Rigorous Global Search: Continuous Problems
This work grew out of several years of research, graduate seminars and talks on the subject. It was motivated by a desire to make the technology accessible to those who most needed it or could most use it. It is meant to be a self-contained introduction,
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Nonconvex Optimization and Its Applications Volume 13 Managing Editors: Panos Pardalos
University of Florida. U.S.A.
Reiner Horst University of Trier. Germany
Advisory Board: Ding-ZhuDu University of Minnesota. U.S.A.
C.A. Floudas
Princeton University. U.S.A.
G. Infanger Stanford University. U.S.A.
J. Mockus Lithuanian Academy of Sciences. Lithuania
P.D. Panagiotopoulos Aristotle University. Greece
H.D. Sherali
Virginia Polytechnic Institute and State University. U.S.A.
The titles published in this series are listed at the end of this volume.
Rigorous
Global Search: Continuous Problems
by
R. Baker Kearfott University of Southwestern Louisiana
Springer-Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4762-8 ISBN 978-1-4757-2495-0 (eBook) DOI 10.1007/978-1-4757-2495-0
Printed on acid-free paper
All Rights Reserved © 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996. Softcover reprint of the hardcover 1st edition 1996
No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
CONTENTS
LIST OF FIGURES
vii
LIST OF TABLES
xi
PREFACE
xiii
1
PRELIMIN ARIES
2
SOFTWARE ENVIRONMENTS
71 71 78 102
ON PRECONDITIONING
113 115
3
4
1.1 1.2 1.3 1.4 1.5 1.6
2.1 2.2 2.3
3.1 3.2
Interval Arithmetic Interval Linear Systems Derivatives and Slopes Automatic Differentiation and Code Lists Interval Newton Methods and Interval Fixed Point Theory The Topological Degree
INTLIB Fortran 90 Interval and Code List Support Other Software Environments
The Inverse Midpoint Preconditioner Optimal Linear Programming Pre conditioners
VERIFIED SOLUTION OF NONLINEAR SYSTEMS 4.1 4.2
An Overall Branch and Bound Algorithm Approximate Roots and Epsilon-Inflation v
1 3 18 26 36 50 66
120
145 146 150
vi
RIGOROUS GLOBAL SEARCH: CONTINUOUS PROBLEMS
4.3 4.4 4.5
5
169 170 177 199
NON-DIFFERENTIABLE PROBLEMS
209 210 218
6.1 6.2
7
154 159 167
OPTIMIZATION
5.1 5.2 5.3
6
Tessellation Schemes Description of Provided Software Alternate Algorithms and Improvements
Background and Historical Algorithms Handling Constraints Description of Provided Software
Extensions of Non-Smooth Functions Use in Interval Newton Methods
USE OF INTERMEDIATE QUANTITIES IN THE EXPRESSION VALUES 7.1 7.2 7.3 7.4 7.5 7.6
The Basic Approach An Alternate Viewpoint - Constraint Propagation Application to Global Optimization Efficiency and Practicality Provided Software Exercises
227 227 230 230 232 233 234
REFERENCES
235
INDEX
255
LIST OF FIGURES
Chapter 1
The united solution set I:: (A, B) for the system (1.19) E(A, B) n X can still be bounded when AX = B is underdetermined. 1.3 The difference between the interval slope SU(j, z, x) and the derivative range lu for z =
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