Finding largest small polygons with GloptiPoly

TL;DR

This paper demonstrates that GloptiPoly, a global polynomial optimization tool, can effectively find the largest small polygons with 10 and 12 vertices, advancing solutions for open cases in geometric optimization.

Contribution

The paper applies GloptiPoly to solve previously open instances of the largest small polygon problem for even n ≥ 10, providing new solutions and demonstrating the method's effectiveness.

Findings

Successfully found largest small polygons for n=10 and n=12.

Improved upon existing results in geometric optimization.

Provided numerical and rigorous guarantees of optimality.

Abstract

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many instances are already solved in the literature, namely for all odd $n$, and for $n=4, 6$ and 8. Thus, for even $n\geq 10$, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for $n=10$ and $n=12$. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic.