Finding largest small polygons with GloptiPoly

• Published in: Journal of global optimization Vol. 56; no. 3; pp. 1017 - 1028
• Main Authors: Henrion, Didier, Messine, Frédéric
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-07-2013; Springer Nature B.V; Springer Verlag
• Subjects: Computer Science; Conics; Mathematical models; Mathematics; Mathematics and Statistics; Matlab; Operations Research/Decision Theory; Optimization; Optimization algorithms; Optimization and Control; Packages; Polygons; Polynomials; Quadratic programming; Real Functions; Semidefinite programming; Solvers; Studies; Symmetry; Extremal convex polygons; Global optimization; Semidefinite programming; Nonconvex quadratic programming
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-011-9818-7

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
 ≥ 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.