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 prop...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Author: Bingane, Christian
Format: Article
Language:English
Subjects:
Algorithms
Apexes
Ascent
Computational geometry
Convergence
Convex analysis
Convexity
Optimization
Polygons
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Summary: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 an 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.
ISSN:2331-8422
DOI:10.48550/arxiv.2009.07893