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Physical Models and Equilibrium Methods in Programming and Economics PDF

364 Pages·1984·15.446 MB·English
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Physical Models and Equilibrium Methods in Programming and Economics Mathematics and Its Applications (Soviet Series) Managing Editor: M. HAZEWINKEL Centre tor Mathematics and Computer Science, Amsterdam, The Netherlands Editorial Board: L. D. FADDEEV,LOMI, Leningrad, U.S.S.R. A. A. KIRILLOV, MGU, Moscow, U.S.S.R. Yu.l. MANIN, Steklov Inst. of Math. Moscow, U.S.S.R. N. N. MOISEEV, Camp. Centre, Acad. of Sci. Moscow, U.S.S.R. S. P. NOVIKOV,Landau Inst. Theor. Phys., Moscow, U.S.S.R. M. C. POLYVANOV, Steklov Inst. of Math., Moscow, U.S.S.R. Yu. A. ROZANOV, Steklov Inst. of Math., Moscow, U.S.S.R. B. S. Razumikhin Institute fOT Systems Studies, Academy of Sciences, Moscow Physical Models and Equilibrium Methods in Programming and Economics (Revised and augmented compared to the original Russian edition and the French translation) * D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP Dordrecht / Boston / Lancaster Ubrary of Congress Cataloging in Publication Data Ratumikhin, Boris Sergoevich. Physicalmodeis and equilibrium methods in programming and economics. (Mathematics and its applications) Translation of: Fizicheskie modeli i metody teorii ravnovesiia v programmirovanii i ekonomike. "Revised and augmented English edition"- Pref., p. xi. Includes bibliography and index. I. Economics, MathematicaL 2. Economics- Mathematical models. 3. Equilibrium (Economics) 4. Programming (Mathematics) 5. Equilibrium. I. Title. II. Series: Mathematics and its applica' tions (D. Reidel Publishing Company) HBI35.R3913 1984 330'.0124 83-11191 ISBN·13: 918-94-009-6216-1 o-ISBN-13: 978·94-009-6214·3 001: 10.10011978·94-009-6214-3 Publish.cd by D. Reidel Publishing Company P.O. Box 11,3300 AA Dordrecht, Holland Sold and distributed in the U.S.A. and Canada by Kiuwer Academic Publishers 190 Old Derby Street, Hingham, MA 02043. U.S.A. In all other countries, sold and distributed by Kluwer Academic PUblishers Group, P.O. Box 322, 3300 AH Dordreeht, Holland Original title: Fisifeskie modeli i metody teodi ravnovesiya v programmirovanil i ekonomike Translated from the Russian by M. Hazewinkel All Rights Reserved "" 1984 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover lSI edition 1984 No pan of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronie or me(:hanical, including photocopying. recording or by any information storage and retIieval system, without written permission from the (:opytight owner. TABLE OF CONTENTS Editor's preface ix Preface to the revised and augmented English edition xi Preface to the original Russian edition xiii Introduction Chapter I: Equilibrium of mechanical systems with linear constraints and linear programming problems 9 1.1. Introduction 9 1.2. Linear equations and inequalities 10 1.3. Systems of linear equations and inequalities 16 1.4. Linear programming problems. Duality theorems 22 Chapter II: Equilibrium of physical systems and linear programming problems 38 2.1. Introduction 38 2.2. Some concepts from thermodynamics 40 2.3. Physical models of dual pairs of systems of linear equations and inequalities. Alternative theorems 48 2.4. A physical model for linear programming problems. Equilibrium conditions 64 2.5. Penalty methods 72 2.6. Some properties of approximate solutions of dual problems of linear programming problems 87 2.7. Models for transport type problems 91 Chapter III: The method of redundant constraints and iterative algorithms 109 3.1. Introduction 109 3.2. The method of redundant constraints 109 3.3. The first iterative algorithm for solving linear programming problems and for solving systems of linear equations and inequalities 113 3.4. The second algorithm 120 vi TABLE OF CONTENTS 3.5. Reduction of the general linear programming problem to a sequence of inconsistent systems. The third algorithm 132 Chapter IV: The principle of removing constraints 142 4. I. Introduction 142 4.2. The method of generalized coordinates 144 4.3. The method of multipliers 146 4.4. Elastic constraints. Penalty function methods 151 4.5. Discussion 155 Chapter V: The hodograph method 161 5. I. Introduction 161 5.2. The hodograph method for linear programming problems 162 5.3. Solution of the dual problem 174 5.4. Results of numerical experiments 175 Chapter VI: The method of displacement of elastic constraints 179 6. I. Introduction 179 6.2. The first algorithm 180 6.3. The second algorithm 184 6.4. Combining the algorithms 191 Chapter VII: Decomposition methods for linear programming problems 195 7.1. Introduction 195 7.2. Decomposition algorithms 197 7.3. Allocation of resources problems 217 Chapter VIII: Nonlinear programming 227 8. I. Introduction 227 8.2. The principle of virtual displacements and the Kuhn-Tucker theorem 227 8.3. Numerical methods for solving nonlinear programming problems 234 Chapter IX: The tangent method 250 9.1. Introduction 250 9.2. Constrained minimization problems 251 9.3. Linear programming 257 9.4. Dynamic problems of optimal control 263 TABLE OF CONTENTS vii Chapter X: Models for economic equilibrium 268 10.1. Introduction 268 10.2. Equilibrium problems for linear exchange models 269 10.3. An algorithm for solving numerically equilibrium problems for linear exchange economies 278 10.4. Discussion. The Boltzmann principle 285 10.5. Equilibrium of linear economic models 289 10.6 Physical models for economic equilibrium. The equilibrium theorem 291 10.7. An algorithm for solving equilibrium problems for linear economic models 298 10.8. A generalization of the economic equ111br1urn problem 302 Chapter XI: Dynamic economic models 305 11.1. Introduction 305 11.2. The Von Neumann-Gale model. Growth rates and interest rates 306 11.3. A method for solving the problem of maximum growth rates 309 11.4. Duality and problems of growth rates and interest rates 314 11.5. The minimal time problem 319 11.6. A time optimal control problem economic growth 322 11.7. A physical model for solving optimal control problems 324 11.8. Decomposition for time optimal control problems 329 11.9. Optimal balanced growth problems 331 Chapter XII: Optimal control problems 336 Bibliography 344 Index 349 EDITOR'S PREFACE Approach your problems from It isn't that they can't see the right end and begin with the solution. the answers. Then one day, It is that they can't see the perhaps you will find the problem. final qu~stion. G. K. Chesterton. The Scandal 'The Hermit Clad in Crane of Father Brown ITh~ Point of Feathers' in R. van Gulik's a Pin'. The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. HowQvQr, thQ "tree" of knowledge of mathematics and related field does not grow only by putting forth new branches. It also happ~ns, quit~ often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathe matics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. This program, Mathematics and Its Applications, is devoted to such (new) interrelations as exempla gratia: a central concept which plays an important role in several different mathematical and/or scientific specialized areas; - new applications of th~ results and ideas from one area of scientific edeavor into another; influences which the results, problems and concepts of one field of enquiry have and have had on the development of another. The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimul ating rather than definitive, intriguing rather than encyclo- ix x EDITOR'S PREFACE paedic, we hope to contribute something towards better commu nication among the practitioners in diversified ~ields. Because of the wealth of scholarly research being under taken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to work emanating from these particular regions. Thus it was decided to start three rep.ional series under the umbrella of the main MIA programme. The present book is concerned with the mathematics central to the decision sciences, i.e. linear and nonlinear programming, economic equilibrium and growth problems, and optimal control; from an unusual interdisciplinary point of view, however. Namely these matters are studied by means of gas- and fluid mechanical models and this means that the principles of ana lytical dynamics and thermodynamics can be applied. This turns out to be quite remarkably fruitful both in terms of suggestion of algorithms (e.p,. decomposition algorithms) and in terms of conceptual understanding. Thus e.g. duality in the sense of linear programming turns out to be duality between intensive and extensive variables in thermodynamics, the principle of virtual displacements of analytical mechanics is by and large the same as the Kuhn-Tucker theorem and it turns out that there is some sort of economic potential which is minimal at equilibrium. Also this setting gives a natural and suggestive interpre tation to all kinds of penalty function ideas. It seems to me that the ideas in this book have so far been exploited in only a modest way and that much more can be derived from them (notably the penalty function ones and the economic potential one). The unreasonable effectiveness As long as algebra and geo of mathematics in science .,. metry proceeded alonp, separate paths, their advance was slow Eugene Wigner and their applications limited. But when these sciences Well, if you knows of a joined company they drew from better 'ole, go to it. each other fresh vitality and Bruce Bairnsfather thenceforward marched on at a rapid pace towards perfection. What is now proved was once Joseph Louis Lagrange only imagined. William Blake Bussum Michiel Hazewinkel April, 1983 PREFACE TO THE REVISED AND AUGMENTED ENGLISH EDITION The English edition of this book is the result of substantial revisions of the Russian and French editions. The author has not only used the opportunity to remove defects from these editions, he has also introduced essential changes and additions. The most important of these revisions consists of the replacing of chapter IV (finite methods) of the Russian edition with three new chapters: "Principle of removing constraints" (Ch. IV), "The Hodograph method" (Ch. V) and "Method of displacement of elastic constraints" (Ch. VI). In chapter II there is a new section 2.7 "Models for transport problems" because the mechanical and physical models for such problems are elegant, diverse and, especially yield simple constructive devices for solving eigenvalue problems. The book is devoted to analogies which have always played and will always play a most important role in the never ending progress of science. It seems, therefore, fitting to conclude this short preface with the words of Johannes Kepler: "And above all I value Analogies, my most faithful readers. They pertain to all secrets of nature and can be least of all neglected.1I xi

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