Introduction to the Theory of Nonlinear Optimization

259 Pages·1996·4.391 MB·English

by  Prof. Dr. Johannes Jahn (auth.)

Introduction to the Theory
of Nonlinear Optimization
Springer-Verlag Berlin Heidelberg GmbH
Johannes Ja hn
Introduction
to the Theory
of Nonlinear
Optimization
Second Revised Edition
With 30 Figures
i
Springer
Praf. Dr. Johannes Jahn
Universitiit Erlangen-Niirnberg
Institut fUr Angewandte Mathematik
MartensstraBe 3
91058 Erlangen, Germany
Llbrary of Congre •• Cataloglng-ln-Publlcatlon Data
Jahn. Jahannes. 1951-
Introductlon ta the theory of nonllnaar optlmlzatlon I Johannes
Jahn. -- 2nd rav. ed.
p. cm.
Includas blbl10graphlcal referancas and Index.
ISBN 978-3-662-03273-2 ISBN 978-3-662-03271-8 (eBook)
DOI 10.1007/978-3-662-03271-8
1. Hatha.atlcal optl.'zatlon. 2. Nonl1near thaorles. 1. Tltla.
QA402.5.J335 1996
519.7'S--dc20 96-29218
CIP
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Originally published by Springer-Verlag Berlin Heidelberg New York in 1996
Softcover reprint oftlte hardcover 2.nd edition 1996
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Preface
This book presents an application-oriented introduction to the the
ory of nonlinear optimization. It describes basic notions and concep
tions of optimization in the setting of normed or even Banach spaces.
Various theorems are applied to problems in related mathematical
areas. For instance, the Euler-Lagrange equation in the calculus of
variations, the generalized Kolmogorov fondition and the alternation
theorem in approximation theory as well as the Pontryagin maximum
principle in optimal control theory are derived from general results of
optimization.
Because of the introductory character of this text it is not intended
to give a complete description of all approaches in optimization. For
instance, investigations on conjugate duality, sensitivity, stability, re
cession cones and other concepts are not included in the book.
The bibliography gives a survey of books in the area of nonlinear
optimization and related areas like approximation theory and optimal
control theory. Important papers are cited as footnotes in the text.
This second edition is a revised and corrected version containing
answers to all exercises and an updated bibliography.
I am grateful to S. GeuB, S. Gmein~r, S. Keck, Prof. Dr. E.W.
Sachs and H. Winkler for their support, and I am especially in
debted to D.G. Cunningham, Dr. F. Hettlich, Dr. J. Klose, Prof. Dr.
E.W. Sachs and Dr. T. Staib for fruitful discussions.
Johannes J ahn
Contents
Preface v
1 Introduction and Problem Formulation 1
2 Existence Theorems for Minimal Points 7
2.1 Problem Formulation . 7
2.2 Existence Theorems. . . . . . . . . . . . 8
2.3 Set of Minimal Points .......... . 18
2.4 Application to Approximation Problems 19
2.5 Application to Optimal Control Problems 24
Exercises .......... . 30
3 Generalized Derivatives 33
3.1 Directional Derivative ..... . 33
3.2 Gateaux and Frechet Derivatives 39
3.3 Subdifferential.. 51
3.4 Quasidifferential. 59
3.5 Clarke Derivative 69
Exercises ..... 77
4 Tangent Cones 81
4.1 Definition and Properties. 81
4.2 Optimality Conditions 91
4.3 A Lyusternik Theorem 98
Exercises .......... . . 106
5 Generalized Lagrange Multiplier Rule 109
5.1 Problem Formulation . . . . . . . . . . . 109
Contents
YIll
5.2 Necessary Optimality Conditions ..... · 112
5.3 Sufficient Optimality Conditions . . . . . . 130
5.4 Application to Optimal Control Problems 140
Exercises. 159
6 Duality 163
6.1 Problem Formulation · 163
6.2 Duality Theorems . . · 168
6.3 Saddle Point Theorems · 172
6.4 Linear Problems .... 176
6.5 Application to Approximation Problems 179
Exercises .................... . 188
7 Direct Treatment of Special Optimization Problems 191
7.1 Linear Quadratic Optimal Control Problems . 191
7.2 Time Minimal Control Problems. . 199
Exercises ....... . · 216
A Weak Convergence 219
B Reflexivity of Banach Spaces 221
C Hahn-Banach Theorem 223
D Partially Ordered Linear Spaces 227
Bibliography 231
Answers to the Exercises 247
Index 255
Chapter 1
Introduction and Problem
Formulation
In optimization one investigates problems of the determination of a
minimal point of a functional on a nonempty subset of a real linear
space. To be more specific this means: Let X be a real linear space,
let S be a nonempty subset of X, and let f : S - 1R be a given
functional. We ask for the minimal points of f on S. An element
xES is called a minimal point of f on S if
f(x) ~ f(x) for all xES.
The set S is also called constraint set, and the functional f is called
objective functional.
In order to introduce optimization we present various typical op
timization problems from Applied Mathematics. First we discuss a
design problem from structural engineering.
Example 1.1. As a simple example consider the design of a beam
with a rectangular cross-section and a given length I (see Fig. 1.1 and
1.2). The height and the width have to be determined.
Xl X2
The design variables and have to be chosen in an area which
Xl X2
makes sense in practice. A certain stress condition must be satisfied,
i.e. the arising stresses cannot exceed a feasible stress. This leads to
the inequality
(1.1)
2 Chapter 1. Introduction and Problem Formulation
Figure 1.1: Longitudinal section. Figure 1.2: Cross-section.
Moreover, a certain stability of the beam must be guaranteed. In
order to avoid a beam which is too slim we require
(1.2)
and
(1.3)
Finally, the design variables should be nonnegative which means
(1.4)
and
(1.5)
Among all feasible values for Xl and X2 we are interested in those
which lead to a light construction. Instead of the weight we can also
take the volume of the beam given as IXIX2 as a possible criterion
(where we assume that the material is homogeneous). Consequently,
we minimize IXIX2 subject to the constraints (1.1), ... ,(1.5).
With the next example we present a simple optimization problem
from the calculus of variations.
Example 1.2. In the calculus of variations one investigates, for
f
instance, problems of minimizing a functional given as
Jb
f(x) = l(x(t),x(t),t)dt
a
Chapter 1. Introduction and Problem Formulation 3
where -00 < a < b < 00 and I is argumentwise continuous and
x.
continuously differentiable with respect to x and A simple problem
of the calculus of variations is the following: Minimize f subject to
the class of curves from
S := {x E ella, b]1 x(a) = Xl and x(b) = X2}
where Xl and X2 are fixed endpoints.
In control theory there are also many problems which can be for
mulated as optimization problems. A simple problem of this type is
given in the following example.
Example 1.3. On the fixed time interval [0,1] we investigate
the linear system of differential equations
( ) )
!~~~~ = (~ ~) ( ~~g~ + ( ~ ) u(t), t E (0,1)
with the initial condition
With the aid of an appropriate control function u E e[0,1] this dy
namical system should be steered from the given initial state to a
terminal state in the set
In addition to this constraint a control function u minimizing the cost
functional
I
f(u) = ~(u(t))2dt
o
has to be determined.
Finally we discuss a simple problem from approximation theory.
Example 1.4. We consider the problem of the determination of
a linear function which approximates the hyperbolic sine function on