Leonid Kantorovich

"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 ofjkbni semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form."

"I discovered that a whole range of problems of the most diverse character relating to the scientific organization of production (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i.e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion. I have succeeded in finding a comparatively simple general method of solving this group of problems which is applicable to all the problems I have mentioned, and is sufficiently simple and effective for their solution to be made completely achievable under practical conditions."

"Once some engineers from the veneer trust laboratory came to me for consultation with a quite skilful presentation of their problems. Different productivity is obtained for veneer-cutting machines for different types of materials; linked to this the output of production of this group of machines depended, it would seem, on the chance factor of which group of raw materials to which machine was assigned. How could this fact be used rationally?"

"The university immediately published my pamphlet, and it was sent to fifty People’s Commissariats. It was distributed only in the Soviet Union, since in the days just before the start of the World War it came out in an edition of one thousand copies in all."

"In our time mathematics has penetrated into economics so solidly, widely and variously, and the chosen theme is connected with such a variety of facts and problems that it brings us to cite the words of which are very popular in our country: "One can not embrace the unembraceable". The appropriateness of this wise sentence is not diminished by the fact that the great thinker is only a pen-name."

"In planning the idea of decentralization must be connected with routines of linking plans of rather autonomous parts of the whole system. Here one can use a conditional separation of the system by means of fixing values of flows and parameters transmitted from one part to another. One can use an idea of sequential recomputation of the parameters, which was successfully developed by many authors for the scheme of Dantzig-Wolfe and for aggregative linear models."

"A solution of newly appearing economic problems, and in particular those connected with the scientific-technical revolution often cannot be based on existing methods but needs new ideas and approaches. Such one is the problem of the protection of nature. The problem of economic valuation of technical innovations efficiency and rates of their spreading cannot be solved only by the long-term estimation of direct outcomes and results without accounting peculiarities of new industrial technology, its total contribution to technical progress."

"The accounting methods based on mathematical models, the use of computers for computations and information data processing make up only one part of the control mechanism, another part is the control structure."

"Linear programming was developed as a discipline in the 1940's, motivated initially by the need to solve complex planning problems in wartime operations. Its development accelerated rapidly in the postwar period as many industries found valuable uses for linear programming. The founders of the subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947, and John von Neumann, who established the theory of duality that same year. The Nobel prize in economics was awarded in 1975 to the mathematician Leonid Kantorovich (USSR) and the economist Tjalling Koopmans (USA) for their contributions to the theory of optimal allocation of resources, in which linear programming played a key role. Many industries use linear programming as a standard tool, e.g. to allocate a finite set of resources in an optimal way."

"It is difficult to distinguish another scholar in the history of the twentieth century who contributed as much as him to the fusion of mathematics and economics, the sciences with the antipodal standards of scientific thought. pointed out that he can list only John von Neumann and alongside Leonid Kantorovich among those few of his contemporaries who synthesized the mathematical and humanitarian cultures."