I
M
Mathematical Programming Society Newsletter
Workshop on
Computational
Integer
Programming
The first "Workshop on Computational Integer
Programming" was held in November 1997 at the
KonradZuseZentrum in Berlin, Germany, and
was organized by Bob Bixby, Martin Gr6etschel
and Alexander Martin (who was mostly in charge of
the tedious and flawless organization) with the
support of the "Gottfried Wilhelm Leibniz
Programm" of the German Science Foundation
(DFG). More than 50 researchers and several
practitioners from at least 10 countries followed a
total of 19 talks and participated in the numerous
discussions on the future of the field. SEE PAGE EIGHT 
Optimization: An essential tool for decision support
Plenary address by
John Dennis at the
XVIth International
Symposium
(an edited version)
Tom Liebling asked me to give this talk. He said that I should give a talk that would
appeal to our members, would motivate our official guests to support mathematical pro
gramming research, and would not bore anyone. Furthermore, I would share the podium
with George Dantzig, the father of us all, as well as with our distinguished prize winners.
So I said, "Sure."
Then, when I started to prepare the talk that would accomplish all these things, I re
membered one of my favorite sayings from Mark Twain: A man is about to be ridden
out of town on a rail, and he is asked if he has any last words. The man says, "If it
weren't for the honor of the thing, I'd sooner walk."
This invitation certainly is an honor for which I thank the organizers and my fellow
MPS members. Furthermore, I welcome a ride on this rail because it is a great opportu
nity to tell our story to our guests and our story is ripe for the telling.
PAGE TWO 
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DECEMBER 1997
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John Dennis' Address
You will notice that many of my examples are
from the aerospace industry, as will be the case in
my research talk later today. I spent last year at
Boeing Research and Technology, and I owe a
great deal to my Boeing colleagues for helping to
crystallize these ideas.
We have distinguished guests among us who
may wonder what Mathematical Programmers do
and why we deserve support. I am including our
spouses in this category.
For our guests, I will give a personal overview
of the kinds of contributions we can make to
computeraided decision making. Key parts of
the required technology are in hand, but much
more research is needed to add crucial functional
ity.
For my colleagues, my message will be that
there are some fascinating research issues we
should address, and that interdisciplinary research
is a fertile field for our talents.
The Mathematical Programming Society has
about 1200 members with only about 35% from
North America. Our symposia are unique among
professional societies in that we always have more
attendees and more lectures than we have mem
bers. This underscores our emphasis on research.
I hasten to add that our name does not mean
that all members of MPS program computers.
The research done by our members spans the
spectrum in a healthy and appropriate way from
purest to most applied. A major thrust of our
members' research is in computer implementa
tion of optimization algorithms. We have had a
hand in all the most successful optimization soft
ware.
My thesis in this talk is based on the following
assumptions: In the modern world, decisions
must be made more and more quickly, and deci
sion side effects must be understood in advance.
Availability of high performance computing has
caused common use of computers as decision aids
(spreadsheets); huge progress in computational
optimization; and a rich infrastructure of simula
tion models. People use computers in their work
without any expectation that it will be necessary
to learn programming or computer organization.
Spreadsheet applications are among the most
widespread uses of computers because they allow
decision makers to ask valuable "whatif' ques
tions and examine the consequences of specific
decisions.
My thesis is that the basic notion of a spread
sheet can be modified in fundamental ways by us
ing optimization research to produce new tools
required by a wide range of the most complex de
cisions.
A major reason we have such an exciting op
portunity is because of the success of some other
areas of applied mathematics. Interdisciplinary
teams have provided us with a rich infrastructure
of simulations of important phenomena.
Decision
Variables
Decision Context Variables 
I
Next, I will show how these simulations are
being used now to support decision makers; then,
I will introduce a more powerful paradigm made
possible by incorporating optimization.
I visited a chemical plant where an analyst had
been trying for two years to improve a process by
manipulating two decision variables. He would
choose values for the variables each evening be
fore leaving work and start the process simulation
program. The next morning he would see if he
had made an improvement. He was a very patient
man, and so was his boss.
I made a gentle suggestion not in front of his
boss that pattern search methods could help. He
did not take my suggestion. It was clear that
thinking about how to set the variables for the
nightly run had become an enjoyable part of his
day.
It is common for decision makers to enjoy
twiddling the decision variables, but they are no
match for our algorithms at that piece of the
problem. I will advocate tools that free the deci
sion maker to concentrate on the essentially hu
man part of the process: using judgement.
Consequence Decision
Variables Merit
Decision
Requirements
I
2.
.
Figure 1. "Cut and Try" Optimization.
DECEMBER 1997
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10 i sP T IMA5
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DECEMBER1997 PAGE
Decision Context Variables 
I
Consequence Decision
Variables Merit
Decision
Requirements
Optimization
Solver
Figure 2. Exploratory Optimization.
Think of a fluid dynamics simulation as a for
mula attached to some cells on a spreadsheet. Fill
in cells to specify design conditions like altitude,
speed of flight, angle of attack for the wing; then
fill in a couple of hundred cells with parameters
that specify a particular wing shape, and after two
hours of supercomputer time, the spreadsheet has
calculated the airflow patterns around the hypo
thetical wing. It writes these in some cells con
nected to a formula to calculate the "drag" associ
ated with the wing. That "drag" number is the
figure of merit for the design. Less drag is better.
Of course, there are extra constraints or require
ments on the design. The aircraft should be able
to fly a certain distance without refueling, etc.
The designer looks at these results and either
alters the problem specifications or decides on
some new design variables or, most likely, does
both.
This approach was fine back when there was a
lot of room for improvement and the simulations
were not very accurate anyway. Wing design and
many other engineering problems are at the point
now where even small improvements are difficult
and time consuming to make by "cut and try"
(see Figure 1).
The point is to help you ask the right questions
by showing you the answer to what you think is
the right question. In simulation based decision
support, the simulations can be complex and ex
Figure 3. Back to the Drawing Board.
... pen
S '" sive to
J . r,. ,,.,n. In
S .'*, .:. irrast, the
figure of merit is
often simple and tentative. Important require
ments may be omitted initially because they are
so obvious to the decision maker that he forgets
to specify them.
We should think of ourselves as providing
tools for a client who is exploring decision vari
able space. The idea is to use optimization of
these tentative formulations as a guide to where it
would be interesting to look. Thus, we need to
make it easy to change problem formulations and
retain useful algorithmic information.
Notice that we have replaced the human deci
sion maker only in the part of the process he
wasn't suited for (see Figure 2). But only the deci
sion maker is qualified to evaluate the decision
context and specifications. A futile and misguided
attempt was made in engineering 20 years ago to
'' '. replace
the human
altogether and do
automatic design. Now
this is scoffingly called "pushbutton" design. I
mean the term, "exploratory optimization" to dis
tance these ideas from automatic design.
Figure 3 shows a wing designed to minimize a
sensible measure of drag using a sophisticated
CFD simulation called a 3D thin layer Navier
Stokes solver. It requires two Cray hours to run
for a given wing in given operating conditions.
Look how wavy the surface is! So, the decision
criteria were not the right ones to use, even if
they do produce an efficient wing, because this
wing would be too expensive to manufacture. So
phisticated designers did not know this would be
the consequence of the design problem formula
tion they used.
How should the problem be changed? Does
one change the way to calculate a single number
that represents how good the particular design is,
or does one add a requirement that bounds the
manufacturing cost of the wing shape chosen?
Both have their place. In this case the designer
did the latter, but his calculation of drag also
evolved, so really he did both.
Decision
Variables
II
0 P I A 5 6
Simulatio
S DECEMBER 1997 PAGE
$tr'
Figure 4.
Figure 4 (above) has been around Boeing so
long that no one knows who produced it. It illus
trates the different disciplines and notions of how
to measure merit in a design.
This speaks to our point that the proper role of
a decision maker is to make these compromises or
"trade offs," not to twiddle design parameters.
In mathematical programming we have a no
tion of efficient frontier for a multiobjective
problem that can help produce tools for the team
making these trade offs. I expect wonderful cross
fertilization from math programmers watching
how real decision makers make these trades.
Technology transfer is a twoway street.
The decision makers will want to know which
design variables most influence which trades.
They will want some notion of how the merit
measures change globally with the most impor
tant variables.
In fact, it is common that the different disci
plines involved in these trade offs will have their
own simulations that must be coupled together to
have a meaningful simulationbased tool.
As I begin my concluding remarks, I have one
more new point to make. When the decision
maker uses our tools to arrive at a decision, that
decision has to be "sold" before it can be imple
mented. One can design a wing (or a transporta
tion system), but there will always be an oversight
group to convince that the right compromises
have been made. In the case of the wing, the boss
will want to be satisfied that the right trade off
between manufacturing cost and aerodynamic
performance has been made, and that the right
trade off has been made between "cruise" perfor
mance of the wing and performance during take
off, climb, and landing.
Tools that document the decision process
would be invaluable not just here but in any open
decision process. There will generally be the need
to convince others that one has made a defensible
compromise between competing concerns.
For the 50 years that linear optimization has
been around, people recognized that integer pro
gramming was a powerful conceptual model. Un
til recently, many important problems could be
formulated but not solved. The fact that they can
be solved now is as much due to algorithmic im
provements we have made as to faster computers.
Of course, we would not have been able to make
these improvements without faster computers, so
it is a chickenandegg situation.
An oil company using a computer to control
online optimization on one piece of equipment at
a single refinery estimated a $5,000,000/year sav
ings.
A Boeing problem involves designing an al
most invisible part of an airplane to have less
drag. They estimate that 0.1% improvement
would save $60,000,000 in fuel for each airplane
over its life; but, the client group would be quite
happy with a tool that did no better than they
can do now, but did it faster. This part has to be
redesigned with every tweak of the airplane de
sign. Time is money.
Airlines have built OR groups to schedule
crews and planes.
Boeing has a large OR group and a sizable
group building high level design tools. Both are
busy and growing. The design tools group I work
with is swamped with requests for help from
manufacturing groups.
Optimization has reached the point in com
merce where there are companies making silly
claims about solving all the client's optimization
problems.
When I speak of customers, I do not mean to
imply that everyone should be interacting directly
with users and doing immediately applied work.
There is an interesting tradition at Boeing, and
probably other companies as well, in which every
one has a customer whose needs one should meet.
The point at Boeing is that not everyone builds
airplanes, and a device of a chain of customer re
lations between oneself and the purchaser of the
product is useful to help keep focus.
I think this interesting mindset could create a
valuable sense of community in mathematical
programming if we were to adopt it. It is always a
good idea to keep in mind who will use our re
search and how they will use it. This leads to a
better understanding of the state of the art and
points to good research questions.
PAGE
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DECEMBER 1997
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Don Hearn
Don Hearn founded OPTIMA in 1980 and was the
editor until the Symposium in Lausanne. During this
time period, OPTIMA has developed into a newsletter of
high quality that reflects the characteristics of the Society.
In the following interview, conducted in Lausanne and
via email, Don tells us about how he got involved in
mathematical programming, his research interests
through the years, and how OPTIMA has developed.
 KAREN AARDAL
OPTIMA: How did you get inter
ested in Mathematical Program
ming?
DH: Manny Bellmore, a young
professor at Johns Hopkins in the
1960s, got me interested in the
field. He left academia for consult
ing many years ago, but at that
time he was working on integer
programming and network algo
rithms. I liked his course in optimi
zation theory, and I became even
more interested when I took non
linear programming from George
Nemhauser and Jack Elzinga.
Then, I had the good fortune to
spend a summer at IBM with
Harlan Mills, who introduced me
to a "transmitter location prob
lem," which led to my dissertation
topic. Mills, by the way, wrote one
of the very early papers on optimal
value functions for nonlinear pro
grams.
OPTIMA: Tell us about your re
search interests and mention
some of your papers of which
you are particularly fond.
DH: The problem Mills posed was
essentially that of finding a circle of
minimum radius to cover a point
set in the plane. Jack Elzinga and I
came up with a geometrical algo
rithm for it and then we studied
various extensions, to nspace, with
weighted points, covering a polyhe
dron, etc. These topics made up
my dissertation and were later pub
lished [14]. While working on the
dissertation, I kept looking for the
mathematical history of the mini
mum circle problem and finally
found a reference in Fritz John's fa
mous paper on optimality condi
tions. It turned out that the prob
lem originated with the British
mathematician J. J. Sylvester, who,
coincidentally, had moved to
Hopkins late in his career. The his
tory included some algorithms that
had been developed in the late
1800s, and I found them in the old
journals that Sylvester had brought
with him. Fortunately, our algo
rithm was new. The problem is still
of interest; I continue to get papers
for review and requests for com
puter codes. For those interested,
my paper with Jim Vijay [6] gives a
survey and synthesis of the algo
rithms up through 1980.
This got me into location theory
for a few years, and there were
other papers on, for example,
multifacility location problems [5].
From location theory I moved to
nonlinear networks, because in the
mid 1970s Harold Kuhn recruited
me to work on the Transportation
Advanced Research Project (TARP)
at Mathematica. (Mathematica, lo
cated in Princeton, was one of the
first OR consulting firms; it was
not related to the software of the
same name.) Our part of the
project was called "network aggre
gation," but I found that what
transportation planners called "ag
gregation" was actually decomposi
tion. I developed several decompo
sition methods based on simplicial
decomposition and Benders de
composition with Russ Barton at
Mathematica and later with Toi
Lawphongpanich and Jose Ventura
[811], both of whom won disserta
tion prizes. In looking at error
bounds for these algorithms, I came
up with the idea of a "gap func
tion" for convex programs which
led to a paper that I like a lot [7].
Those interested in nonlinear net
works might want to read the sur
vey that Mike Florian and I did for
the Handbooks in Operations Re
search and Management Science
[13].
More recently I have collabo
rated on continuous state DP algo
rithms for lotsizing [12] and con
tinuous formulations of the maxi
mum clique problem [14, 15]. This
led to some very effective algo
rithms in both cases, thanks to the
efforts of the two students, Hsin
Der Chen and Luana Gibbons. At
the moment, I am working with
DECEMBER 1997
PAGE 5
1 DEEBR I 9 PAGE6
References
Motakuri Ramana and a student on
congestion toll pricing of traffic
networks. The idea of congestion
tolls has been around a long time,
but we have new results on charac
terizing the set of all tolls that will
force a useroptimal (equilibrium)
solution to yield the system optimal
solution of the untolled problem
[16].
OPTIMA: How did you get in
volved in MPS and, in particular,
what motivated you to start up
OPTIMA?
DH: I joined MPS as a charter
member. Starting a newsletter for
the Society originated with Mike
Held and Phil Wolfe, who were
then chairman of the executive
committee and chairman of MPS,
respectively. George Nemhauser
suggested that I be editor. The con
cern of all of us was that newsletters
tended to come and go, and we
wanted one that had some staying
power while reflecting the quality
emphasis of the Society. I'm happy
to say we have achieved that. An
other key factor was the financial
support from our College of Engi
neering, which paid half the ex
penses for about 10 years.
OPTIMA: Which have been the
key developments of OPTIMA?
DH: Key to the early development
was the involvement of the
College's publications group, par
ticularly Elsa Drake, who has been
the designer for a long time. She is
very creative and we give her a free
hand. The result is that OPTIMA
won a local prize for best newsletter
in its class in 1994.
Also important were the contri
butions of Phil Wolfe, Walter
Murray, Bob Jeroslow and other
leading researchers who wrote nice
expository articles for the early is
sues. That helped define what OP
TIMA was all about.
OPTIMA: What is your vision on
how the field of Mathematical
Programming is developing?
DH: Applications are exploding be
cause computers and algorithms
have evolved to the point that opti
mization models can be used every
where, even in smaller operations.
This demand will continue to jus
tify the research, especially in algo
rithm development.
OPTIMA: How do you think OP
TIMA can continue improving?
DH: I think it will remain low key
and scholarly, like the MPS itself,
and improvements should come
since the council is providing more
support, especially honoraria for as
sociate editors and authors of fea
ture articles. It would be good to
have more news about individuals,
especially as people take sabbaticals
or start up research efforts.
OPTIMA: What did you like most
about working with OPTIMA?
What was the most difficult part
of it?
DH: For me, it has just been the
satisfaction of producing a newslet
ter that members like and working
with staff here who also enjoy do
ing it. The most difficult part used
to be getting feature articles some
issues were published without one
 but now with the efforts you and
the associate editors have been
making, that has improved greatly.
[1] "The Minimum Covering Sphere Problem," Management Science
19, 96104, 1972 (with D.J. Elzinga).
[2] "Geometrical Solutions for Some Minimax Location Problems,"
Transportation Science 6, 379394, 1972 (with D.J. Elzinga).
[3] "A Note on a Minimax Location Problem," Transportation Science
7, 100103, 1973 (with D. J. Elzinga).
[4] "The Minimum Sphere Covering a Convex Polyhedron," Naval
Research Logistics Quarterly 21, 715718, 1974 (with D.J. Elzinga).
[5] "Minimax Multifacility Location with Euclidean Distances,"
Transportation Science 10, 321336, 1976 (with D.J. Elzinga and
W.D. Randolph).
[6] "Efficient Algorithms for a (Weighted) Minimax Location Prob
lem," Operations Research 30, 777795, 1982 (with J. Vijay).
[7] "The Gap Function of a Convex Program," Operations Research
Letters 1, 6771, 1982.
[8] "Simplicial Decomposition of the Asymmetric Traffic Assignment
Problem," Transportation Research 18B, 123133, 1984 (with S.
Lawphongpanich).
[9] "Restricted Simplicial Decomposition: Computation and Exten
sions," Mathematical Programming Study 31, 1987, 99118 (with S.
Lawphongpanich and J. Ventura).
[10] "On the Equivalence of Transfer and Generalized Benders
Decomposition," Transportation Research Vol. 23B, No. 1, 6173,
1989 (with R.R. Barton and S. Lawphongpanich).
[11] "Benders Decomposition for Variational Inequalities," Mathemati
cal Programming 48, 231248, 1990 (with S. Lawphongpanich).
[12] "A Dynamic Programming Algorithm for Dynamic Lot Size Models
with Piecewise Linear Costs," Journal of Global Optimization 4,
397413, 1994 (with H.D. Chen and C. Y. Lee).
[13] "Network Equilibrium Models and Algorithms," Chapter 6 of
Handbooks in Operations Research and Management Science, 8:
Network Routing, M.O. Ball, T.L. Magnanti, C. L. Monma and G.
L. Nemhauser (Eds.), NorthHolland, 1995 (with M. Florian).
[14] "A Continuous Based Heuristic for the Maximum Clique Problem,"
DIMACS Series in Discrete Mathematics and Theoretical Computer
Science, Vol. 28, 103124, 1996 (with L.E. Gibbons and P.
Pardalos).
[15] "Continuous Characterizations of the Maximum Clique Problem,"
Mathematics of Operations Research, 754768, 1997 (with L.E.
Gibbons, P. Pardalos and M. Ramana).
[16] "Congestion Toll Pricing of Traffic Networks," Network Optimiza
tion, SpringerVerlag series Lecture Notes in Economics and
Mathematical Systems, 5171, 1997 (with P. Bergendorff and M.
Ramana).
DECEMBER1997
PAGE 6
Editors
A the Lausanne Symposium, OPTIMA got an almost
completely new editorial staff. Fortunately, Don Hearn, the
founding editor, has agreed to continue giving advice and act
as a link between the new staff and the publisher. Below, each
editor introduces him or herself briefly.
During the coming threeyear period, we will try a slightly
different structure of the board with a Continuous and a
Discrete "area editor" to make it easier to cover the new
developments of our field. We, of course, still have a Book
Review Editor. The main responsibility for the Features
articles will rest with the Editor, but all editors will assist in
attracting feature articles.
We appreciate that the MPS members are very busy, but we
still hope that you will take the time to provide OPTIMA
with material and comments. The addresses and URLs of the
editors can be found on the last page of OPTIMA.
Karen Aardal, Editor
I am working at the Department of
Computer Science at Utrecht Uni
versity as Associate Professor. I ob
tained my Ph.D. degree in 1992
from C.O.R.E., Universite
Catholique de Louvain, Belgium,
under the supervision of Laurence
Wolsey. The topic of my thesis,
and some of my later projects, was
the solution of various facility loca
tion problems using polyhedral
techniques. Since then I also
worked on frequency assignment
and routing problems. My main
current interest is algorithms for
general integer programs. Some of
these problems seem almost hope
less to tackle, even in low dimen
sion, using standard branchand
bound, so new methods are needed.
In 1994 I became Features Editor
of OPTIMA. As Editor, I will con
tinue to be responsible for the Fea
tures Department, and with the
help of the other editors I hope we
will be able to attract a variety of
Sebastian Ceria, Discrete
Optimization Editor
I have an appointment as Associate
Professor in the Management Sci
ence/Operations Management Di
vision of the Columbia Business
School. I was born in Buenos Aires,
Argentina. After obtaining a
Licenciate in Applied Mathematics
at the University of Buenos Aires, I
attended the Graduate School of
Industrial Administration at
Carnegie Mellon University. In
1993 I completed my Ph.D. degree
in Industrial Administration. In my
Ph.D. thesis under the supervision
ofEgon Balas and Girard
Cornudjols, I developed the "lift
andproject method," a disjunctive
programmingbased algorithm for
tackling general mixedinteger pro
gramming problems.
I teach several courses in the
MBA curriculum that are related to
Operations Research and Manage
ment Science. My main research
interest is the solution, both theo
retically and computationally, of
Mary Elizabeth Hribar,
Continuous Optimization
Editor
I am currently a Research Scientist
at Rice University in Houston,
Texas. I was born in Detroit, Michi
gan, and received a Bachelor's de
gree in Mathematics at Albion Col
lege in Albion, Michigan. Inspired
by an internship at Oak Ridge Na
tional Laboratory where I imple
mented algorithms on a hypercube,
I decided to pursue an advanced de
gree in Computer Science at North
western University in Evanston, Illi
nois. I received a Master's degree
and a Ph.D. under the direction of
Jorge Nocedal. As part of my disser
tation work, I developed software
which solves the general nonlinear
programming problem using an in
terior point, trust region method.
Currently, I am working in the
area of multidisciplinary optimiza
tion (MDO). I am investigating
methods as well as developing a so
lution environment for MDO prob
lems. I am also looking forward to
teaching my first course in the
spring.
Robert Weismantel, Book
Review Editor
I was born in Miinchen, Germany
in 1965. After studying mathemat
ics in the years 19841988 at the
University of Augsburg, I moved to
Berlin in 1991 and obtained my
Ph.D. from the Institute of Tech
nology in Berlin in 1992. In the
years 19891991, I was an assistant
of Martin Grotschel at the Univer
sity of Augsburg. Since 1991 I have
been working at the research insti
tute ZIB in Berlin. In 1995 I was
appointed at ZIB as an associate
head of the Department of Optimi
zation. I am currently acting pro
fessor at the University of
Magdeburg.
My area of research is algorith
mic discrete mathematics, in par
ticular, theory and application of
integer programming.
In 1993 I was awarded a Carl
Ramsauer prize for my dissertation.
This year I received the Gerhard
Hess Forschungsforderpreis of the
German Science Foundation
(DFG).
~  
10SP I MWA 56
t! rr., i Fr. 01.tiroc no )d
Karen Aardal, Editor
highquality articles. At the
Lausanne Symposium I became
Council MemberatLarge of MPS.
Before arriving in Utrecht in the
fall of 1995, I held positions at the
University of Essex, Colchester;
Erasmus University, Rotterdam;
and Tilburg University.
Sebastian Ceria, Discrete
Optimization Editor
general discrete optimization prob
lems with a special emphasis on in
teger programming problems. I am
developing new methodologies and
practical implementations of effi
cient algorithms. For the last six
years I have been working on the
"liftandproject method."
In 1994 I implemented a
branchandcut code that uses
Gomory cuts to solve general inte
ger programs. I have also been
working on crewscheduling for
the railways, cutting plane algo
rithms for general integer programs
and a semidefinite programming
approach for the clique problem in
graphs.
There were a wide variety of topics,
including theory of integer program
ming, computational implementation
of efficient algorithms and practical ap
plications of integer programming to
difficult realworld problems.
Dan Bienstockopened the workshop
with a discussion on how to solve diffi
cult network design problems arising
from various problems in the telecom
munications industry. Dan seems to be
able to keep finding relevant practical
problems that lead to very difficult in
teger programming problems. Other
talks that afternoon included the semi
nars by Denis Naddef on the traveling
salesman problem and the semidefinite
cluster, with talks by Michel Goemans,
Franz Rendl and Christoph Helmberg.
The natural question, "Is semidefinite
programming useful for integer pro
gramming?" was raised, but it was very
hard to find a general answer to this
provoking question. The next day,
Laurence Wolsey presented the latest
results with bcopt, a branchandcut
system being developed jointly between
CORE and XPRESS. He once again
pleaded for more mixedinteger pro
gramming data, but in a modelform so
that researchers can understand the con
strains of the problem better (we will
expand on this topic in a future article).
Next, Alexander Martin talked about
mixed integer cutting planes associated
with mixedinteger (feasible) sets, and
Rudiger Schultz showed us some inter
esting applications ofdecomposition for
solving integer programs arising from
Stochastic Programming.
In the afternoon we had a session on
nonlinear approaches to integer pro
gramming problems, with talks by Kurt
Anstreicher and John Mitchel, and a
final cluster with speakers from indus
try. Ulrich Lauter from Siemens demon
strated how preprocessing can consider
ably help in speeding up computations
in the calculations ofshortest paths (with
applications to traffic and vehicle guid
ance systems), andJeanFrancois Puget,
from ILOG, described to us the world
of Constrained Logic Programming and
its relation to general integer program
ming.
The last day also included many in
teresting talks, like Ed Rothberg's de
scription of mathematical programming
from a computer scientist's viewpoint,
Robert Weismantel's primal approach
to integer programming, and Bernd
Bank's description of real equation solv
ing and integer polynomial optimiza
tion. Finally, Lex Schrijver presented
some difficult integer programs arising
from timetabling in the Dutch railways;
I presented the latest computational
results with the liftandproject method,
and Thomas Wintler an application in
dispatching vehicles.
The social program included a won
derful party at Martin Groetschel's
house near Berlin. His wife delighted us
with her cooking; but, nevertheless, we
managed to generate some heated dis
cussions on as varied topics as the trav
eling salesman problem, the future of
integer programming, and the difference
in the educational systems (especially for
children) between Europe and the U.S.
On Sunday morning, we visited "Sans
Souci" (no problem), the wonderful
summer castle of Friedrich the Great,
where we were shown, among other
amenities, various styles ofRococo deco
rations. On Monday night, some of us
had the pleasure of finding an Argentin
ian restaurant, not recommended for
vegetarians. For the last evening the or
ganizers prepared a banquet at the "Cafe
of 100 beers" where, unfortunately, the
drinks were not included.
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PAGE 9
SAlgorithms and Experiments (ALEX98) Building Bridges
Between Theory and Applications
Trento, Italy
February 911, 1998
) Symposium on Combinatorial Optimization, CO98
April 1517,1998
Brussels, Belgium
Email: bfortz@ulb.ac.be
) Internation Conference on Interval Methods and Their Application
in Global Optimization (INTERVAL'98)
April 2023, 1998
Nanjing, China
URL: http://cs.utep.edu/intervalcomp/china.html
) INFORMS National Meeting
April 2629, 1998
Montr6al, Quebec, Canada
U RL: http://www.informs.org/Conf/Montreal98/
) Sixth Conference on Integer Programming and Combinatorial Op
timization, IPCO '98
June 2224, 1998
Houston, TX
URL: http://www.hpc.uh.edu/~ipco98
) INFORMS International Meeting
June 28July 1, 1998
Tel Aviv, Israel
U RL: http://www.informs.org/Conf/TelAviv98/
) Fourth International Conference on Optimization
July 13, 1998
Perth, Australia
URL: http://www.cs.curtin.edu.au/maths/icota98
) Optimization 98
July 2022, 1998
Coimbra, Portugal
URL: http://www.it.uc.pt/~opti98
) ICM98
Berlin, Germany
August 1827, 1998
URL: http://elib.zib.de/ICM98
) Second WORKSHOP ON ALGORITHM ENGINEERING, W A E' 98
August 1921, 1998
Saarbruecken, Germany
URL: http://www.mpisb.mpg.de/~wae98/
) INFORMS National Meeting
October 2528, 1998
Seattle, WA
) International Conference on Nonlinear Programming and Varia
tional Inequalities
Hong Kong
December 1518, 1998
) Sixth SIAM Conference on Optimization
May 1012, 1999
Atlanta, GA
) 19th IFIP TC7 Conference on System Modelling and Optimization
July 1216, 1999
Cambridge, England
Email: tc7con@amtp.cam.ac.uk
Ninth Annual
ACMSIAM Symposium on
Discrete Algorithms
January 2527, 1998
San Francisco, California
URL: http://www.siam.org/
meetings/da98/da98home.htm
The plenary talks will given by:
Laszl6 Lovisz, Yale University:
"Algorithms and Geometric
Representations of Graphs;"
Arjen K. Lenstra, Citibank:
"Factoring: Facts and Fables;"
Thomas L. Magnanti, MIT:
"Four Decades of Optimal
Network Design"
From the Nordic Section
The 5th meeting of the Nordic Sec
tion of the Mathematical Program
ming Society will take place in
Molde, Norway, May 910, 1998.
It is open to all Nordic members of
MPS and, of course, to all others
with similar interests.
For details, please look at the con
ference home page (http://
www.himolde.no/arnel/
mpsnordic98). Also, the fourth issue
of our newsletter, covering the time
from July 1, 1995, to December 31,
1996, is in preparation. Once it is
completed, it can be accessed via the
Nordic MPS home page (http://
www.mai.liu.se/Opt/MPS/
index.html).
DECEMBER1997
International Congress of Mathematicians (ICM'98)
August 1827, 1998
Berlin, Germany
URL: http://elib.zib.de/ICM98
Plenary talks will be given by:
JeanMichel Bismut: Differential Geometry and Global Analysis
Christopher Deninger: Arithmetic Algebraic Geometry, LFunctions of Motives
Persi Diaconis: Statistics, Probability, Algebraic Combinatorics
Giovanni Gallavotti: Dynamical Systems, Statistical Mechanics, Probability
Wolfgang Hackbusch: Numerical Analysis, Scientific Computing
Helmut H. W. Hofer: Global Analysis, Dynamical Systems
Ehud Hrushovski: Logic
I. G. Macdonald: Lie Groups, Algebraic Combinatorics
Stiphane Mallat: Applied Mathematics, Signal Processing
Dusa McDuff: Symplectic Topology
Tetsuji Miwa: Integrable Systems, Infinite Dimensional Algebras
Jiirgen Moser: Dynamical Systems, Partial Differential Equations
George C. Papanicolaou: Applied Mathematics, Probability
Gilles Pisier: Functional Analysis
Peter Sarnak: Number Theory
Peter W. Shor: Computer Science
Karl Sigmund: Mathematical Ecology, Evolutionary Game Theory
Michel Talagrand: Probability, Statistical Mechanics, Functional Analysis, Measure Theory
Cumrun Vafa: String Theory, Quantum Field Theory and Quantum Gravity
PAGE 10
Marcelo Viana: Dynamical Systems, Ergodic Theory
Vladimir Voevodsky: Algebraic Cycles and Motives
CALL FOR PRESENTATIONS
Mathematical Software Session
International Congress of Math
ematicians 1998 (ICM'98)
Berlin, Germany
August 1827, 1998
The International Congresses of
Mathematicians, taking place
roughly every four years since 1897,
belong to the most important math
ematical events in the world. One
distinguishing feature, among oth
ers, is the award of the Fields Med
als and the Nevanlinna Prize (the
"mathematical Nobel Prizes") dur
ing the opening ceremony.
The ICM'98 will take place at the
Technische University in Berlin,
Germany, from August 18 to 27,
1998. In addition to the scientific
program (with plenary and invited
speakers chosen by the IMUap
pointed ICM'98 Program Commit
tee), a "Section of Special Activities"
is planned. One of these activities
will be a session on mathematical
software, to be held on two after
noons during the congress. The fo
cus of this session will be the presen
tation of a broad spectrum of math
ematical software systems ranging
from general purpose systems to spe
cialized systems, e.g., systems from
numerical analysis, computer alge
bra, optimization, mathematical vi
sualization, or mathematical educa
tion. The presentations should in
clude typical applications.
This session is planned to attract
a broad audience including ICM at
tendees, students and teachers, with
a special interest in mathematical
software. The session will take place
at the conference site.
Program Committee
A program committee for this par
ticular session has been appointed. It
will be chaired by Johannes
Grabmeier of IBM Germany, who is
speaker of the special interest group
for computer algebra of DMV (Ger
man Mathematical Society),
GAMM and GI.
Organization
Winfried Neun
KonradZuseZentrum fir
Informationstechnik
Berlin, Germany
Email: neun@zib.de
0 DEEME 199 PAG M1156
Call for Presentations
The systems to be presented should
meet the highest standards with re
spect to mathematical content.
Mathematical originality, new solu
tions, or uncommon fields of appli
cation will be highly appreciated.
The technical quality in design and
implementation is also an important
issue. Submissions for the Session
on Mathematical Software are en
couraged from all fields of math
ematics where software systems are
used. Systems which are available
free of charge (e.g., public domain)
are especially desired and will be
given preference during the selection
process.
There will be a software exhibi
tion and a book fair in connection
with ICM'98 too. This may be
more suitable for the demands of
vendors of commercial software sys
tems. Please contact the chairman of
the local arrangements committee,
Professor Rolf H. Moehring (email:
moehring@math.tuberlin.de), for
details about the exhibition. Talks
are also sought in which various
commercial packages are compared
from an independent viewpoint,
pointing out particular strengths
and weaknesses of the systems.
The program committee, a group
of internationally renowned math
ematicians and experts on math
ematical software, will evaluate the
entries and select a number of con
tributions according to quality and
thematic balance. To aid the com
mittee in judging the submissions,
contributors should include material
(either in paper form or an elec
tronically readable format, e.g., a
URL) which explains to the com
mittee the mathematical back
ground of the systems, the fields of
application and the software design
and techniques.
Submissions
Submissions should be sent, prefer
ably by electronic mail, to:
ICM'98 Session on Math. Soft
ware, c/o W. Neun, KonradZuse
Zentrum (ZIB), Takustr. 7,
D14195 Berlin, Germany
EMAIL: neun@zib.de
and must be received by March 1,
1998. Submissions that arrive after
this deadline will not be considered.
Some guidelines that will help the
program committee to review the
submissions are:
1. For a first glance a URL is usually
very helpful.
2. For each system it should be very
clear where information about the
mathematical content can be found.
This is usually not trivial if the sub
mission consists, say, of
uncommented pictures.
3. The special features and the tar
geted user community should be
identified.
4. The availability of the software
and the terms and conditions for
distribution should be easily acces
sible.
The scheduled length of the pre
sentations including discussion is 30
minutes. This allows the organizers
to put approximately 12 lectures
into the time available for the ses
sion. Financial support for presenta
tions is not available. Presenters are
required to register for ICM'98.
Upon Acceptance
Contributors will be notified of the
acceptance or rejection of their sub
mission by the program committee.
Based on this selection, the organiz
ing committee will arrange a time
table in cooperation with the pre
senters.
Requests for special equipment
needed for presentations can be dis
cussed at this time, but the resources
will be limited. Therefore, it is not
advisable to rely on any special hard
ware and software support from the
session organizers.
It is the contributor's responsibil
ity to secure any necessary permis
sions and licenses for any material
contained in the presentation or
handouts. The organizers of
ICM'98 would appreciate it if the
commercial attitude of the system
providers were modest.
Deadlines
Submission of Presentations:
March 1, 1998
Notification of Acceptance:
April 1, 1998
MARTIN GROETSCHEL, PRESIDENT OF THE
ICM'98 ORGANIZING COMMITTEE
FIRST ANNOUNCEMENT AND
CALL FOR PAPERS
International Conference on
Operations Research (OR98)
31 August 3 September 1998
ETH Zurich, Switzerland
Uptodate information on the con
ference can be found at the OR 98
web site (URL: http://
www.or98.ethz.ch). The Program
Committee invites papers of presen
tations in all areas of Operations Re
search. The conference will give par
ticular attention to the following
topics followed by chairperson of
each section:
1. Mathematical Optimization
A) Continuous (Feichtinger, Horst,
Vial)
B) Discrete (Burkard, Hertz,
Reinelt)
2. Stochastic Modelling, Optimiza
tion and Simulation (Rieder,
Stadlober)
3. Econometrics and Statistics
(Deistler, Garbers, Schmitz)
4. Mathematical Economics, Game
Theory and Decision Theory
(Brachinger, Ulrike Leopold
Wildburger)
5. Banking and Finance (Buehler,
Frauendorfer, Zechner)
6. Operations and Production
Management (Guenther,
Jammernegg, Tempelmeier)
7. Energy and Ecology (Haurie,
Kalliauer)
8. Telecommunication (Martine
Labbe, Mechthild Stoer)
9. Logistics and Transportation
(Domschke, Fleischmann, Staehly)
10. Fuzzy Systems and Neural
Networks (Rommelfanger, Brigitte
Werners)
Conference Languages:
English and German
Deadlines:
Deadline for submission of extended
abstracts: 15 January 1998
Notice of acceptance: 1 April 1998
Submission of Papers:
Authors wishing to contribute pa
pers are requested to submit
a) full name(s), affiliations) and
addresses) (including email) of the
authorss.
b) an extended abstract of two
pages (indicating intended section).
The extended abstract should be
submitted either as hard copy (four
copies) or by email as ASCII/TeX/
LaTeXfile to:
Institute fuer Operations Research
der Universitaet Zuerich
OR 98
Moussonstrasse 15
CH8044 Zuerich
Email: kall@ior.unizh.ch
Extended abstracts will be refereed
and accepted papers will be subdi
vided for
a) presentation in a session (30
minutes including discussion)
b) presentation within special
poster sessions."
About 50 full papers will be selected
for publication in the Proceedings of
the Conference.
Conference Chairman: H.J. Luethi
Chairman Program Committee:
P. Kall
Plenary Speakers:
M. Groetschel, Berlin
Th. L. Magnanti, MIT
F. J. Radermacher, Ulm
F. Delbaen, ETH Zurich
F. Jensen, Aalborg
PAGE 11
DECEMBER 1997
PAGE 12
 W 4 .. 'r .g w
p.r
4 = o ;' MMF
Linear Programming:
Foundations and Extensions
by Robert J. Vanderbei
Kluwer Academic Publishers
Boston, 1996
ISBN 0792398041
Summary
This book presents a thoroughly modern treatment
of linear programming that achieves a healthybalance
between theory, implementation, computation, and
between the simplex method and interiorpoint meth
ods. Its most novel feature is that it is written in a
delightful and refreshing conversational manner that
bespeaks the author's teaching style and relaxed wit.
It is a pleasure to read. Students will find the book to
be friendly and engaging, while professors will find
in the book a wealth of teaching material, nicely or
ganized and packaged for classroom use. The book is
also meant to be used in conjunction with a public
available website that contains software for various al
gorithms, additional exercises, and demos of algo
rithms.
The Need for New Linear Programming
Textbooks
The world of linear programming has changed
dramatically in the last 10 years. For one thing, the
incredible changes in computer technology have made
it easy to solve truly huge LPs, and routine LP prob
lems solve in fractions of a second, even on a personal
computer. As a result, the study of linear program
ming algorithms is of less interest to the casual stu
dent. (In a similar vein, we usually do not teach stu
dents how to efficiently compute square roots; we
simply presume they can press the right buttons on
their calculator.) On the other hand, because we can
now solve truly gigantic linear programs, issues of
computer implementation, numerical stability, and
software architecture, etc., are as important for the
serious optimizer as is, say, duality theory. Further
more, the development and recognition of the impor
tance of interior point methods has changed the land
scape of linear programming significantly, so that lin
ear programming is no longer synonymous with the
simplex method, and a modern treatment of LP must
also present an indepth treatment of the most impor
tant interior point methods.
Vanderbei's Book Is Thoroughly Modern
Vanderbei's book is completely uptodate. Aside
from a nice treatment of the simplex method, it also
contains a very uptodate treatment of interior point
methods, including the homogeneous selfdual formu
lation and algorithm (which might soon become the
dominant algorithm in practice and theory). It con
tains extensive material on issues of implementation
of both the simplex algorithm and interior point al
gorithms. A politician might call it a "book for the 21st
century.
Vanderbei's Book Has Many Novel Features
This book is quite different from most other text
books on LP in a number of important ways. For start
ers, the "standard form" of linear program in the book
is the symmetric form of the problem (max cx Ax
b, t 0), as opposed to the usual form (min cx Ax
Sb, x t 0). This difference allows for an easier treat
ment of duality, and allows one to see the geometry of
linear programming more easily as well. The symmet
ric form also makes it easier to set up the homogeneous
selfdual interior point algorithm. However, this form
has the drawback that discussions of bases, basic fea
sible solutions, and some of the mechanics of the sim
plex method are all a bit more awkward. (The book uses
 5
0 DEEME 199 PAG 1356
the language of"dictionaries" to describe the essential
information in a simplex method iteration.) The book
has more of a focus on engineering applications than
does the more typical LP textbook (which tends to rely
on business problems). For example, there is a nice
chapter on optimization of engineering structures such
as trusses. The book gives a very broad treatment of
interior point methods, including several topics that
are not usually found in textbooks, such as the homo
geneous selfdual formulation and algorithm, qua
dratic programming via interior point methods, and
general convex optimization via interior point meth
ods.
These novel features are good in that the author has
clearly tried to be innovative and to build an LP text
from the ground up, without regard for past texts.
Some Nice Features
There are some particularly nice features in the book.
The book contains a muchsimplified variant of the
KleeMinty polytope that allows for a more straight
forward proof that the simplex method can visit expo
nentially many extreme points. In addition to proving
strong duality, the book also presents Tucker's strict
complementarity theorem, which has become impor
tant in the new view of sensitivity analysis, optimal
partitions, and interior point methods. The book also
contains a nice treatment ofthe steepest edge pivot rule,
which has recently emerged as an important compo
nent in speeding up the performance of the simplex al
gorithm. In the treatment of interior point methods,
the author spends very little time on polynomial time
bounds and guarantees (as a theorist, I like to see this
material), instead adding value by discussing impor
tant computational and implemention issues, includ
ing ordering heuristics, strategies for solving the KKT
system byNewton's method, etc. The booksometimes
has an engineer's feel for the proofs, which is good for
students but is a bit frustrating to hardcore math types
such as myself. There are many instances where the
"proof" is just a proof via an example. This is consis
tent with the conversational and informal style of the
text, and this informalityspills over into the mathemat
ics on occasion.
This Book Has Style
As mentioned earlier, the book has a wonderfully
appealing conversational style. While the author does
not purposely go out of his way to be cute and corny,
he succeeds in leaving the reader grinning with his
humor. There are some passages that are downright
funny, but the style succeeds mostly by default. One
section on the issue of modeling the anchoring of truss
design problems is called "Anchors Away." The sub
section on updating factorizations to reduce fillin is
aptly called "Shrinking the Bump," and there is the hint
of a racy discussion of an application of Konig's Theo
rem involving boys and girls that the curious reader
might enjoy.
Overall, I greatly enjoyed reviewing this book, and
I highly recommend it as a textbook for an advanced
undergraduate or master's level course in linear pro
gramming, particularly for courses in an engineering
environment. In addition, it also is a good reference
book for interior point methods as well as for imple
mentation and computational aspects of linear pro
gramming. This is an excellent new book.
 ROBERT M. FREUND, MIT
Geometry of Cuts and Metrics
by M. M. Deza and M. Laurent
SpringerVerlag
Berlin, 1997
ISBN 354061611X
This book is definitely a milestone in the literature of
integer programming and combinatorial optimization.
It draws from the interdisciplinarity of these fields as
it gathers methods and results from polytope theory,
geometry of numbers, probability theory, design and
graph theory around two objects, cuts and metrics.
Deza and Laurent do not only write but with their
work actually prove the correctness of the statement,
"Research on cuts and metrics profits greatly from the
variety of subjects where the problems arise. Observa
tions made in different areas by independent authors
turn out to be equivalent, facts are not isolated and
views from different perspectives provide new inter
pretations, connections and insights."
Every researcher in integer programming and com
binatorial optimization will find his fields of research
and interest represented in this book. This is one, but
not the only aspect that makes the book unique.
The book has five parts, each of which is fairly self
contained.
Part 1 treats relations between cuts and metrics.
Every generator of the cut cone (the generators of the
cut cone are all incidence vectors of cuts of a given
graph) defines asemimetric, i.e., asymmetric function
fon the pairs ofvertices, satisfying the triangle inequali
ties andf(i,i) = 0 for all vertices i. (Of course, not every
semimetric is a cut.) Of major interest in this part are
the characterizations of cuts by means of measure
theory and i embeddability including, in particular,
the following theorem: a semimetric belongs to the cut
cone if and only if it is isometrically i embeddable.
Part 2 studies socalled hypermetric spaces.
Hypermetric inequalities are inequalities of the form
Sbix,, < 0 with b Z Z", b, = 1.
One can prove that every semimetric in the cut cone
satisfies the family of hypermetric inequalities, yet not
every semimetric satisfying the family of hypermetric
inequalities is a member of the cut cone. Hypermetric
spaces, the hypermeytric cone and the connections to
point lattices and Delauny polytopes are the central is
sue in Part 2.
Part 3 is devoted to investigations of graphs whose
path metric is i,embeddable or hypercube
embeddable. It is shown in the book that a graph is ,
embeddable if and only if a nonnegative multiple of
its path metric is hypercubeembeddable. Of particu
lar beauty is the fact that ,embeddable graphs can be
recognized in polynomial time.
Part 3 is directly connected to Part4 ofthe bookthat
treats questions of the form: given a distance function
on a finite number of points, decide whether this dis
tance function is hypercubeembeddable. There are
some distance functions for which this problem is easy
to solve. For others, the decision about hypercube
embeddability is NPhard. For various other classes of
metrics, there are conditions available that can be tested
in polynomial time and ensure hypercube
embeddability.
Part 5 deals with the geometry of the cut cone and
the cut polytope. It surveys extensively polyhedral
material, including the fundamental facetmanipulat
ing operations such as switching, the family of triangle
inequalities and more general hypermetric inequalities.
Very appealing is the detour to cycle polyhedra of
binary matroids and the questions that the authors
discuss in this context about linear relaxations by the
triangle inequalities and Hilbert bases. Also very inter
esting are the discussions about the completion prob
lem and the connections to geometric questions such
as the partitioning of a set in the ndimensional space
into n+1 sets of smaller diameter.
The book is very nicely written, although it is quite
dense and requires a lot of knowledge to understand
the details. Startingwith the important definitions that
it resorts to, each of the chapters is selfcontained. I
found it helpful to read Chapter 1, the outline of the
PAGE 13
DECEMBER 1997
1 DCME I 9 P 146
book, in the beginning. It really helps in getting
through the advanced parts. The book is also very well
structured. With knowledge about the relevant terms,
one can enjoy special subsections without being en
tirely familiar with the rest of the chapter. This makes
it not only an interesting research book but even a dic
tionary. The material is uptodate, and there are vari
ous sections that contain enough open questions for
a couple of Ph.D. theses.
In my opinion, the books a beautiful piece ofwork.
The longer one works with it, the more beautiful it
becomes.
 ROBERT WEISMANTEL, BERLIN
Lectures on Polytopes
by G. Ziegler
SpringerVerlag
Berlin, 1995
ISBN 0387943293
During the last 30 years, the theory of (convex)
polytopes has drawn growing attention. As the convex
hull of finite point sets in euclidean spaces, polytopes
are very natural objects; therefore, it is not surprising
that they have a great number of applications in such
diverse mathematical areas as Linear and Combinato
rial Optimization, Functional Analysis, Algebraic
Geometry and Semialgebraic Geometry. This book
does not concentrate so much on these fields of appli
cations as on the theory of polytopes itself, which has
by now obtained an enormous scope and depth. The
reader, however, will still find numerous references to
related areas. Avery motivatingand exampleoriented
introduction is presented in Chapter 0, which gives the
reader a first impression of the interesting subject and
introduces the basic terminology at the same time. This
chapter explains in detail the different ways of repre
senting polytopes which are important in Computa
tional Geometry and Optimization.
Chapters 1 and 2 present the foundations of con
vex geometry and the most important facts about face
lattices ofpolytopes. Chapter 3 studies the edge graphs
of polytopes and extensively discusses the newest re
sults on the diameter of such graphs. These are of
particular importance for Linear Optimization as they
reflect the worst possible behavior of best possible edge
following LPalgorithms. This chapter also includes
Kalai's extremely elegant proofofthe fact that the edge
graph of a simple polytope already determines its
complete face lattice. The edgegraphs of 3dimen
sional polytopes are characterized by planarity and 3
connectedness. This is the famous theorem ofSteinitz
which is the basis for many further results about 3di
mensional polytopes. A new proof of this theorem is
presented in Chapter 4. This proof is based on agraph
reduction technique due to Truemper, and it avoids
some of the complications of earlier proofs.
The two following chapters are devoted to realizabil
ity problems for higherdimensional polytopes. In
analogy to the theorem of Steinitz, the question is
whether cellcomplexes with given geometric or com
binatorial properties are isomorphic to the facelattice
ofpolytopes. For such problems, oriented matroids and
Galediagrams have proven very useful. As an appli
cation of this theory, the reader is presented with a 5
dimensional polytope which has a 2dimensional face
whose shape cannot be arbitrarily preassigned. Mean
while, RichterGebert have constructed a 4polytope
with this property, thereby solving a problem posed in
the book. The part of the theory of oriented matroids
that is needed in polytope theory is described verywell.
In Chapter 7, this theory is studied in depth and is
applied to zonotopes and other objects related to
polytopes like arrangements ofhyperplanes and tilings
of space.
Chapter 8 introduces the spectacular results on the
numbers of faces of polytopes, the "UpperBound
Theorem" and the "gTheorem." The concept of
shellability and the related hvectors, which can be
defined by it, are essential for these results. Both are
explained in detail and applied to the first construc
tion ofa polytope having a partial shelling which can
not be extended to a complete shelling.
The last chapter studies fiberpolytopes which are
important for Grobnerbases. As an application, the
author presents a construction of the permuto
associahedron. The book ends with an extensive list of
references. All chapters contain a useful collection of
problems, beginning with "warmups" and ending
with important open problems. The book excels be
cause of its lucid presentation, which is supported by
many helpful illustrations. The careful descriptions of
the results provide an excellent motivation for students
and make the book a valuable basis for a course on
polytopes.
The publication of the book has obviously led to the
solution of some of the open problems described in it.
T I .... ..i., L l.. . .. i .. .... ,I,.l....
established a web site (http://winnie.math.tu
berlin.de/ziegler) which, in addition to the correction
of minor errors, has all the information on these inter
esting new developments. These updates will be con
tinued in a revised edition to appear soon.
As the book contains all important techniques of
polytope theory and also many new results, it is most
useful both for the expert and for other mathematicians
and computer scientists who use polytopes in one of
the application areas mentioned. I very much enjoyed
reading it.
 PETER KLEINSCHMIDT. PASSAU
PAGE 14
DECEMBER1997
PAGE 15
L(
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Deadline for the next
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February 28, 1998
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