Table Of ContentPhysical 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
