Integer Programming and the Theory of Grouping
Integer Programming and the Theory of Grouping
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DOI：
10.1080/01621459.1969.10500990
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发表时间：
1969-06
影响因子：
3.7
通讯作者：
Hrishikesh D. Vinod Mathematica
中科院分区：
文献类型：
--
作者：
Hrishikesh D. Vinod Mathematica
摘要本文的写作有三个目标。首先，指出分组的问题，其中大量的元素n被组合成m个相互排斥的组（m < n）应该被认为是可编程的问题，并且这种认识可以帮助我们避免完全枚举从n到n - 1...到m的分组阶段，以及每个阶段的替代可能性。第二，在数学上给出了相关的可编程问题的一些简单形式，以便现有的计算机代码可以解决它.当分组试图最小化组内平方和时，证明了所谓的弦性质对于最小值是必要的（引理1）。它表明，字符串属性可以被利用来写的非线性组内的平方和作为一个线性函数（引理2）。尝试推广的字符串属性的高维情况下。最后，给出了一些数值例子，说明了该数学模型的正确性。
Abstract This paper is written with three objectives in mind. First, to point out that the problem of grouping, where a larger number of elements n are combined into m mutually exclusive groups (m < n) should be recognized as a problem in Integer Programming, and that such recognition can help us in avoiding complete enumeration of stages in grouping from n to n — 1 … to m, and of alternative possibilities in each stage. Second, to formulate mathematically some simple versions of the relevant Integer Programming Problem, so that the available computer codes can solve it. When the grouping attempts to minimize the within groups sums of squares, the so-called string property is proved to be necessary for the minimum (Lemma 1). It is shown that the string property can be exploited to write the non-linear within group sums of squares as a linear function (Lemma 2). An attempt is made to generalize the string property for the higher dimensional case. Third, to give some numerical examples to clarify the mathem...