Solving the Heilbronn Triangle Problem using Global Optimization Methods

TL;DR

This paper introduces new global optimization formulations and enhancements to solve the Heilbronn triangle problem, achieving certified optimal solutions for up to 9 points significantly faster than previous methods.

Contribution

The authors develop MIQCP and QCP formulations with enhancements that enable efficient global optimization of the Heilbronn triangle problem, including for instances where no prior solutions were known.

Findings

Certified optimal solutions for n=8 points in seconds.

Established a certified optimal value for n=9 points in one day.

Reduced computation time from 31 days to one day for n=9.

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).