多椭球的最小包围半径
Minimal Radius Enclosing Polyellipsoids
摘要：
在本文中，我们分析了经典的最小封闭磁盘问题的情况下，一个多椭球的位置，以完全覆盖$\mathbb{R}^d$中的一组需求点的扩展。我们证明了该问题在固定维数下是多项式可解的，并分析了它的数学规划公式，在多椭球的焦点已知的情况下，我们还考虑了该问题的一些几何方法. Elzinga-Hearn的经典算法的扩展也来自这个新问题。此外，我们解决几个扩展的问题，作为情况下，封闭polyellipsoid的焦点没有给出，必须确定一个潜在的一组点或诱导覆盖问题时，而不是polyellipsoid，一个使用有序的中值polyellipsoid。对于这些问题，我们还提出了混合的非（非）线性规划的战略，导致有效的方法来解决它。
英文摘要：
In this paper we analyze the extension of the classical smallest enclosing disk problem to the case of the location of a polyellipsoid to fully cover a set of demand points in $\mathbb{R}^d$. We prove that the problem is polynomially solvable in fixed dimension and analyze mathematical programming formulations for it. We also consider some geometric approaches for the problem in case the foci of the polyellipsoids are known. Extensions of the classical algorithm by Elzinga-Hearn are also derived for this new problem. Moreover, we address several extensions of the problem, as the case where the foci of the enclosing polyellipsoid are not given and have to be determined among a potential set of points or the induced covering problems when instead of polyellipsoids, one uses ordered median polyellipsoids. For these problems we also present Mixed Integer (Non) Linear Programming strategies that lead to efficient ways to solve it.
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