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Solving the Heilbronn Triangle Problem using Global Optimization Methods

Amirhossein Monji, Amirali Modir, Burak Kocuk

TL;DR

This work casts the Heilbronn triangle problem as a global-optimization challenge in the unit square and develops three formulations (two MIQCP and one QCP) to obtain certified optima using standard solvers. It introduces bound-tightening, symmetry-breaking, boundary-point constraints, and local packing strategies to dramatically prune the search space. The authors certify $H_9^*=0.0548767$ for nine points and report substantial improvements over the prior 31-day computation by Chen et al., while also obtaining strong lower bounds for $n=10$ under plausible structural assumptions. Collectively, the results demonstrate a viable, solver-based path to tight certificates for Heilbronn numbers and highlight directions for extending certification to larger instances. The work also provides a comprehensive framework for comparing formulations, solvers, and instance-specific enhancements in nonlinear combinatorial geometry problems.

Abstract

We study the Heilbronn triangle problem, which involves placing n points in the unit square such that the minimum area of any triangle formed by these points is maximized. A straightforward maximin formulation of this problem is highly non-linear and non-convex due to the existence of bilinear terms and absolute value equations. We propose two mixed-integer quadratically constrained programming (MIQCP) and one QCP formulation, which can be readily solved by any global optimization solver. We develop several formulation enhancements in the form of bound tightening and symmetry breaking inequalities that are prevalent in the global optimization literature in addition to other enhancements that exploit the problem structure. With the help of these enhancements, our models reproduce proven optimal values for instances up to n = 8 points with certified optimality in the order of seconds. In the case of n = 9 points, for which no analytical proof is known, we establish a certified optimal value by a computational effort of one day. This is a significant improvement over the previous benchmark established in 31 days of computations by Chen et al. (2017).

Solving the Heilbronn Triangle Problem using Global Optimization Methods

TL;DR

This work casts the Heilbronn triangle problem as a global-optimization challenge in the unit square and develops three formulations (two MIQCP and one QCP) to obtain certified optima using standard solvers. It introduces bound-tightening, symmetry-breaking, boundary-point constraints, and local packing strategies to dramatically prune the search space. The authors certify for nine points and report substantial improvements over the prior 31-day computation by Chen et al., while also obtaining strong lower bounds for under plausible structural assumptions. Collectively, the results demonstrate a viable, solver-based path to tight certificates for Heilbronn numbers and highlight directions for extending certification to larger instances. The work also provides a comprehensive framework for comparing formulations, solvers, and instance-specific enhancements in nonlinear combinatorial geometry problems.

Abstract

We study the Heilbronn triangle problem, which involves placing n points in the unit square such that the minimum area of any triangle formed by these points is maximized. A straightforward maximin formulation of this problem is highly non-linear and non-convex due to the existence of bilinear terms and absolute value equations. We propose two mixed-integer quadratically constrained programming (MIQCP) and one QCP formulation, which can be readily solved by any global optimization solver. We develop several formulation enhancements in the form of bound tightening and symmetry breaking inequalities that are prevalent in the global optimization literature in addition to other enhancements that exploit the problem structure. With the help of these enhancements, our models reproduce proven optimal values for instances up to n = 8 points with certified optimality in the order of seconds. In the case of n = 9 points, for which no analytical proof is known, we establish a certified optimal value by a computational effort of one day. This is a significant improvement over the previous benchmark established in 31 days of computations by Chen et al. (2017).

Paper Structure

This paper contains 37 sections, 12 theorems, 22 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation and Solution Approaches
  3. Problem Formulation
  4. Solution Approaches
  5. Approach 1
  6. Approach 2
  7. Approach 3
  8. Enhancements
  9. First Group: Bound Tightening and Symmetry Breaking
  10. Bounds for the Objective Value $H_n$
  11. Symmetry Breaking
  12. Second Group: Points on the Boundary
  13. At Least One Point on Each Edge ($n \ge 4$)
  14. Two Points on Some Edge ($n\in\{7,8\}$)
  15. Approximately Eight Boundary Points ($n=9$)
  16. ...and 22 more sections

Key Result

Proposition 1

For $n>3$, the optimal minimum‑triangle area is non-increasing in $n$; in particular,

Figures (5)

  • Figure 1: Optimal point placements for $n=3$ to $n=5$ for the Heilbronn triangle problem.
  • Figure 2: The unit square partitioned into four strips of height $s$ and a final strip of height $1-(m-1)s<s$. Each strip has area less than $2\underline{H}_n$, so it can contain at most two points.
  • Figure 3: Partitioning the unit square into an $m \times m$ grid. If $1/m < \underline{H}$, then each subsquare can contain at most one point.
  • Figure 4: Applying the Heilbronn triangle problem in smaller rectangles to derive cardinality constraints. The height of the restricted strip is $1/\kappa$, where $\kappa$ is the number of strips.
  • Figure 5: Point placements for $n=6$ to $n=10$. One of the smallest triangles is highlighted.

Theorems & Definitions (26)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Corollary 1
  • proof
  • Proposition 4
  • Remark 1
  • ...and 16 more