Groupe d’études et de recherche en analyse des décisions

# Largest small polygons: A sequential convex optimization approach

## Christian Bingane

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