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A prescient quote from Tjalling Koopmans in the introduction to [64] reads: \It has been found so far that, for any computation method which seems useful in relation to some set of data, another set of data can be constructed for which that method is obviously unsatisfactory." (This compares strikingly with the quote from Bixby et al. [13] at the end of this section.)
In [30], Dantzig writes: \Luckily the particular geometry used in my thesis was the one associated with the columns of the matrix instead of its rows. This column geometry gave me the insight which led me to believe that the simplex method would be an efficient solution technique. I earlier had rejected the method when I viewed it in the row geometry because running around the outside edges seemed so unpromising."
Since much has been written about the early history (and pre-history) of linear programming, for example in [29], Chapter 2, [30], and [83], pp. 209 225, this paper will concentrate more on developments since the seventies. I hope to intrigue the reader enough to investigate some of the byways and alleys associated with linear programming as well as the more well-travelled highways. We will look at simplex, ellipsoid, and interior-point methods, and also at least mention some other approaches. Of course, I hope the reader will forgive my personal bias in the topics selected. (Let me mention here Megiddo's article [75], which also surveys some recent developments from a different viewpoint.)
Following the development of the simplex method in 1947 [27], the '50s had been the decade of developing the theoretical underpinnings of linear programming, of extending its applicability in industrial settings and to certain combinatorial problems, and of the first general-purpose codes. The '60s saw the emergence of large-scale linear programming, of exploitation of special structure (again pioneered by Dantzig and Dantzig-Wolfe in [28,31]), and of extensions to quadratic programming and linear complementarity. If the '50s and the '60s were the decades of unbridled enthusiasm, the '70s were the decade of doubt, as the theory of computational complexity was developed and Klee and Minty [60] showed that the simplex method with a common pivot rule was of exponential complexity. We will concentrate on the developments since that time; hope has been restored by new polynomial-time algorithms, by bounds on the expected number of pivot steps, and by amazing computational studies on problems with numbers of variables ranging up to the millions.
Linear programming studies the optimization of a linear function over a feasible set defined by linear inequalities, hence a polyhedron. The problem is in some sense trivial, since it is only necessary to examine a finite number of vertices (and possibly edges), but if one is interested in efficient computation, the topic is wonderfully rich and has been the subject of numerous surprising new insights.

来自对 Tjalling Koopmans 的介绍预知的引证 [64] 阅读:\它已经在到现在为止被发现那,为关于数据的一些组合有用的任何计算方法,数据的另外组合可能被构造哪一那一个方法明显令人不满意对。"(醒目地以来自 Bixby et al 的引证这比较。 [在这一个区段结束的时候 13].)
在 [30],Dantzig 写: \幸运地被用于我的论题特别的几何学是被和那纵队的点阵式联合而不是它的排那一个。 这专栏几何学给予我领导我相信单纯的方法会是有效率的解决技术的洞察力。 当我在排几何学方面看了它的时候 , 我已经更早地拒绝方法因为在边缘如此不有希望的外面周围跑。"
因为很多已经被写有关线性规划的早历史 ( 和前历史),举例来说在 [29], 第 2 章,[30], 和 [83] , pp。 209225, 这一张纸将会在发展上多集中自那以后七十。 我希望密谋读者充足调查一些旁道和被和线性规划和那更井- 到处旅行的公路联合的小路。我们将会看着单纯的,椭圆体和内部- 点方法, 以及至少提到一些另外地接近。 当然,我希望读者将会原谅被选择的主题我个人的偏见。 ( 让我在这里提到 Megiddo's 的文章 [75], 也审视来自一个不同的观点一些最近的发展.)
在 1947[27] 的单纯方法的发展之后, 那 '50 年代已经是发展线性规划的理论上的支撑十年,扩充工业的设定它适用性而且对某 combinatorial 问题, 和第一个泛用型的密码。 那 '60 年代看见了大规模的线性规划的出现, 特别的结构 ( 被 Dantzig 和 Dantzig- 沃尔夫再一次提倡在 ) 的开发, 和对二次的规画和线的补充延长。 如果 '50 年代和那 '60 年代是十年的无缰辔的狂热, 那 '70 年代是十年的怀疑, 如计算的复杂理论被发展和 Klee 和 Minty[60]由于一条通常的枢规则的单纯方法是有指数的复杂。我们自从那次以后将会专注于发展;希望已经被新的多名- 时间运算法则回复,被在枢步骤的预期数字上的范围, 而且被在令人惊异的计算研究的问题方面的变数数字直到数百万排列。
线性规划学习在被线的不平等定义的能实行的组合上的线功能的最佳化,一个多面体由此而来。问题是在一些感觉中琐细的,因为它是唯一的必需品调查顶点 ( 和可能地边缘) 的一个有限的数字,但是如果一对有效率的计算感兴趣,主题令人惊奇富有并且已经是很多的令人惊讶的新洞察力的主题。
提问者:匿名
参考技术A 一明引述tjalling库普曼斯在导言[ 64 ]内容如下: \它已发现到目前为止,对于任何的计算方法,这似乎有用,在有关的一些数据集,另一套数据可兴建该方法显然是不能令人满意。 “ (这比较突出,与引述比克斯比等人。 [ 13 ]在本节结尾) 。
在[ 30 ] , dantzig写道: \幸运的特定几何用在我的论文是一相关栏目矩阵,而非其行。此栏几何给了我的洞察力,而导致我相信,单纯形法将是一个有效的解决方法技术。我刚才已拒绝的方法,当我认为这是在该行几何,因为四处外界的优势,似乎使没出息“ 。
由于大部分已经写入有关的早期历史(和前历史)线性规划,例如在[ 29 ] ,第2章, [ 30 ] , [ 83 ] ,聚丙烯。 209 ( 225 ,这份文件将集中更多的发展,自七十年代。我希望阴谋读者足够的调查部分的小道和胡同与线性规划以及更完善的旅游公路。我们将看看简单,椭球,和内部点的方法,也至少提及一些其他办法。当然,我希望读者会原谅我的个人偏见,在选定的专题。 (让我在这里提到米吉多的文章[ 75 ] ,这也调查,最近的一些事态发展从一个不同的观点。 )
以下的发展,单纯形法在1947年[ 27 ] ,五十年代已十年发展的理论基础的线性规划,扩大其适用于工业设置和某些组合的问题,和第一的一般用途守则。六十年代,看到出现的大型线性规划,被剥削的特殊结构(再次开创dantzig和dantzig -沃尔夫在[ 28,31 ] ) ,并扩展到二次规划和线性互补。如果五十年代和六十年代人几十年来肆无忌惮的积极性,七十年代人十年的疑问,作为理论的计算复杂性是发达国家和克利和minty [ 60 ]表明,单纯形法与一个共同的支点规则指数的复杂性。我们将专注于发展自那时起;希望已恢复新的多项式时间算法,由界对预期的数目枢轴的步骤,以及惊人的计算研究,对问题有多少变数,高达数百万。
线性规划的研究优化的线性函数,超过一个可行的设置所定义的线性不等式,因此多面体。问题是,在一定意义上微不足道的,因为它是不仅是必要的审查有限数目的顶点(和可能的优势) ,但如果一个是有兴趣在高效率的计算,题目是完美的丰富和一直受到众多令人惊讶的新的见解。本回答被提问者采纳
参考技术B too long for me

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