列昂尼德·坎托罗维奇Leonid Kantorovich

列昂尼德·坎托罗维奇的父亲是维塔利·Moiseevich Kantorovich，一位专治性传播疾病的受欢迎医生，母亲是保利娜·格里戈里耶夫娜·扎克斯。坎托罗维奇·维塔利耶维奇有两个姐姐莉迪亚和娜杰日达，以及两个哥哥尼古拉和格奥尔基，他们跟随父亲成为医生。影响幼年坎托罗维奇的最早事件是1917年2月在圣彼得堡爆发的大罢工和骚乱，随后是1917年10月的暴力革命，工人冲进冬宫并解散了政府。1918-20年间内战席卷全国，坎托罗维奇一家前往白俄罗斯（当时称为白俄罗斯），在这个困难时期度过了一年。坎托罗维奇是个神童，他写道自己对科学的兴趣始于八岁时[52]：-

我对科学的最初兴趣和独立思维的最初表现大约在1920年显现。

1922年，他十岁时，父亲去世，从那时起由母亲抚养长大，母亲在他的成长中发挥了重要作用。1926年，他年仅14岁就进入列宁格勒国立大学数学系。他听了弗拉基米尔·Ivanovich Smirnov、格里戈里·米哈伊洛维奇·菲赫金哥尔茨（1888-1959，列宁格勒实分析学派创始人之一）和鲍里斯·德劳内的讲座。在同学中，他与伊西多尔·帕夫洛维奇·纳坦松（1906-1964）、舍盖·索伯列夫、所罗门·格里戈里耶维奇·米赫林（1908-1990）、德米特里·康斯坦丁诺维奇·法捷耶夫和薇拉·尼古拉耶夫娜·扎米亚京（1930年结婚后被称为Vera Nikolaevna Faddeeva）交好。在大学第二年，他仍然只有十五岁，坎托罗维奇作为格里戈里·米哈伊洛维奇·菲赫金哥尔茨的描述函数论研讨班成员开始了研究，他写道[52]：-

我认为那些年里我最重大的研究是关于集合和射影集合上的解析运算（1929-30），在那里我解决了尼古拉·卢津的一些问题。我把这些结果报告给了在哈尔科夫召开的第一届全联盟数学大会（1930）。

哈尔科夫大会于1930年6月24日至30日举行，约有500名参加者，尽管只有14人来自苏联世界之外，包括雅克·阿达马、威海姆·布拉希开、奥托·布卢门塔尔、阿尔诺·当茹瓦、索勒姆·曼德尔布罗伊特、埃利·嘉当和保罗·蒙泰尔。给坎托罗维奇留下最深刻印象的两场演讲是艾尔哈德·施密特的开幕致辞“数学在社会主义建设中的作用”和谢尔盖·伯恩斯坦的深刻而广泛的演讲“单实变函数多项式逼近理论的现状与问题”。坎托罗维奇于6月25日在由迪米特里·缅绍夫主持的“函数论与级数论”分会上发言。他作了题为“论射影集合”的演讲，但后来评论说，他意识到它没有达到大会所设定的极高标准。

他于1930年毕业，年仅十八岁，已达到相当于博士的水平，并继续在列宁格勒国立大学物理数学系数学教研室进行研究。请注意，在这个阶段，苏联已经废除了博士学位，他只有在1935年这种学位的授予恢复时才会正式获得该学位。1930年，他被任命为海军工程学校的助理。次年，他被任命为列宁格勒国立大学数学与力学研究所的研究助理，并从1932年起任计算数学系的副教授。因此，到1932年，他担任了这三个职位，而且由于他仍然只有20岁，他年轻的外表在他的学生中引起了一些惊讶，他们起初拒绝相信这个“年轻人”是他们的讲师，而不是同学。当他来上第一堂课时，几个学生向他喊叫，让他坐下等待教授到来。他在[52]中谈到了他此时的研究：-

1930年从大学毕业时，我在高等学校教育机构从事教学活动的同时，开始了应用问题的研究。国家不断扩大的工业化为此类发展创造了适当的氛围。正是在那个时候，我的著作《一种新的近似共形映射方法》和《新的变分方法》发表了。

他于1933年出版了他的第一本书，与弗拉基米尔·伊万诺维奇·克雷洛夫（1902-1994）和弗拉迪米尔·斯米尔诺夫合著，题为Calculus of variations。1934年，第二届全联盟数学大会在列宁格勒举行，吸引了约700名参加者。坎托罗维奇作了两场演讲，“论区域的共形映射”和“论偏微分方程的一些近似解法”。他的研究也在弗拉迪米尔·斯米尔诺夫所作的演讲“列宁格勒的分析研究”中得到了突出。1934年，他取得了教授资格，次年，他参加了莫斯科拓扑大会。在那里，他遇到了冯·诺伊曼、乔治·戴维·伯克霍夫、阿尔伯特·W·塔克、莫里斯·弗雷歇和其他数学家，并就他在偏序空间方面的工作与这些数学家保持联系。他写道[52]：-

三十年代是泛函分析密集发展的时期，它成为现代数学的基本部分之一。我自己在这一领域的努力主要集中在一个新方向上。那就是对有某些元素对定义了序的函数空间进行系统研究。这种偏序空间理论被证明非常富有成果，并且大约在同一时间在美国、日本和荷兰得到发展。

1934年至1960年，他是列宁格勒国立大学的数学教授。1935年，他取得了重大突破，定义了现在称为$K$-空间的概念。他在论文中这样引入这一概念：-

在本注记中，我定义了一种新类型的空间，我称之为半序线性空间。引入这样的空间使我们能够像研究线性泛函一样研究一个抽象类（取值于这些空间中的那些）的线性运算。

$K$-空间是向量格，其中每个非空序有界子集都有下确界和上确界：-

坎托罗维奇空间为发展线性不等式理论提供了自然的框架，而线性不等式理论在当时实际上是一个未开拓的研究领域。

1936年，他发表了On one class of functional equations（俄文），其中他将半序空间应用于数值方法。他在这篇论文中写道：-

逐次逼近法常被用于证明各类函数方程解的存在性；此外，这些逼近收敛性的证明依赖于所研究的方程可以被另一类简单方程所控制这一事实。类似的证明可以在无穷多个联立线性方程理论以及积分和微分方程理论中遇到。考虑半序空间及其间的运算，使我们能够以抽象形式轻松地发展出这类函数方程的完整理论。

1937年，他在列宁格勒青年研究工作者数学竞赛中获得一等奖；1938年，在全苏青年研究工作者竞赛中，他凭借论文Functional analysis using the theory of semi-ordered spaces获奖。同样在1938年，他与Natalya Ilyina结婚，她像他的两个兄弟和父亲一样，是一名医生；他们有两个孩子，一个儿子和一个女儿，两人都成为了数学经济学家。

他对经济学的兴趣始于1938年。Leon Smolinski解释了这种兴趣是如何开始的[72]：-

1938年，列宁格勒大学的一位年轻数学教授被当地胶合板托拉斯找去，帮助解决一个看似微不足道但令人困惑的生产问题：如何为八台车床制定工作日程，以最大化给定品种的五种胶合板的产量。该托拉斯的实验室似乎无法得出一个令人满意且无法进一步改进的解决方案。坎托罗维奇教授能否告诉他们哪里出了问题？

J M Montias解释道[57]：-

1938-1939年，列宁格勒数学家L V 坎托罗维奇与胶合板行业的一个研究实验室合作，研究在需要不同机器时间的多种产品之间分配多台不同机器的问题，以按某些期望比例最大化产品产量。1939年7月，他在列宁格勒国立大学发行的一本小册子[《组织和计划生产的数学方法》（俄文）]中发表了他的成果。除了对机器问题的详细处理外，他还概述了选择最优轮作方案、合理规划运输货物路线以及最小化从标准形状切割材料浪费的方法。这些本质上是线性规划问题，其中之一——金属切割问题——是最一般的一类，可以证明等价于矩阵博弈问题。

坎托罗维奇的背景完全在数学领域，但他对 underlying 经济学表现出了相当大的感觉，他将数学技术应用于经济学。他是最早将线性规划用作经济学工具的人之一，这出现在上述引文中提到的出版物Mathematical methods of organising and planning production中。坎托罗维奇的学生之一Valery Makarov在[55]中写道：-

这可以被视为一份历史文献，包含了线性规划发现的事实。最优规划的生产问题的数学表述在这里首次被提出，并且提出了求解这些问题的有效方法以及经济分析。

坎托罗维奇将许多新概念引入数学规划的研究中，例如在生产空间的解点处基于支撑超平面给出必要且充分的最优性条件、原始-对偶方法的概念、乘子的经济学解释，以及线性规划中使用的列生成法。他在经济学方面最基础性的著作之一是The best use of economic resources，该书于1942年写成，但直到1959年才出版。在这部著作中，坎托罗维奇将优化技术应用于经济学中的广泛问题。他还提出了一种处理技术创新经济学的理论。该理论有三个组成部分，即对生产者的影响、对消费者的影响，以及该理论的新颖部分——由创新带来的经济潜力增长所产生的影响。当然，当坎托罗维奇最初撰写这部著作时，第二次世界大战正在对研究和科学出版物产生重大影响。他被征召入伍，被授予少校军衔，并且在1941年，他任教所在的工业建筑学院变成了军事工程高等技术学校。该校从列宁格勒迁至莫斯科以北300公里的雅罗斯拉夫尔，坎托罗维奇也被疏散到那里。除了在军事工程高等技术学校任教外，他还承担了与国防有关的各项任务。他讲授概率论课程，这些课程构成了Theory of Probability（俄文）（1946）一书的基础。由于他当时所从事教学工作的军事性质，该书强调概率在军事问题中的应用。他写道[52]：-

在那些日子里，我的理论研究和应用研究毫无共同之处。但后来，特别是在战后时期，我成功地将它们联系起来，并展示了在数值数学中使用泛函分析思想的广泛可能性。我在我的论文中证明了这一点，这篇论文的标题本身——‘泛函分析与应用数学’（俄文）——在当时看来似乎是自相矛盾的。1949年，这项工作获得了国家奖，后来被收录在与G P Akilov合著的《赋范空间中的泛函分析》（俄文）（1959）一书中。

Edwin Hewitt在评论他与G P Akilov合著的1959年那本书时写道：-

作者们特别关注泛函分析在逼近论以及微分方程和积分方程（线性和非线性）解的存在性与唯一性理论中的应用。这种观点使他们纳入了大量空间和算子的具体例子，其中一些在其他地方不易获得，许多涉及复杂的计算，许多具有相当大的意义。这种对具体应用的强调是受欢迎的，应当使这本书具有广泛的实用性。

他在新西伯利亚的苏联科学院西伯利亚分院担任数学与经济学讲席（1961-1971），然后在莫斯科的国民经济计划研究所主持研究（1971-76）。坎托罗维奇是1975年诺贝尔经济学奖的共同获得者。该奖项由坎托罗维奇和Tjalling C 特亚林·科普曼斯共同获得，其颁奖词是：-

……因其对资源最优配置理论的贡献。作为他们在此领域工作的起点，两人都研究了这一对所有经济活动都至关重要的问题——如何将可用的生产资源用于商品和服务生产中以获得最大利益。这一领域涵盖的问题包括：应生产哪些商品、应采用哪些生产方法、当前生产中有多少应被消费、多少应留作储备以为未来的生产和消费创造新资源。……坎托罗维奇教授今天是苏联经济研究中数学学派的主要代表人物。他早在1939年就在经济研究领域做出了最初的贡献，当时他写了一篇关于个别企业中资源有效利用的含义和意义的文章。在许多出版物中，其中之一是他的著作《经济资源的最佳利用》，坎托罗维奇教授分析了整个经济的类似效率条件，并在那里特别展示了资源配置与价格体系之间的联系，无论是在某一时点上还是在增长的经济中。这一分析中的一个重要内容是表明计划经济中决策分散化的可能性如何取决于一个合理价格体系的存在，包括一个统一的核算利率，以构成投资决策的基础。

正如Herbert E Scarf所说[71]：-

这段简短的引文无法揭示他们的工作所引发的经济理论与方法上的科学革命的规模。

文章[26]是坎托罗维奇必须提交给正在考虑授予他奖项的诺贝尔奖委员会的自传。在[13]中，Belykh考察了西方学者对作为数学经济学家的坎托罗维奇的看法，并得出结论：-

尽管存在意见分歧，并且有人试图将坎托罗维奇归入某一经济学派或另一学派，但这里所考虑的所有科学家都强调他对经济科学发展的杰出贡献。

坎托罗维奇最著名的是将数学方法，特别是数学规划，应用于经济学，然而，正如我们所看到的，他还在许多其他数学领域工作过。这些其他领域包括泛函分析和数值分析，在这些主题中，他发表了关于函数论、复变量理论、逼近理论的论文，其中他对使用Bernstein多项式特别感兴趣，还有变分法、求偏微分方程近似解的方法以及描述集合论。在他职业生涯的后期，他还对计算机体系结构产生了兴趣。坎托罗维奇描述了他在1940年代末与Mark Konstantinovich Gavurin（1911-1992）和Vera Nikolaevna Faddeeva一起从事的计算机项目（见[69]）：-

它们有效使用的基本原则是相似计算的并行化，这使得可以在插线板上引入简单的程序变更（当然是手工操作）。例如，有人提出了几种从表格中快速采样的方法，以及一种在制表机上不通过乘法而是通过加法来计算标量积的方法，其中一个乘数不是以十进制而是以二进制形式构成。一个重大的实际成就是借助这种原始设备，在大区间上计算了高达120阶的弗里德里希·威廉·贝塞尔函数表。这里最有趣的是在这些机器上对Bessel functions的微分方程进行积分的计算的并行化。并行化是通过将积分区间分割成若干区间，并在每个区间上同时计算不同指标的函数来实现的；这样就得到了足够大量的相同运算，可以在这些机器上高效执行。

在[19]中，Yakov Il'ich Fet写道：-

我们简要描述基于坎托罗维奇的建议而创建的计算机。我们注意到坎托罗维奇提出的计算过程大块组织概念的重要性，以及这一概念对计算机系统体系结构发展的影响。

Valery Makarov在[55]中写到坎托罗维奇的：-

……数学天才以及他兴趣和知识的广泛范围。

以下是Valery Makarov的另一段引文，这次出自[3]：-

他在分析、函数论、计算数学方面取得了一流成果。他有许多关于集合论、计算机程序设计理论等方面的伟大著作。他出版了十几部享有盛誉的数学专著。显然，坎托罗维奇 是一位彻头彻尾的数学家。……实际上，这并非全部真相。坎托罗维奇 的独特之处恰恰在于，他同时是一位杰出的经济学家，一位从根本上改变了人们对经济事件的理解、改变了整个经济思维的科学家，并成为一门原创经济学派的创始人。

他对数学、经济学和计算机的卓越贡献发表在 300 多篇论文和著作中。有趣的是，在 20 世纪 80 年代，坎托罗维奇 提出他的贡献可以分为以下九个不同领域：(1) 描述函数论与集合论；(2) 构造函数论；(3) 分析的近似方法；(4) 分析；(5) 分析与应用数学；(6) 线性规划；(7) 硬件与软件；(8) 最优规划与最优价格；(9) 计划经济的经济问题。然而，正如 Semen Samsonovich Kutateladze 在 [37] 中指出的，他的个性却不太容易定义：-

辉煌成就与对实际生活阴暗面适应不良的实例之间的对比，被 坎托罗维奇 列为戏剧性的谜团之一。他的一生成为一个难以置信、令人困惑的人道主义现象。坎托罗维奇 在私人交往中明显的内向，却莫名其妙地伴随着公开场合的彻底外向。毫无演说才能，却与他深邃的逻辑和论辩中的特殊造诣相邻。他天生的自由与自足，与有目的、不知疲倦的忍耐共存，在必要时这种忍耐能达到“铁腕”般的力量。坎托罗维奇 的自由几乎不会让任何人困惑，因为它源于他的本质，即数学天赋。他的善良与温和是天生的。坚韧和巨大的洞察力是他为理性而自觉选择并培养的后天特质。

坎托罗维奇 因其卓越贡献获得了许多荣誉，其中最负盛名的是我们已经提到的诺贝尔奖。他当选为许多科学院和科学学会的成员，包括国际计量经济学会（1966 年）、Hungarian Academy of Sciences（1967 年）、American Academy of Arts and Sciences（1969 年）、德意志民主共和国科学院（1977 年）、墨西哥国家工程科学院（1977 年）、南斯拉夫科学与艺术学院（1980 年）、爱尔兰国际控制研究所（1984 年）。他被以下大学授予名誉博士学位：格拉斯哥大学（1966 年）、华沙大学（1966 年）、格勒诺布尔大学（1966 年）、尼斯大学（1968 年）、赫尔辛基大学（1969 年）、慕尼黑大学（1970 年）、巴黎大学（索邦）（1975 年）、剑桥大学（1976 年）、宾夕法尼亚大学（1976 年）、加尔各答印度统计研究所（1978 年）以及哈勒-维滕贝格马丁·路德大学（1984 年）。他获得了斯大林奖（1949 年）、美国运筹学会证书（1960 年）、列宁奖（1965 年）、布拉格高等经济学院铜质奖章（1981 年）以及伯明翰 运筹学 学会银质奖章（1986 年）。

1986年4月，他因癌症去世后，被安葬在莫斯科的新圣女公墓。

Leonid Vital'evich Kantorovich's father was Vitaliy Moiseevich Kantorovich, a popular medical doctor specialising in sexually transmitted diseases, and his mother was Paulina Grigoryevna Zaks. Leonid Vital'evich had two elder sisters Lidiya and Nadezhda, and two elder brothers Nikolay and Georgiy who followed their father in becoming medical doctors. The first events to impact on the young child Leonid was the general strike and disorder that broke out in St Petersburg in February 1917 and then the violent revolution of October 1917 when workers stormed the Winter Palace and disposed the government. Civil war then raged throughout the country during 1918-20 and the Kantorovich family went to Belarus (then known as Byelorussia) where they spent a year of this difficult period. Kantorovich was a child prodigy and he writes about his interest in science beginning when he was eight years old [52]:-

My first interest in sciences and the first displays of self-dependent thinking manifested themselves about 1920.

In 1922, when he was ten years old, his father died and from that time on he was brought up by his mother who played an important role in his upbringing. He entered the Mathematics Faculty of Leningrad State University in 1926, when he was only 14 years old. He attended lectures by Vladimir Ivanovich Smirnov, Grigorii Mickhailovich Fichtengolz (1888-1959), one of the founders of the Leningrad school of real analysis, and Boris Nikolaevich Delone. Among his fellow students, he was friends with Isidor Pavlovich Natanson (1906-1964), Sergei Lvovich Sobolev, Solomon Grigor'evich Michlin (1908-1990), Dmitrii Konstantinovich Faddeev and Vera Nikolaevna Zamyatin (who was known as Vera Nikolaevna Faddeeva after her marriage in 1930). In his second year at university, still only aged fifteen, Kantorovich began research as a member of Grigorii Mickhailovich Fichtengolz's descriptive function theory seminar and he writes [52]:-

I think my most significant research in those days was that connected with analytical operations on sets and on projective sets (1929-30) where I solved some of Nikolai Nikolaevich Luzin's problems. I reported these results to the First All-Union Mathematical Congress in Kharkov (1930).

The Kharkov Congress took place from 24 to 30 June 1930 and had around 500 participants although only 14 came from outside the Soviet world including Jacques Hadamard, Wilhelm Blaschke, Otto Blumenthal, Arnaud Denjoy, Szolem Mandelbrojt, Élie Cartan, and Paul Montel. The two lectures that made the greatest impression on Kantorovich were the opening address by Otto Yulyevich Schmidt on "The role of mathematics in construction of socialism" and Sergei Natanovich Bernstein's deep and wide-ranging address "State of the art and problems of the theory of approximations of functions of one real variable by polynomials". Kantorovich spoke on 25 June in the session on 'Theory of functions and theory of series' chaired by Dmitrii Evgenevich Menshov. He gave the lecture "On projective sets" but later remarked that he realised it was not up to the extraordinarily high standards that the Congress set.

He graduated in 1930 at the age of eighteen having reached the level equivalent to a doctorate and continued to undertake research in the Mathematical Department of the Faculty of Physics and Mathematics of Leningrad State University. Note that at this stage, the Soviet Union had abolished doctoral degrees and he would only formally receive the degree in 1935 when the award of such degrees was reinstated. He was appointed as an assistant in the Naval Engineering School in 1930. In the following year he was appointed as a research associate in the Research Institute of Mathematics and Mechanics of Leningrad State University and, from 1932, an associate professor in the Department of Numerical Mathematics. By 1932, therefore, he held these three positions and, being still only 20 years of age, his youthful appearance caused some surprise among his students who at first refused to believe that the "youngster" was their lecturer and not a fellow-student. When he came to give his first lecture, several students shouted to him to sit down and wait for the professor to arrive. He spoke of his research at this time in [52]:-

On graduating from the university in 1930, simultaneously with my teaching activities at the higher school educational institutions, I started my research in applied problems. The ever expanding industrialisation of the country created the appropriate atmosphere for such developments. It was precisely at that time such works of mine, 'A New Method of Approximate Conformal Mapping', and 'The New Variational Method' were published.

He published his first book in 1933, coauthored with Vladimir Ivanovich Krylov (1902-1994) and Vladimir Ivanovich Smirnov, entitled Calculus of variations. In 1934 the Second All-Union Mathematical Congress was held in Leningrad and attracted around 700 participants. Kantorovich gave two lectures, "On conformal mappings of domains" and "On some methods of approximate solution of partial differential equations". His research was also highlighted in the lecture "Leningrad studies in analysis" given by Vladimir Ivanovich Smirnov. In 1934 he qualified as a professor and, in the following year, he participated in the Moscow Topological Congress. There he met John von Neumann, George David Birkhoff, Albert William Tucker, Maurice Fréchet and other mathematicians, and he maintained his contacts with these mathematicians concerning his work on partially-ordered spaces. He writes [52]:-

The Thirties was a time of intensive development of functional analysis which became one of the fundamental parts of modern mathematics. My own efforts in this field were concentrated mainly in a new direction. It was the systematical study of functional spaces with an ordering defined for some of pairs of elements. This theory of partially-ordered spaces turned out to be very fruitful and was being developed at approximately the same time in the USA, Japan and the Netherlands.

From 1934 to 1960 he was a professor of mathematics at Leningrad State University. In 1935 he made a major breakthrough when he defined what are now called $K$-spaces. He introduces the notion in his paper as follows:-

In this note, I define a new type of space that I call a semiordered linear space. The introduction of such a space allows us to study linear operations of one abstract class (those with values in these spaces) in the same way as linear functionals.

$K$-spaces are vector lattices in which every nonempty order bounded subset has an infimum and supremum:-

Kantorovich spaces have provided the natural framework for developing the theory of linear inequalities which was a practically uncharted area of research those days.

In 1936 he published On one class of functional equations (Russian) in which he applied semiordered spaces to numerical methods. He writes in this paper:-

The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form.

He was awarded first prize in mathematics at the Leningrad competition for young research workers in 1937 and at the All-Union competition of young research workers in 1938, he received the prize for his paper Functional analysis using the theory of semi-ordered spaces. Also in 1938 he married Natalya Ilyina who, like his two brothers and father, was a medical doctor; they had two children, a son and a daughter, who both became mathematical economists.

His interest in economics began in 1938. Leon Smolinski explains how this interest began [72]:-

A young professor of mathematics at the Leningrad University was approached by the local plywood trust in 1938 to help with a seemingly trivial but puzzling production problem: how to draw a work schedule for eight lathes so as to maximize output of five varieties of plywood of a given assortment. The trust's laboratory seemed unable to arrive at a satisfactory solution that could not be further improved upon. Could Professor Kantorovich tell them where they had gone wrong?

J M Montias explains [57]:-

In 1938-1939, L V Kantorovich, a Leningrad mathematician, worked in consultation with a research laboratory of the plywood industry on the problem of allocating a number of different machines among products requiring various amounts of machine-time so as to maximize the output of the products in certain desired proportions. In July 1939, he published his results in a pamphlet issued by the Leningrad State University [Mathematical Methods of Organizing and Planning Production (Russian)]. In addition to his detailed treatment of the machine problem, he sketched out methods for selecting an optimum crop-rotation scheme, for the rational routing of transported goods, and for minimizing the waste of materials cut from standard forms. These were essentially linear programming problems, one of them - the metal-trimming problem - being of the most general sort that can be shown to be equivalent to a matrix-game problem.

Kantorovich's background was entirely in mathematics but he showed a considerable feel for the underlying economics to which he applied the mathematical techniques. He was one of the first to use linear programming as a tool in economics and this appeared in a publication Mathematical methods of organising and planning production mentioned in the above quote. Valery Makarov, one of Kantorovich's pupils, writes in [55]:-

This may be considered a historic document, containing the facts about discovery of the linear programming. The mathematical formulation of production problems of optimal planning was presented here for the first time and the effective methods of their solution and economic analysis were proposed.

Kantorovich introduced many new concepts into the study of mathematical programming such as giving necessary and sufficient optimality conditions on the base of supporting hyperplanes at the solution point in the production space, the concept of primal-dual methods, the interpretation in economics of multipliers, and the column-generation method used in linear programming. One of his most fundamental works on economics was The best use of economic resources which he wrote in 1942 but was not published until 1959. In this work Kantorovich applies optimisation techniques to a wide range of problems in economics. He also proposed a theory to handle the economics of technological innovations. This had three components namely the effect on the producer, the effect on the consumer and, the novel part of the theory, the effect derived from the increasing economic potential arising from the innovation. Of course when Kantorovich first wrote this work World War II was having a major impact on research and scientific publications. He was drafted into the armed forces, given the military rank of a major and, in 1941, the Institute of Industrial Construction where he was teaching became the Higher Technical School of Military Engineering. This was moved from Leningrad to Yaroslavl, 300 km north of Moscow and Kantorovich was evacuated there. As well as teaching at the Higher Technical School of Military Engineering he also undertook various tasks relating to defence. He taught courses on probability which formed the basis for the book Theory of Probability (Russian) (1946). Because of the military nature of the teaching that he was doing at this time the book emphasises applications of probability to military problems. He writes [52]:-

In those days, my theoretical and applied research had nothing in common. But later, especially in the postwar period, I succeeded in linking them and showing broad possibilities for using the ideas of functional analysis in Numerical Mathematics. This I proved in my paper, the very title of which, 'Functional Analysis and Applied Mathematics' (Russian), seemed, at that time, paradoxical. In 1949, the work was awarded the State Prize and later was included in the book, 'Functional Analysis in Normed Spaces' (Russian), written with G P Akilov (1959).

Edwin Hewitt reviewing his 1959 book with G P Akilov writes:-

The authors are particularly concerned with applications of functional analysis to the theory of approximation and the theory of existence and uniqueness of solutions of differential and integral equations (both linear and non-linear). This point of view has led them to include a great many specific examples of spaces and operators, some not readily accessible elsewhere, many involving intricate computations, and many of considerable interest. This emphasis on concrete applications is welcome and should make the book of wide usefulness.

He held the chair of mathematics and economics in the Siberian branch of the USSR Academy of Sciences in Novosibirsk (1961-1971), then directed research at Moscow's Institute of National Economic Planning (1971-76). Kantorovich was a joint winner of the 1975 Nobel Prize for economics. The citation for the award, made jointly to Kantorovich and Tjalling C Koopmans, was:-

... for their contributions to the theory of optimum allocation of resources. As the starting point of their work in this field, both have studied the problem - fundamental to all economic activity - of how available productive resources can be used to the greatest advantage in the production of goods and services. This field embraces such questions as what goods should be produced, what methods of production should be used and how much of current production should be consumed, and how much reserved to create new resources for future production and consumption. ... Professor Kantorovich is today the leading representative of the mathematics school in Soviet economic research. He made his first contributions in the field of economic research as early as 1939 when he wrote an essay on the meaning and significance of an efficient use of resources in individual enterprises. In a number of publications, one being his book, 'The Best Use of Economic Resources', Professor Kantorovich has analysed similar efficiency conditions for an economy as a whole, and there, particularly demonstrated the connection between the allocation of resources and the price system, both at a certain point in time and in a growing economy. An important element in this analysis was to show how the possibility of decentralising decisions in a planned economy is dependent on the existence of a rational price system, including a uniform accounting interest rate to form a foundation for investment decision.

As Herbert E Scarf remarks [71]:-

This brief citation cannot reveal the magnitude of the scientific revolution in economic theory and methods initiated by their work.

The article [26] is the autobiography which Kantorovich had to submit to the Nobel Prize committee who were considering him for the award. In [13] Belykh examines the opinions of Western scholars on Kantorovich as a mathematical economist and concludes that:-

Despite the differences of opinion and attempts to assign Kantorovich to one economic school or another, all the scientists under consideration here emphasise his outstanding contribution to the development of economic sciences.

Although Kantorovich is most famous for applications of mathematical methods, particularly mathematical programming, to economics, however, as we have seen, he also worked in many other areas of mathematics. These other areas include functional analysis and numerical analysis and within these topics he published papers on the theory of functions, the theory of complex variables, approximation theory in which he was particularly interested in using Bernstein polynomials, the calculus of variations, methods of finding approximate solutions to partial differential equations, and descriptive set theory. Later in his career he also became interested in computer architecture. Kantorovich described the computer projects he was working on with Mark Konstantinovich Gavurin (1911-1992) and Vera Nikolaevna Faddeeva in the late 1940s (see [69]):-

The basic principle of their effective use was the paralleling of similar calculations, which made it possible to introduce simple program changes on the plugboard (of course, by hand). For example, several methods were suggested for fast sampling from tables and a method for calculating a scalar product not by multiplication, but by addition performed on the tabulator, with one of the multipliers being formed not in the base-ten system, but in the binary one. A serious actual achievement was the calculation of tables of Bessel functions up to the 120th order on a large interval with the help of this primitive equipment. The most interesting thing here was the paralleling of calculations for integrating the differential equation for the Bessel functions on these machines. The parallelism was achieved by splitting the integration interval into several intervals and simultaneously calculating functions of different indices on each of the intervals; thus one obtained a sufficiently large number of identical operations that could be efficiently performed on these machines.

In [19] Yakov Il'ich Fet writes:-

We describe briefly computers created on the basis of Kantorovich's suggestions. We note the significance of the concept, put forth by Kantorovich, of the large-block organisation of computing processes and the influence of this concept on the development of the architecture of computer systems.

Valery Makarov writes in [55] of Kantorovich's:-

... mathematical genius and the vast range of his interests and knowledge.

Here is another quote by Valery Makarov, this time from [3]:-

He is the author of first-class results in functional analysis, in the theory of functions, in computational mathematics. He has a number of great works on the theory of sets, the theory of computer programming, etc. He published a dozen of reputable monographs on mathematics. It seems clear that Leonid Kantorovich is a mathematician to the core. ... In reality, this is not the whole truth. The uniqueness of Kantorovich is precisely in that he is at the same time an outstanding economist, a scientist who changed fundamentally the understanding of economic events, the whole economic thinking, and became the founder of an original economic school.

His remarkable contribution to mathematics, economics and computers was published in over 300 papers and books. It is interesting to note that, in the 1980s, Kantorovich suggested that his contributions might be divided into the following nine distinct areas: (1) descriptive function theory and set theory; (2) constructive function theory; (3) approximate methods of analysis; (4) functional analysis; (5) functional analysis and applied mathematics; (6) linear programming; (7) hardware and software; (8) optimal planning and optimal prices; and (9) the economic problems of a planned economy. However, his personality was less easy to define as pointed out by Semen Samsonovich Kutateladze in [37]:-

The contradistinction between the brilliant achievements and the instances of poor adaptation to the practical seamy side of life is listed among the dramatic enigmas by Kantorovich. His life became a fabulous and puzzling humanitarian phenomenon. Kantorovich's introvertness, obvious in personal communications, was inexplicably accompanied by outright public extravertness. The absence of any orator's abilities neighboured his deep logic and special mastery in polemics. His innate freedom and self-sufficiency coexisted with the purposeful and indefatigable endurance that reached the power of a "iron grip" in the case of necessity. The freedom of Kantorovich can hardly bewilder anyone as stemming from his essence, the gift of mathematics. His kindness and mildness were inborn. The tenacity and tremendous force of penetration were the acquired traits that he selected and cultivated conscientiously for the sake of rationality.

Kantorovich received a great many honours for his remarkable contributions, the most prestigious being the Nobel prize which we have already mentions. He was elected to many academies and scientific societies including the International Econometric Society (1966), the Hungarian Academy of Sciences (1967), the American Academy of Arts and Sciences (1969), the Academy of Sciences of German Democratic Republic (1977), the National Engineering Academy of Mexico (1977), Yugoslavian Academy of Science and Arts (1980), the International Control Institute of Ireland (1984). He was awarded an honorary doctorate by the universities of Glasgow (1966), Warsaw (1966), Grenoble (1966), Nice (1968), Helsinki (1969), Munich (1970), Paris (Sorbonne) (1975), Cambridge (1976), Pennsylvania (1976), the Indian Statistical Institute in Calcutta (1978), and Martin-Luther University, Halle-Wittenberg (1984). He received the Stalin prize (1949), a diploma of the Operations Research Society of America (1960), Lenin Prize (1965), Bronze medal of the Prague Higher Economic School (1981), and the Silver medal of Operational Research Society of Birmingham (1986).

Following his death from cancer in April 1986, he was buried at Novodevichy Cemetery in Moscow.