论文标题
最大的小多边形：顺序凸优化方法
Largest small polygons: A sequential convex optimization approach
论文作者
论文摘要
小多边形是单位直径的多边形。当$ m \ ge 7 $时，不知道具有$ n = 2M $顶点的小多边形的最大面积。找到给定数量$ n \ ge 3 $的最大的小$ n $ gon可以作为非convex二次约束四次优化问题。我们建议通过顺序凸优化方法解决此问题，该方法是一种上升算法，可确保收敛到本地最佳解决方案。对多边形的数值实验最多$ n = 128 $侧面表明获得的最佳解决方案几乎是全球的。的确，即使以$ 6 \ le n \ le 12 $，这项工作中提出的算法会收敛于文献中发现的已知全球最佳解决方案。
A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically constrained quadratic optimization problem. We propose to solve this problem with a sequential convex optimization approach, which is an ascent algorithm guaranteeing convergence to a locally optimal solution. Numerical experiments on polygons with up to $n=128$ sides suggest that the optimal solutions obtained are near-global. Indeed, for even $6 \le n \le 12$, the algorithm proposed in this work converges to known global optimal solutions found in the literature.
