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Proceedings of a Workshop
Bad Honnef August 30 - September ,1 1978
Edited by .R Hettich
IIIIIIIII
galreV-regnirpS
Berlin Heidelberg New kroY 1979
Series Editors
A.V. Baiakrishnan • M. Thoma
Advisory Board
.L D. Davisson • A. G. .J MacFarlane • H. Kwakernaak
Ya. Z. Tsypkin
Editors
Prof. Dr. .R Hettich, Bonn
fSBN 3-540-09479-2 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-38?-094?9-2 Springer-Verlag NewYork Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, re-
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Under § 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to
be determined by agreement with the publisher.
© Springer-Verlag Berlin Heidelberg 1979
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr,
2060/3020-543210
Preface
This book contains the proceedings of a Workshop on Semi-Infinite
Programming which was held in the "Honnefer Haus" of the Elly-HSlter-
hoff-B6cking Stiftung, Bad Honnef, August 30 - September ,I 1978, and
which was organized within the scope of the activities of the Sonder-
forschungsbereich 72 at the University of Bonn, sponsored by the
Deutsche Forschungsgemeinschaft.
Aim of the workshop was to get an impression of the state of the art
in the field of semi-infinite programming with special emphasis on
the practical aspects, i.e. numerical methods and applications. I
thank all participants for their contributions to discussions and to
these proceedings. Furthermore, I thank .I Kreuder for her assistance
in organizing the workshop and the Springer-Verlag for the good
cooperation.
Bonn, March 1979
Rainer Hettich
TABLE OF CONTENTS
THEORY
Glashoff, K.
Duality Theory of Semi-Infinite Programming . . . . . . . . . .
Ben-Tal, A., Teboulle, M, Zowe, J.
Second Order Necessary Optimality Conditions
for Semi-Infinite Programming Problems 17
Kornstaedt, H.-J.
Necessary Conditions of Higher Order for
Semi-Infinite Programming 31
METHODS FOR LINEAR PROBLEMS
Gustafson, S.-A.
On Numerical Analysis in Semi-Infinite Procramming 51
Gribik, P.R°
A Central-Cutting-Plane Algorithm for
Semi-Infinite Programming Problems 66
Roleff, K.
A Stable Multiple Exchange Alqorithm for Linear SIP ...... 83
METHODS FOR NONLINEAR PROBLEMS
Hettich, R., Van Honstede, W.
On Quadratically Convergent Methods for
Semi-Infinite Programming 97
Hettich, R.
A Comparison of some Numerical Methods for
Semi-Infinite Programming . . . . . . . . . . . 112
Van Honstede, W.
An Approximation Method for Semi-Infinite Problems ....... 126
APPLICATIONS
Gustafson, S.-A.
On Semi-Infinite Programming in Numerical Analysis ....... 137
Dahlquist, Go, Golub, G.H., Nash, St.G.
Bounds for the Error in Linear Systems . . . . . . . . . . . . 154
Krabs, W.
One Sided L1-Approximation as a Problem of
Semi-Infinite Linear Programming . . . . . . . . . . . 173
lV
P@pers presented ' at the workshop but not ' received for the Proceedings:
Eckhardt, U.
Semi-lnfinite Systems of Linear Inequalities
Hofmann, K.H.
Methods of Approximation Theory in Semi-Infinite Programming
and Applications
Klostermaier, A.
A Semi-Infinite Linear ProgramMing Procedure
LIST OF PARTICIPANTS
A. BEN-Tal Technion - Israel Institute of Technology -
Computer Science Department
Haifa, Israel
H.G. BOCK Institut .f Angewandte Mathematik, SFB 72,
Universit~t Bonn, Wegelerstr. 6
5300 Bonn
D. BRAESS Institut f~r Mathematik, Ruhruniversit~t
Universit~tsstr. 150, 4630 Bochum
U. ECKHARDT Inst. .f Angew. Math. der Universit~t
Bundesstr. 55, D-2000 Hamburg 13
K. GLASHOFF Inst. .f Angew. Math. der Universit~t
Bundesstr. 55, D-2000 Hamburg 13
G.H. GOLUB Serra House, Serra St. Computer Science Dept.
Stanford University
Stanford, C.A. 94305, USA
P.R. GRIBIK Department of Mathematics,
carnegie-Mellon University
Pittsburgh, P.A. 15213, USA
S.A. GUSTAFSON The Royal Institute of Technology,
Numerical Analysis
S-1OO44 Stockholm 70, Schweden
R. HETTICH Institut .f Angewandte Mathematik, SFB 72,
Universit~t Bonn, Wegelerstr. 6,
5300 Bonn
H. K. HOFF~/qN Freie Universit~t Berlin
Institut fur Mathematik III
Arnimallee 2-6, 1OOO Berlin 33
W. V~N HONSTEDE Institut .f An~ewandte Mathematik, SFB 72,
Universit~t Bonn, Wegelerstr. 6,
5300 Bonn
A. KLOSTERMAIER Freie Universit~t Berlin
Institut fHr Mathematik III
Arnimallee 2-6, 1000 Berlin 33
H.J. KORNS ~H@T DT Freie Universit~t Berlin
Institut fHr Mathematik III
Arnimallee 2-6, 1000 Berlin 33
W. KRABS Fachbereich Mathematik, Arbeitsqruppe 10
Technische Hochschule Darmstadt
Kantplatz ,1 6100 Darmstadt
K. ROLEFF Institut fHr Angewandte Mathematik der
Universit~t,
Bundesstr. 55, 2000 Hamburg 13
IIIV
W.W.E. WETTERLING Technische Hochschule Twente/TW,
Postbus 217, Enschede, Niederlande
P. ZENCKE Institut fGr Angewandte Mathematik, SFB 72,
Universit~t Bonn,
Wegelerstr. 6, 5300 Bonn
J. ZO~.~ Institut fGr Angewandte Mathematik u. Statistik
Universit~t W~rzburg
Am Hubland, 8700 WGrzburg
Introductory remarks on semi-infinite programming
Semi-infinite programming deals with the problem of minimizing a func-
tion f(xl,...,Xn) of variables xi, which, for every y in some set Y,
are required to meet an inequality g(x I .... ,Xn;Y) ~ r(y). In case of
a finite set Y this is a usual optimization problem with a finite
number of inequality constraints on the variables x i. In many appli-
cations infinite sets Y occur quite naturally:
For instance, in controlling air-pollution in some region Y, let i x
be costs incident to provisions for reducin~ the emission of some sub-
stance. Then the total cost f(xl,...x n) = l~IXi is to be minimized
subject to the restriction that in each point Y e y the concentration
g(xl,...Xn;Y) remains below some given standard r(y).
Obviously, there are many problems in numerous applications the con-
straints of which depend on the time or the space coordinates and
therefore may be formulated as semi-infinite problems. Anotherq uestion
is, whether a semi-infinite model should be used in practice instead
of a discretized one, the solution of the latter bein~ close to that
of the first if sufficiently many points are taken into account. In
fact, if it were true that generally a discrete problem could be
treated more easily and efficiently, from a practical point of view
it would not be worth-while to consider semi-infinite programming.
However, there are two aspects which make the problem interesting in
practice:
A model with constraints depending on some parameter y (time, space,
-
etc.) given by means of one inequality g(xl,...,Xn;Y) ~ r(y) rather
than a big number of unrelated inequalities gi(Xl,...,Xn) i ~ r is
preferable with respect to storage requirements as well as simpli-
city of handling. Even in cases where the second description is
given initially (for instance due to measurement restrictions) it
may be advantageous to transform the data by some approximation Pro-
cedure into the first form.
- At the optimal point in general the inequalities are binding (i. e.
equality holds) only in a finite number of points yl,...,yr, with
r ~ n as a rule. In many cases, especially if Y is a subset of
Euclidian space E m, m Z 2, this leads to a much higher efficiency
of continuous methods, because it is sufficient to consider only a
small number of points in each iteration and, moreover, the points
for the next step can be computed very efficiently by using deriva-
tives with respect to y for instance.
Together with the great variety of applications(cf, for instance
Krabs: Optimierung und Approximation, Teubner, 1975, and the last Dart
of this volume) including problems in fields like
- Chebyshev- and Ll-approximation (including constrained problems)
Operations Research (cf. the example given above)
-
- Optimal Control
Computing bounds for solutions of monotonic boundary value problems
-
Free boundary value problems
-
this justifies a growing interest in semi-infinite programming.
Rainer Hettich
DUALITY THEORY OF SEMI-INFINITE
PROGHAMMING
Klaus Glashoff
University of Hamburg
Institute for Applied Mathematics
Bundesstra6e 55
D-2000 Hambur~ 13
West-Germany
O. Summary and c0ntents
Duality theory of semi-infinite programming (SIP) has its roots in the
theory of systems of linear inequalities, in the theory of uniform
approximation of functions, and in the classical theory of moments.
In this paper we present the main duality theorems of SIP in such a
way that the historical development of this theory is shown and the
connections between moment theory and SIP become evident.
Semi-infinite programming is such an important tool in many fields of
applications that there is a need for a simple but complete 'standard'
way to the main duality theorems - this is what we try to give in our
paper,
Our classical 'geometrical' approach to the duality theory of SIP is the
natural generalization of the theory of Linear Programming. It is very
useful for the understanding and development of numerical methods for
SIP (d.f.GLASHOFF and GUSTAFSON (1978), ROLEFF (1979)).
