Finding largest small polygons with GloptiPoly
Year: 2011
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Henrion, Didier, Messine, Frederic
Institutions: Centre National de la Recherche Scientifique, Université Toulouse III - Paul Sabatier, Laboratoire d'Analyse et d'Architecture des Systèmes, Institut National des Sciences Appliquées de Toulouse, Université de Toulouse, Czech Technical University in Prague, Institut de Recherche en Informatique de Toulouse, École Nationale Supérieure d'Électrotechnique, d'Électronique, d'Informatique, d'Hydraulique et des Télécommunications
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2953044678