Finding the Conjectured Sequence of Largest Small n-Polygons by Numerical Optimization
| Autor: | János D. Pintér, Frank J. Kampas, Ignacio Castillo |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Mathematical and Computational Applications, Vol 27, Iss 3, p 42 (2022) |
| Druh dokumentu: | article |
| ISSN: | 2297-8747 1300-686X |
| DOI: | 10.3390/mca27030042 |
| Popis: | LSP(n), the largest small polygon with n vertices, is a polygon with a unit diameter that has a maximal of area A(n). It is known that for all odd values n≥3, LSP(n) is a regular n-polygon; however, this statement is not valid even for values of n. Finding the polygon LSP(n) and A(n) for even values n≥6 has been a long-standing challenge. In this work, we developed high-precision numerical solution estimates of A(n) for even values n≥4, using the Mathematica model development environment and the IPOPT local nonlinear optimization solver engine. First, we present a revised (tightened) LSP model that greatly assists in the efficient numerical solution of the model-class considered. This is followed by results for an illustrative sequence of even values of n, up to n≤1000. Most of the earlier research addressed special cases up to n≤20, while others obtained numerical optimization results for a range of values from 6≤n≤100. The results obtained were used to provide regression model-based estimates of the optimal area sequence {A(n)}, for even values n of interest, thereby essentially solving the LSP model-class numerically, with demonstrably high precision. |
| Databáze: | Directory of Open Access Journals |
| Externí odkaz: |
|