Introduction to the Theory of Nonlinear Optimization

249 Pages·1994·7.209 MB·English

by  Prof. Dr. Johannes Jahn (auth.)

Johannes Jahn
Introduction
to the Theory
of Nonlinear
Optimization
With 30 Figures
Springer-Verlag
Berlin Heidelberg GmbH
Prof. Dr. Johannes Jahn
Universităt Erlangen-Ni.imberg
Institut fUr Angewandte Mathematik
MartensstraBe 3
D-91058 Erlangen
ISBN 978-3-662-02987-9 ISBN 978-3-662-02985-5 (eBook)
DOI 10.1007/978-3-662-02985-5
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© Springer-Verlag Berlin Heidelberg 1994
Originally published by Springer-Verlag Berlin Heidelberg New York Tokyo in 1994
Softcover reprint of the hardcover 1s t edition 1994
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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 condition 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.
I am grateful to S. GeuB, S. Gmeiner, 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 Jahn
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
Vlll Contents
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
Index 247
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 --+ lR be a given
functional. We ask for the minimal points of f on S. An element
x E S is called a minimal point of f on S if
f(x) :S: 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 l (see Fig. 1.1 and
1.2). The height x and the width x have to be determined.
1 2
The design variables x and x have to be chosen in an area which
1 2
makes sense in practice. A certain stress condition must be satisfied,
i.e. the ari"sing 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 x and x we are interested in those
1 2
which lead to a light construction. Instead of the weight we can also
take the volume of the beam given as lx x as a possible criterion
1 2
(where we assume that the material is homogeneous). Consequently,
we minimize lx x subject to the constraints (1.1), ... ,(1.5).
1 2
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
instance, problems of minimizing a functional f given as
Jb
f(x) = l(x(t), x(t), t)dt
a
Chapter 1. Introduction and Problem Formulation 3
where -oo < a < b < oo and 1 is argumentwise continuous and
continuously differentiable with respect to x and i:. A simple problem
of the calculus of variations is the following: Minimize f subject to
the class of curves from
S := {x E C1[a, b] I x(a) = Xt and x(b) = x2}
where x and x are fixed endpoints.
1 2
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
( ~1(t) ) = ( 0 1 ) ( Xt(t) ) + ( 0 ) u(t) t E (O, 1)
x2(t) 0 0 x2(t) 1 '
with the initial condition
-2J2).
( Xt(O) ) = ( sJ2
x2(0)
With the aid of an appropriate control function u E C[O, 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
1
f(u) = J(u(t))2dt
0
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
4 Chapter 1. Introduction and Problem Formulation
4
f3 =sinh a:
f3 = xa
3
2
x 1.600233
:::::J
1
0 1 2
Figure 1.3: Best approximation of sinh on [0, 2).
the interval [0, 2) with respect to the maximum norm in a best way
(see Fig. 1.3). So, we minimize
max la:x -sinh a:l.
a-E(0,2)
This optimization problem can also be written as
min >.
subject to the constraints
>. = maxa-E(0,2Jia:x - sinh a:l
(X' ). ) E JR2•
The preceding problem is equivalent to the following optimization
problem which has infinitely many constraints:
min >.
subject to the constraints
ax - sinh a: ~ ). } for all a: E [0, 2)
ax- sinh a: ? ->.
( x, ). ) E JR2.