arXiv subjects

Nathan Sudermann-Merx

Publications and source records attributed to Nathan Sudermann-Merx.

5 recordsLinked to original sources

arXiv2026

Heilbronn's Problem in the Unit Triangle: Certified Optimal Configurations for up to $n\le 8$

We study Heilbronn's triangle problem in the unit right triangle, where $n$ points are placed to maximize the smallest of the $\binom{n}{3}$ triangle areas they span. We prove a boundary-structure result: unless all three vertices are occupied, some optimal configuration with $n \ge 5$ has at least four points on the boundary, one edge carrying two of them. With the affine $S_3$ symmetry this fixes four boundary points and $n$ orientation variables in a mixed-integer model that certifies global optimality for all $n \le 8$: for $n = 8$ apparently the first proof, and for $n = 7$ an independent confirmation of the symbolic-computation proof of Zeng and Chen. For $n \le 7$ we obtain exact optima with explicit configurations. For $n = 8$ the optimum is conjectured to be the real root of a septic obtained by Chen, Zeng and Zhou, which our reconstruction confirms to $250$ digits. We show its Galois group is $S_7$, so on that conjecture no expression in radicals exists.

math.OC↗

arXiv2026

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

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.

math.OC↗

arXiv2022

Tree ensemble kernels for Bayesian optimization with known constraints over mixed-feature spaces

Tree ensembles can be well-suited for black-box optimization tasks such as algorithm tuning and neural architecture search, as they achieve good predictive performance with little or no manual tuning, naturally handle discrete feature spaces, and are relatively insensitive to outliers in the training data. Two well-known challenges in using tree ensembles for black-box optimization are (i) effectively quantifying model uncertainty for exploration and (ii) optimizing over the piece-wise constant acquisition function. To address both points simultaneously, we propose using the kernel interpretation of tree ensembles as a Gaussian Process prior to obtain model variance estimates, and we develop a compatible optimization formulation for the acquisition function. The latter further allows us to seamlessly integrate known constraints to improve sampling efficiency by considering domain-knowledge in engineering settings and modeling search space symmetries, e.g., hierarchical relationships in neural architecture search. Our framework performs as well as state-of-the-art methods for unconstrained black-box optimization over continuous/discrete features and outperforms competing methods for problems combining mixed-variable feature spaces and known input constraints.

stat.ML↗

arXiv2021

Multi-Objective Constrained Optimization for Energy Applications via Tree Ensembles

Energy systems optimization problems are complex due to strongly non-linear system behavior and multiple competing objectives, e.g. economic gain vs. environmental impact. Moreover, a large number of input variables and different variable types, e.g. continuous and categorical, are challenges commonly present in real-world applications. In some cases, proposed optimal solutions need to obey explicit input constraints related to physical properties or safety-critical operating conditions. This paper proposes a novel data-driven strategy using tree ensembles for constrained multi-objective optimization of black-box problems with heterogeneous variable spaces for which underlying system dynamics are either too complex to model or unknown. In an extensive case study comprised of synthetic benchmarks and relevant energy applications we demonstrate the competitive performance and sampling efficiency of the proposed algorithm compared to other state-of-the-art tools, making it a useful all-in-one solution for real-world applications with limited evaluation budgets.

stat.ML↗

arXiv2021

ENTMOOT: A Framework for Optimization over Ensemble Tree Models

Gradient boosted trees and other regression tree models perform well in a wide range of real-world, industrial applications. These tree models (i) offer insight into important prediction features, (ii) effectively manage sparse data, and (iii) have excellent prediction capabilities. Despite their advantages, they are generally unpopular for decision-making tasks and black-box optimization, which is due to their difficult-to optimize structure and the lack of a reliable uncertainty measure. ENTMOOT is our new framework for integrating (already trained) tree models into larger optimization problems. The contributions of ENTMOOT include: (i) explicitly introducing a reliable uncertainty measure that is compatible with tree models, (ii) solving the larger optimization problems that incorporate these uncertainty aware tree models, (iii) proving that the solutions are globally optimal, i.e. no better solution exists. In particular, we show how the ENTMOOT approach allows a simple integration of tree models into decision-making and black-box optimization, where it proves as a strong competitor to commonly-used frameworks.

stat.ML↗