synergy between symbolic optimization and learning-based code generation.

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.

Global Optimization for Combinatorial Geometry Problems

TL;DR

The paper investigates whether standard global NLP solvers can match or exceed results from LLM-driven algorithm discovery on challenging geometric/combinatorial problems by reformulating them as nonlinear programs. It solves three AlphaEvolve benchmark problems—the min-max distance ratio, circle packing with variable radii, and hexagon packing inside a hexagon—using off-the-shelf solvers (SCIP and FICO Xpress) without solver modifications, achieving matching or improved best-known solutions. Key modeling choices include using squared distances to avoid square roots, exploiting scale-invariance by fixing or , and employing Farkas-based separation for hexagon packing, all within nonconvex quadratically constrained programs. The results demonstrate that modern NLP solvers provide a robust, industry-ready baseline that can complement LLM-driven discovery, enabling rapid, provably high-quality solutions and suggesting a productive synergy between symbolic optimization and learning-based code generation.

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.
Paper Structure (16 sections, 6 equations, 3 figures, 3 tables)

This paper contains 16 sections, 6 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Stronger solutions with unmodified solvers.
  3. Lessons learned.
  4. Minimizing the Ratio of Maximum to Minimum Distance
  5. Optimization model
  6. Circle Packing Formulation
  7. Dual Formulation.
  8. Computational Results
  9. Packing Circles inside a Square or a Rectangle
  10. Optimization model
  11. Computational Results
  12. Packing Unit Regular Hexagons inside a Regular Hexagon
  13. Optimization model
  14. Computational Results
  15. Conclusion
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Graphical representation of the solutions for the distance ratio problem in 2D. Red lines indicate maximum distance pairs, blue lines indicate minimum distance pairs.
  • Figure 2: Graphical representation of the solutions for the circle packing problem. Left: square variant (n=32). Middle and right: rectangle variant (n=26, n=27).
  • Figure 3: Graphical representation of the solutions for the hexagon packing problem