Projected Dynamical Systems and Variational Inequalities with Applications
Equilibrium is a concept used in operations research and economics to understand the interplay of factors and problems arising from competitive systems in the economic world. The problems in this area are large and complex and have involved a variety of m
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INTERNATIONAL SERIES IN OPERATIONS RESEARCH & MANAGEMENT SCIENCE
Frederick S. Hillier, Series Editor Department of Operations Research Stanford University Stanford, California Saigal, Romesh. The University of Michigan LINEAR PROGRAMMING: A Modem Integrated Analysis
Projected Dynamical Systems and Variational Inequalities with Applications
Anna Nagumey
and
Ding Zhang School of Management University of Massachusetts Amherst, Massachusetts
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Springer Science+Business Media, LLC
ISBN 978-1-4613-5972-2 ISBN 978-1-4615-2301-7 (eBook) DOI 10.1007/978-1-4615-2301-7
Library of Congress Cataloging-in-Publication Data A C.I.P. Catalogue record for this book is available from the Library of
Congress.
Copyright c 1996 by Springer Science+Business Media New York Origina11y published by Kluwer Academic Publishers in 1996 Softcover reprint of the hardcover 1st edition 1996 Ali rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC.
Printed on acid-free paper.
To Lad and Alexandra and June
Contents Preface
xv
Acknowledgments
xx
Glossary of Notation
I
XXI
Theory of Projected Dynamical Systems
1 Introduction and Overview 1.1 Sources and Notes . . . . 2 Projected Dynamical Systems 2.1 The Variational Inequality Problem. 2.2 The Projected Dynamical System. 2.3 The Skorokhod Problem . . . . . 2.3.1 A Discrete Time Example . 2.3.2 The Skorokhod Problem . . 2.3.3 An Equivalent Problem and Uniqueness of Solutions to the ODE(F, K) . . . . . . . . . . . . . .. 2.3.4 Existence of Solutions to the ODE(F, K) and Convergence of Discrete Approximations 2.4 Sources and Notes .. . . . . . . . . . . . . 3 Stability Analysis 3.1 Basic Concepts of Stability 3.1.1 Examples . . . . . . 3.2 Local Properties Under Regularity 3.3 Properties Under Monotonicity Vll
1 3 6 9
12
17 27
28 29 31 34 39 45 47
50 52 67
CONTENTS
viii 3.4 4
II
Sources and Notes . . . . . . . . . . . . . . . . . . . . . .
Discrete Time Algorithms 4.1 The General Iterative Scheme . . . . . . 4.1.1 Examples of Induced Algorithms 4.2 Convergence.... 4.3 Source and Notes . . . . . . . . . . . . .
Applications
72 75 76 79 82 86
91
5 Oligopolistic Market Equilibrium 5.1 Oligopoly Models. . . . . . . . . . . . . . . . . . 5.1.1 The Variational Inequality Formulations . 5.1.2 The Projected Dynamical System Model. 5.2 Stability Analysis . . . . . . . . . . . 5.2.1 Stability Under Monotonicity 5.2.1.1 An Example . . . 5.2.2 Stability Under Regularity. 5.2.2.1 An Example 5.3 A Discrete Time Algorithm 5.3.1 Numerical Examples 5.4 Sources and Notes .. .
93 94 94 101 102 103 111 112 116 118 122 129
6 Spatial Price Equilibrium 133 6.1 The Quantity Model . . . . . . . . . . . . . . . . 135 6.1.1 A Variational Inequality Formulation. . . 135 6.1.2 The Projected Dynamical S~stems Model 138 6.2 Stability................ 139 6.2.1 Stab
