渐近定位问题中的长期规划与短期规划
Long-term planning versus short-term planning in the asymptotical location problem
摘要：
在给定区域$Ω\subset \R^n$上的概率测度$ν$的情况下，考虑由$n$点$\Om$组成的集合$Σ$的最优选址问题，以最小化平均距离$Σ\mapsto \int_\Om \dist(x,Σ) dν$（经典的最优设施定位问题）.本文比较了两种策略，以找到最佳的配置：长期的一个，其中包括在放置所有的$n$点在一次在一个最佳的位置，和短期的一个，其中包括在放置点一个接一个添加在每一步最多一个点，并保持在前几步建立的配置。我们表明，各自的最优化问题表现出定性不同的渐近行为的$n\to\infty$，虽然在这两种情况下的优化成本有相同的渐近消失的顺序。
英文摘要：
Given the probability measure $ν$ over the given region $Ω\subset \R^n$, we consider the optimal location of a set $Σ$ composed by $n$ points $\Om$ in order to minimize the average distance $Σ\mapsto \int_\Om \dist(x,Σ) dν$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all $n$ points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving the configuration built at previous steps. We show that the respective optimization problems exhibit qualitatively different asymptotic behavior as $n\to\infty$, although the optimization costs in both cases have the same asymptotic orders of vanishing.
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