最大小多边形：一种顺序凸优化方法
Largest small polygons: A sequential convex optimization approach
摘要：
小多边形是单位直径的多边形。当计算$m\ge 7$时，不知道具有$n=2m$顶点的小多边形的最大面积。对于给定的数$n\ge 3$，寻找最大的小$n$-边形可以用非凸二次约束二次最优化问题来表示。我们建议解决这个问题的顺序凸优化的方法，这是一个上升算法，保证收敛到局部最优解。在$n=128$多边形上的数值实验表明，所得到的最优解是近全局的。事实上，即使是$6 \le n \le 12$，在这项工作中提出的算法收敛到已知的全球最优解中找到的文献。
英文摘要：
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.
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