On abstract dual linear programs
On abstract dual linear programs
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关于抽象对偶线性规划
DOI：
10.1002/nav.3800100131
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发表时间：
1963
期刊：
影响因子：
--
通讯作者：
A. Hoffman
中科院分区：
文献类型：
--
作者：
A. Hoffman
本文在一般代数背景下研究了线性规划的对偶定理。众所周知，当原程序和对偶程序的常量和变量是实数（或任何有序域）时，则（i）要最大化的函数的任何值不超过要最小化的函数的任何值，并且（ii）max=min。性质（i）是微不足道的，性质（ii）取决于超平面分离定理[3]，单纯形法[2]或其他参数[4]。然而，用于证明 (ii) 的所有论点似乎都取决于域的属性；然而，(i) 的证明则不然。事实上，它的琐碎性将在下一节描述的抽象设置中持续存在。然后我们提出一些问题，这是本文的主要目的是做广告。这些问题有一定的兴趣，将在题为“对偶成立的集合 S 的示例”的部分中进行说明，其中对偶定理将在一些不寻常的环境中证明成立。
This article examines the duality theorem of linear programming in the context of a general algebraic setting. It is well known that, when the constants and variables of primal and dual programs are real numbers (or any ordered field), then (i) any value of the function to be maximized does not exceed any value of the function to be minimized, and (ii) max= min.Property (i) is a triviality, and property (ii) depends on the hyperplane separation theorem [3], the simplex method [2], or some other argument [4]. All of the arguments used to prove (ii), however, seem to depend on the properties of a field; the proof of (i), however, does not. In fact, its triviality will persist in the abstract setting described in the next section. We then formulate some questions, which it is the main purpose of this article to advertise. That these questions have some interest will be illustrated in the section entitled" Examples of Sets S for Which Duality Holds," where the duality theorem will be shown to hold in some unusual surroundings.