# Leonid Vitalyevich Kantorovich

### Quick Info

Born
19 January 1912
St Petersburg, Russia
Died
7 April 1986
Moscow, USSR

Summary
Leonid Kantorovich was a Soviet mathematician and economist who can be regarded as the founder of linear programming.

### Biography

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.

### References (show)

1. Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Leonid-Vitalyevich-Kantorovich
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61. R W Pfouts, Review: The Best Use of Economic Resources, by L V Kantorovich, G Morton and P F Knightsfield, Annals of the American Academy of Political and Social Science 361 (Political Socialization: Its Role in the Political Process) (1965), 168-169.
62. P Rabinowitz, Review: Approximate Methods of Higher Analysis, by L V Kantorovich, V I Krylov and Curtis D Benster, Science, New Series 134 (3487) (1961), 1358.
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65. T J Rivlin, Review: Approximate Methods of Higher Analysis, by L V Kantorovich, V I Krylov and Curtis D Benster, SIAM Review 2 (4) (1960), 299.
66. J V Romanovsky, L V Kantorovich's works in mathematical programming, in Mathematical programming, Tokyo, 1988 (Tokyo, 1989), 365-382.
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70. H Sagan, Review: Approximate Methods of Higher Analysis, by L V Kantorovich, V I Krylov and Curtis D Benster, Amer. Math. Monthly 67 (5) (1960), 485.
71. H E Scarf, The 1975 Nobel Prize in Economics: Resource Allocation, Science, New Series 190 (4215) (1975), 649; 710; 712.
72. L Smolinski, Portrait: Leonid Kantorovich, Challenge 19 (2) (1976), 46-47.
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