We develop a dual simplex-type algorithm for computing the smallest enclosing ball of a set of balls and other closely related problems. Our algorithm employs a pivoting scheme resembling the simplex method for linear programming, in which a sequence of exact curve searches is performed until a new dual feasible solution with a strictly smaller objective function value is found. We utilize the Cholesky factorization and procedures for updating it, yielding a numerically stable implementation of the algorithm. We show that our algorithm can efficiently solve instances of dimension 5000 with 100000 points, often within minutes.

中文翻译：

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最小包围球的对偶单纯形算法

我们开发了一种对偶单纯形算法，用于计算一组球中最小的封闭球以及其他紧密相关的问题。我们的算法采用类似于线性方法的单纯形法的旋转方案，其中执行一系列精确的曲线搜索，直到找到具有严格较小目标函数值的新对偶可行解。我们利用Cholesky分解和程序对其进行更新，从而在算法上实现了数值稳定的实现。我们证明了我们的算法可以在几分钟之内有效地解决具有500000个点的维度5000实例。