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Largest Small n-Polygons: Numerical Results and Conjectured Optima

János D. Pintér (jdp416***at***lehigh.edu)

Abstract: LSP(n), the largest small polygon with n vertices, is defined as the polygon of unit diameter that has maximal area A(n). Finding the configuration LSP(n) and the corresponding A(n) for even values n >= 6 is a long-standing challenge that leads to an interesting class of nonlinear optimization problems. We present numerical solution estimates for all even values 6 <= n <= 80, using the AMPL model development environment with the LGO nonlinear solver engine option. Our results compare favorably to the results obtained by other researchers who solved the problem using exact approaches (for 6 <= n <= 16), or general purpose numerical optimization software (for selected values from the range 6 <= n <= 100) using various local nonlinear solvers. Based on the results obtained, we also provide a regression model based estimate of the optimal area sequence {A(n)} for n >= 6.

Keywords: Largest Small Polygons, Mathematical Model, Analytical and Numerical Solution Approaches, AMPL Modeling Environment, LGO Solver Suite For Nonlinear Optimization, AMPL-LGO Numerical Results, Regression Model Based Optimum Estimates

Category 1: Global Optimization (Applications )

Category 2: Optimization Software and Modeling Systems (Modeling Languages and Systems )

Category 3: Optimization Software and Modeling Systems (Optimization Software Benchmark )

Citation: Research Report, Department of Industrial and Systems Engineering, Lehigh University, Bethlehem PA, USA. Submitted for publication.

Download: [PDF]

Entry Submitted: 11/27/2018
Entry Accepted: 11/27/2018
Entry Last Modified: 12/26/2018

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