计算球覆盖最小半径的神经网络方法

A Neural Network Method for the Minimum Radius Problem of Ball-coverings

Banach空间中的闭球族称为球覆盖,如果任一元素的内部不含原点,且所有元素之并覆盖了单位球面.本文采用神经网络方法研究n中球覆盖最小半径的计算问题,重新给出计算基数为m(≥n+1)的球覆盖最小半径的公式(对于m=2n(对称)和m=n+1给出了解析表达式),然后基于罚函数法建立神经网络模型,该模型的平衡点集具有大范围吸引性且(渐近)稳定平衡点等价于(严格)极大值点.最后给出了数值例子验证该方法的有效性.

A collection of closed balls in a Banach space is called a ball-covering,if its union contains the unit sphere and the interior of each member is off the origin. This paper considers the minimum radius problem of ball-coverings with the cardinality m（≥n＋ 1） in n by the neural network method. It gives a new computing formula for the minimum radius（and the exact minimum radius for m = 2n and n＋ 1）, then, based on the penalty method, presents a neural network which is globally convergent and the solution is approximated. Numerical examples are given to demonstrate further the effectiveness of the method.