Groupe d’études et de recherche en analyse des décisions

G-2020-50

Largest small polygons: A sequential convex optimization approach

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with \(n=2m\) vertices is not known when \(m\ge 7\). Finding the largest small \(n\)-gon for a given number \(n\ge 3\) can be formulated as a nonconvex quadratically constrained quadratic optimization problem. We propose to solve this problem with a sequential convex optimization approach, which is a ascent algorithm guaranteeing convergence to a locally optimal solution. Numerical experiments on polygons with up to \(n=128\) sides suggest that the optimal solutions obtained are near-global. Indeed, for even \(6 \le n \le 12\), the algorithm proposed in this work converges to known global optimal solutions found in the literature.

, 12 pages