From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization

Nathan Sudermann-Merx

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arXiv · 2603.11107

From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization

Nathan Sudermann-Merx

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Abstract

We develop a mixed-integer nonlinear programming (MINLP) approach for the classical Heilbronn triangle problem, demonstrating the capability of modern global optimization solvers to tackle challenging combinatorial geometry problems. A symmetry-breaking strategy based on boundary structure yields a substantially stronger model: for $n=9$, we compute an $\varepsilon$-globally optimal point in 15 minutes on a standard desktop computer, improving upon the previously reported effort of approximately one day. By combining numerical certification with exact symbolic computation, we recover exact coordinates matching all best-known configurations for $n\le 9$, including the $n=9$ configuration of Comellas and Yebra (2002). An analysis of these configurations reveals the clustering of noncritical triangle areas around a small number of distinct values, suggesting rich underlying algebraic structure. All code and data are publicly available.

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Nathan Sudermann-Merx. 2026-03-11. From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization. https://arxiv.org/abs/2603.11107

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