用数值Optimization求最大小n-多项式序列
Finding the Sequence of Largest Small n-Polygons by Numerical Optimization
摘要：
LSP（n）是具有最大面积A（n）的单位直径多边形，是n个顶点的最大小多边形。已知对于所有奇数值$n \geq 3$，LSP（n）是正n多边形;然而，该陈述对于n的偶数值无效。寻找偶数值$n \geq 6$的多边形LSP（n）和A（n）是一个长期存在的挑战。在这项工作中，我们开发了高精度的数值解估计A（n）的偶数值$n \geq 4$，使用Mathematica模型开发环境和IPOPT本地非线性最优化求解引擎。首先，我们提出了一个修订（收紧）LSP模型，大大有助于有效的解决方案的模型类考虑。接着是n的偶数值的说明性序列的数值结果，直到$n \leq 1000$。我们的结果与所有早期研究中报道的最佳结果非常一致，甚至超过了这些结果。大多数这些早期的作品解决了特殊情况下，$n \leq 20$，而其他人从$6 \leq n \leq 100$的值范围内获得数值优化结果。为了完整性，我们还计算数值优化的结果为选择奇数值的n，直到$n \leq 999$：这些结果可以比较相应的理论（精确）值。所获得的结果用于提供最佳面积序列{A（n）}的基于回归模型的估计，对于所有感兴趣的偶数和奇数值n，从而基本上以明显的高精度数值求解整个LSP模型类。
英文摘要：
LSP(n), the largest small polygon with n vertices, is the polygon of unit diameter that has maximal area A(n). It is known that for all odd values $n \geq 3$, LSP(n) is the regular n-polygon; however, this statement is not valid for even values of n. Finding the polygon LSP(n) and A(n) for even values $n \geq 6$ has been a long-standing challenge. In this work, we develop high-precision numerical solution estimates of A(n) for even values $n \geq 4$, 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 the efficient solution of the model-class considered. This is followed by numerical results for an illustrative sequence of even values of n, up to $n \leq 1000$. Our results are in close agreement with, or surpass, the best results reported in all earlier studies. Most of these earlier works addressed special cases up to $n \leq 20$, while others obtained numerical optimization results for a range of values from $6 \leq n \leq 100$. For completeness, we also calculate numerically optimized results for a selection of odd values of n, up to $n \leq 999$: these results can be compared to the corresponding theoretical (exact) values. The results obtained are used to provide regression model-based estimates of the optimal area sequence {A(n)}, for all even and odd values n of interest, thereby essentially solving the entire LSP model-class numerically, with demonstrably high precision.
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