Nonlinear Optimization in Finite Dimensions: Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects

516 Pages·2001·16.99 MB·English

by  Hubertus Th. Jongen

Nonlinear Optimization in Finite Dimensions
Nonconvex Optimization and Its Applications
Volume 47
Managing Editor:
Panos Pardalos
University of Florida, USA.
Advisory Board:
J.R. Birge
Northwestern University, USA.
Ding-Zhu Du
University of Minnesota, USA.
C. A. Floudas
Princeton University, USA.
J. Mockus
Lithuanian Academy ofS ciences, Lithuania
H. D. Sherali
Virginia Polytechnic Institute and State University, USA.
G. Stavroulakis
Technical University Braunschweig, Germany
The titles published in this series are listed at the end of this volume.
Nonlinear Optimization in
Finite Dimensions
Morse Theory, Chebyshev Approximation,
Transversality, Flows, Parametric Aspects
by
Hubertus Th. Jongen
Aachen University of Technology,
Department of Mathematics,
Aachen, Germany
and
Peter Jonker and Frank Twilt
University of Twente,
Department ofM athematical Sciences,
Enschede, The Netherlands
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4613-4887-0 ISBN 978-1-4615-0017-9 (eBook)
DOI 10.1007/978-1-4615-0017-9
Printed on acid-free paper
Ali Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
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
Preface ix
1 Introduction 1
1.1 Local minima, critical points, (Morse-)index . 1
1.2 Convexity . . . . . . . . . . . . . . . . . . . 4
1.3 Asymptotical aspects ............ . 7
1.4 Local minima in lR 2, connection with planar graphs 9
1.5 One parameter deformations of functions 12
1.6 The basic idea of Morse Theory . . . 14
1.7 The (magic) infinite dimensional cell . 18
2 Morse theory (without constraints) 21
cr
2.1 -manifolds . . 21
2.2 Partition of unity ....... . 22
2.3 Vector fields . . . . . . . . . . . 24
2.4 Some concepts from general topology 28
2.5 Some concepts from matrix theory . . 34
2.6 The functions p, and p . . . . . . . . . 38
cr,
2. 7 Morse theory (without constraints) for functions of class
r~4 .............................. 40
cr,
2. 8 Morse theory (without constraints) for functions of class
1 ~ r ~ 3 . . . . . . 55
2. 9 The attaching of cells . . . . 65
3 Morse theory (with constraints) 83
3.1 C"-Manifolds with Generalized Boundary, r ~ 1 83
3.2 Critical points, vector fields . . . . . . . . . . . . 100
3.3 Lower level sets and(+) Karush-Kuhn-Tucker points 125
v
vi CONTENTS
4 Chebyshev approximation, focal points 155
4.1 Chebyshev approximation: Introduction, reduction theorem . 155
4.2 Functions of maximum type . . . . . . . . . . . 162
4.3 Focal point theory in !Rn . . . . . . . . . . . . . . . 167
4.4 Focal point theory for Chebyshev approximation . . 171
4.5 Chebyshev approximation: constraints, local minima 182
5 Homology, Morse relations 207
5.1 Singular homology 207
5.2 Morse relations . . . . 226
6 Stability of optimization problems 237
6.1 On the dependence of critical points (values) on the problem data237
6.2 Stability of critical points . . . . . . . . . . . 252
6.3 Structural stability of optimization problems . 259
7 Transversality 271
7.1 Introduction, Sard's theorem, regular optimization problems 271
7.2 Transversal intersection of manifolds in !Rn 289
7.3 Transversality of mappings . . . . 293
7.4 Jet-extension, jet-transversality . . 321
7.5 Whitney-regularity, final remarks . 332
8 Gradient Flows 337
8.1 Flows, hyperbolic singularities, linearization . 337
8.2 Variable metric and gradient systems . 354
8.3 Gradient systems in general position . . . . 364
8.4 The graphs 0-1-0 and 0-n-0 . . . . . . . . . 376
8.5 Inequality constraints: the min-max digraph 383
8.6 Reflected gradients . . . . . . . . . . . . . 388
9 Newton flows 393
9.1 Introduction; essential and extraneous singularities 393
9.2 Some global results . . . . . 400
9.3 Gradient Newton flows . . . 406
9.4 Meromorphic Newton flows 427
10 Parametric aspects 443
10.1 Critical manifolds . 443
10.2 One-parameter families of functions (!Rn) 450
10.3 One-parameter families of functions (IHln) 464
CONTENTS Vll
10.4 One-parameter families of constraint sets 472
10.5 Final remarks .............. . 490
References 493
Index 505
List of symbols 509
Preface
At the heart of the topology of global optimization lies Morse Theory: The
study of the behaviour of lower level sets of functions as the level varies.
Roughly speaking, the topology of lower level sets only may change when
passing a level which corresponds to a stationary point (or Karush-Kuhn
Tucker point). We study elements of Morse Theory, both in the unconstrained
and constrained case. Special attention is paid to the degree of differentiabil
ity of the functions under consideration. The reader will become motivated
to discuss the possible shapes and forms of functions that may possibly arise
within a given problem framework. In a separate chapter we show how certain
ideas may be carried over to nonsmooth items, such as problems of Chebyshev
approximation type. We made this choice in order to show that a good under
standing of regular smooth problems may lead to a straightforward treatment
of "just" continuous problems by means of suitable perturbation techniques,
taking a priori nonsmoothness into account. Moreover, we make a focal point
analysis in order to emphasize the difference between inner product norms and,
for example, the maximum norm. Then, specific tools from algebraic topol
ogy, in particular homology theory, are treated in some detail. However, this
development is carried out only as far as it is needed to understand the relation
between critical points of a function on a manifold with structured boundary.
Then, we pay attention to three important subjects in nonlinear optimization.
The first is concerned with the dependence of an optimization problem on the
problem data: stability and sensitivity aspects are considered from both a lo
cal and global point of view. Since stability results are intimately related with
some concepts of transversality, we treat transversality theory in detail. A sec
ond subject consists on one hand of gradients, the gradient differential equation
and its use in the search for several local minima. On the other hand we treat
the underlying differential equation for Newton's method for finding critical
points of a function. Again, transversality plays a crucial role. A slight digres
sion is made towards the search for zeros of meromorphic functions. Finally,
we consider optimization problems depending on parameters (critical sets, bi
furcation of the feasible set, etc.).
IX
X PREFACE
This book originates from the lectures "Nichtkonvexe Optimierung I, II", given
by the first author during his stay at the University of Hamburg, in particular at
the Institute of Applied Mathematics, in the beginning of the eighties.
It is an updated version of the two volumes (same authors):
Nonlinear Optimization in Rn:
I. Morse Theory, Chebyshev Approximation, Vol. 29 (1983).
II. Transversality, Flows, Parametric Aspects, Vol. 32 (1986). Methoden
und Verfahren der mathematischen Physik, Peter Lang Verlag, Frankfurt
a.M., Bern, New York.
The first author would like to thank all members of the Hamburg Institute of
Applied Mathematics (in particular Mrs. W. Bergmann, Klaus Glashoff and
Wolf Hofmann) for their kind hospitality during his stays at the University of
Hamburg. Moreover, he is indebted to Dirk Siersma and Floris Takens for
useful discussions.
We all would like to thank Dini Heres and Jitse Niesen for their careful and
excellent preparation of the T_EX-version of this manuscript.
Bert Jongen, Peter Jonker, Frank Twilt
Aachen (D) and Enschede (NL) April2000.