"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." (Leonid V Kantorovich, "On one class of functional equations", 1936)
"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." (Leonid V Kantorovich, "Mathematical Methods of Organizing and Planning Production", Management Science 6(4), 1960)
"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." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture]1975)
"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 Kozma Prutkov 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." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture]1975) [lead paragraph]
"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." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives," 1975)
"In spite of its universality and good precision the linear model is very elementary in its means which are mainly those of linear algebra, so even people with very modest mathematical training can understand and master it. The last is very important for a creative and non-routine use of the analytical means which are given by the model." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives," 1975)
"The hard thing in a model realization is to receive and often to construct necessary data which in many cases have considerable errors and sometimes are completely absent, since none needed them previously. Difficulties of principle lie in the future prediction data and in the estimation of industry development variants." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture] 1975)
"The treatment of the economy as a single system, to be controlled toward a consistent goal, allowed the efficient systematization of enormous information material, its deep analysis for valid decision-making. It is interesting that many inferences remain valid even in cases when this consistent goal could not be formulated, either for the reason that it was not quite clear or for the reason that it was made up of multiple goals, each of which to be taken into account." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture] 1975)
"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." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", [Nobel lecture]1975)
"One aspect of reality was temporarily omitted in the development of the theory of function spaces. Of great importance is the relation of comparison between practical objects, alongside algebraic and other relations between them." (Leonid V Kantorovich, "Functional analysis: Basic Ideas)", Siberian Mathematical Journal 28 (1), 1987)
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