Practical Mathematical Optimization An Introduction to Basic Optimiz
This book presents basic optimization principles and gradient-based algorithms to a general audience, in a brief and easy-to-read form without neglecting rigour. The work should enable the professional to apply optimization theory and algorithms to his ow
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Applied Optimization VOLUME 97 Series Editors: Panos M. Pardalos University of Florida, U.S.A. Donald W. Heam University of Florida, U.S.A.
PRACTICAL MATHEMATICAL OPTIMIZATION An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms
By JAN A. SNYMAN University of Pretoria, Pretoria, South Africa
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Library of Congress Cataloging-in-Publication Data A C.I.P. record for this book is available from the Library of Congress.
AMS Subject Classifications: 65K05, 90C30, 90-00 ISBN 0-387-24348-8
e-ISBN 0-387-24349-6
Printed on acid-free paper.
© 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 springeronline.com
SPIN 11161813
To Alta my wife and friend
Contents PREFACE
XV
TABLE OF NOTATION
xix
1
INTRODUCTION
1
1.1
What is mathematical optimization?
1
1.2
Objective and constraint functions
4
1.3
Basic optimization concepts
6
1.3.1 1.3.2
Simplest class of problems: Unconstrained one-dimensional minimization . . .
6
Contour representation of a function of two variables (n = 2)
7
1.3.3
Contour representation of constraint functions
1.3.4
Contour representations of constrained optimization problems
11
Simple example illustrating the formulation and solution of an optimization problem
12
1.3.6
Maximization
14
1.3.7
The special case of Linear Programming
14
1.3.5
. . 10
viii
CONTENTS 1.3.8 1.4
1.5
15
Further mathematical prerequisites
16
1.4.1
Convexity
16
1.4.2
Gradient vector o f / ( x )
18
1.4.3
Hessian matrix o f / ( x )
20
1.4.4
The quadratic function in R^
20
1.4.5
The directional derivative of / ( x ) in the direction u 21
Unconstrained minimization
21
1.5.1
Global and local minima; saddle points
23
1.5.2
Local characterization of the behaviour of a multivariable function
24
Necessary and sufficient conditions for a strong local minimum at x*
26
General indirect method for computing x*
28
1.5.3
1.5.4 1.6
Scahng of design variables
Exercises
31
2 LINE SEARCH D E S C E N T M E T H O D S FOR U N C O N S T R A I N E D MINIMIZATION 33 2.1
General line search descent algorithm for unconstrained minimization 2.1.1
2.2
33
General structure of a line search descent method . 34
One-dimensional line search
35
2.2.1
Golden section met
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