一个组合几何最优化未解决问题的半机械化解法

本文证明了一个关于凸n边形面积的不等式猜测在n=8时的正确性，并对n=9的情况做了讨论。首先将这个最优化问题转化为多项式不等式方程组的实解的存在性问题；其次通过分析最优图形给出了一些化简不等式方程组和减少系统自由变元的方法；利用符号计算等方法建立了一个半机械化方法求多项式方程组作为约束条件的非线性规划问题准确
解。

A Semi-Mechanization Algorithm for an Unsolved Optimal Problem

This paper aims at proving the validity of an inequality conjecture about convex n-gon when n=8 and gaining headway in proving it when n=9.This conjecture is generally converted into a global optimization problem which is related to the Heilbronn triangular problem.The bottleneck of solving this problem is the complexity increasing very quickly with n.In this paper,we intend to establish a new semi-mechanization method for solving it.In our algorithm,to reduce the dimension of freedom we first analyze the properties of the optimal configurations and try to obtain the strict polynomial inequality and equation conditions as many as possible.After the precondition process,the mechanization method can be implemented to solve this nonlinear optimization problem,so we call the overall approach as semi-mechanization.Our algorithm will be useful for proving the conjecture with a larger value of n.