Mathematical Programming and Control Theory
In a mathematical programming problem, an optimum (maxi mum or minimum) of a function is sought, subject to con straints on the values of the variables. In the quarter century since G. B. Dantzig introduced the simplex method for linear programming, man
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CHAPMAN AND HALL MATHEMATICS SERIES Edited by Professor R. Brown
Head of the Department of Pure Mathematics,
University College of North Wales. Bangor and Dr M. A. H. Dempster,
University Lecturer in Industrial Mathematics
and Fellow of Balliol College. Oxford A Preliminary Course in Analysis R. M. F. Moss and G. T. Roberts Elementary Differential Equations R. L. E. Schwarzenberger
A First Course on Complex Functions G. J. O. Jameson Rings, Modules and Linear Algebra B. Hartley and T. O. Hawkes Regular Algebra and Finite Machines J. H. Conway Complex Numbers
W. H. Cockcroft Galois Theory Ian Stewart Topology and Nonned Spaces
C. J. O. Jameson Introduction to Optimization Methods P. R. Adby and M. A. H. Dempster Graphs, Surfaces and Homology P. J. Giblin Linear Estimation and Stochastic Control
M. H. A. Davis
Mathematical Programming and Control Theory B. D. CRAVEN Reader in Mathemntics University of Melbourne
LONDON
CHAPMAN AND HALL A Halsted Press Book John Wiley & Sons , New York
First published 1978 by Chapman and Hall Ltd 11 New Fetter Late, London EC4P 4EE
© 1978 B. D. Craven Typeset by The Alden Press (London and Northampton) Ltd ISBN-13: 978-0-412-15500-0 e-ISBN-I3: 978-94-009-5796-1 DOl: 10.1007/978-94-009-5796-1
This title is available in both hardbound and paperback editions. The paperback edition is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circuloted without the publisher's prior consent in fonn of binding or cover over than that in which it is published and without a similor condition including this condition being imposed on the subsequent purchaser. All nglfts reserved. No part a/this book may be reprinted. or reproduced or utilized in any [ann or by an electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without pennission in wn'ting [rom the Publisher.
Distributed in the U.S.A. by Halsted Press, a Division of John Wiley & Sons, Inc., New York Library of Congress Cataloging in Publication Data Craven. Bruce Desmond. Mathematical progranuning and control theory. (Chapman and Hall mathematics series) Includes bibliographical references. 1. Progranuning (Mathematics) 2. Control theory. I. Title. QA402.5.C73 519.4 78-8463
Contents
Preface I
1.1 1.2 1.3
1.4 1.5 1.6 !.7
1.8 1.9 1.10
2
2.1 2.2 2.3 2.4 2.5 2.6
page
Optimization problems; introduction Introduction Transportation network
Production allocation model Decentralized resource allocation An inventory model Control of a rocket Mathematical formulation Symbols and conventions
Differentiability Abstract version of an optimal control problem References
Mathematical techniques Convex geometry
Convex cones and separation theorems Critical points Convex functions
Alternative theorems
Local solvability and linearization
References
,
IX
I
2 3 4 4 5 6 9 13 14 18 19 19 22 27 27 31 33 35
,;
CONTENTS
3 3.1 3.2 3.3 3.4
Linear systems
