关于拟合点集的多重超平面定位问题
On the multisource hyperplanes location problem to fitting set of points
摘要：
在本文中，我们研究的问题，定位一个给定数量的超平面最小化的目标函数的最近距离从一组点。我们提出了一个一般框架的问题，其中规范为基础的点和超平面之间的距离聚集通过有序的中值函数。给出了该问题的一个紧凑的混合线性（或非线性）规划公式，并导出了一个指数变量的扩展集划分公式。我们开发了一个嵌入分支和价格算法中的列生成程序，通过充分执行预处理、定价和分支来解决该问题。我们还分析几何的最优解的问题，推导出的属性，利用所提出的算法生成初始解。最后，一个广泛的计算经验的结果报告。可扩展性的问题也得到了解决，显示理论上的上界的错误，假设由聚合版本取代原始数据集。
英文摘要：
In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We propose a general framework for the problem in which norm-based distances between points and hyperplanes are aggregated by means of ordered median functions. A compact Mixed Integer Linear (or Non Linear) programming formulation is presented for the problem and also an extended set partitioning formulation with an exponential number of variables is derived. We develop a column generation procedure embedded within a branch-and-price algorithm for solving the problem by adequately performing its preprocessing, pricing and branching. We also analyze geometrically the optimal solutions of the problem, deriving properties which are exploited to generate initial solutions for the proposed algorithms. Finally, the results of an extensive computational experience are reported. The issue of scalability is also addressed showing theoretical upper bounds on the errors assumed by replacing the original datasets by aggregated versions.
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