Title: Optima
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Permanent Link: http://ufdc.ufl.edu/UF00090046/00056
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Title: Optima
Series Title: Optima
Physical Description: Serial
Language: English
Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: December 1997
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Bibliographic ID: UF00090046
Volume ID: VID00056
Source Institution: University of Florida
Holding Location: University of Florida
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Mathematical Programming Society Newsletter

Workshop on

The first "Workshop on Computational Integer
Programming" was held in November 1997 at the
Konrad-Zuse-Zentrum 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
(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.

interview 5 conference notes 9




reviews 12 gallimaufry 15

A In


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
computer-aided decision making. Key parts of
the required technology are in hand, but much
more research is needed to add crucial functional-
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-

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 "what-if' ques-
tions and examine the consequences of specific
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-
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 Context Variables -


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
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



Figure 1. "Cut and Try" Optimization.



10 i sP T IMA5



Decision Context Variables -


Consequence Decision
Variables Merit



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
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 "push-button" 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.



0 P I A 5 6




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 two-way 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 simulation-based 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 chicken-and-egg 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-
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
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.


10 P I M A 5




0 DEEME 199 PAG M556

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 e-mail, Don tells us about how he got involved in

mathematical programming, his research interests

through the years, and how OPTIMA has developed.


OPTIMA: How did you get inter-
ested in Mathematical Program-
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-
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 n-space, with
weighted points, covering a polyhe-

dron, etc. These topics made up
my dissertation and were later pub-
lished [1-4]. 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
[8-11], 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
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





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 user-optimal (equilibrium)
solution to yield the system optimal
solution of the untolled problem
OPTIMA: How did you get in-
volved in MPS and, in particular,
what motivated you to start up
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, 96-104, 1972 (with D.J. Elzinga).
[2] "Geometrical Solutions for Some Minimax Location Problems,"
Transportation Science 6, 379-394, 1972 (with D.J. Elzinga).
[3] "A Note on a Minimax Location Problem," Transportation Science
7, 100-103, 1973 (with D. J. Elzinga).
[4] "The Minimum Sphere Covering a Convex Polyhedron," Naval
Research Logistics Quarterly 21, 715-718, 1974 (with D.J. Elzinga).
[5] "Minimax Multifacility Location with Euclidean Distances,"
Transportation Science 10, 321-336, 1976 (with D.J. Elzinga and
W.D. Randolph).
[6] "Efficient Algorithms for a (Weighted) Minimax Location Prob-
lem," Operations Research 30, 777-795, 1982 (with J. Vijay).
[7] "The Gap Function of a Convex Program," Operations Research
Letters 1, 67-71, 1982.
[8] "Simplicial Decomposition of the Asymmetric Traffic Assignment
Problem," Transportation Research 18B, 123-133, 1984 (with S.
[9] "Restricted Simplicial Decomposition: Computation and Exten-
sions," Mathematical Programming Study 31, 1987, 99-118 (with S.
Lawphongpanich and J. Ventura).
[10] "On the Equivalence of Transfer and Generalized Benders
Decomposition," Transportation Research Vol. 23B, No. 1, 61-73,
1989 (with R.R. Barton and S. Lawphongpanich).
[11] "Benders Decomposition for Variational Inequalities," Mathemati-
cal Programming 48, 231-248, 1990 (with S. Lawphongpanich).
[12] "A Dynamic Programming Algorithm for Dynamic Lot Size Models
with Piecewise Linear Costs," Journal of Global Optimization 4,
397-413, 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.), North-Holland, 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, 103-124, 1996 (with L.E. Gibbons and P.
[15] "Continuous Characterizations of the Maximum Clique Problem,"
Mathematics of Operations Research, 754-768, 1997 (with L.E.
Gibbons, P. Pardalos and M. Ramana).
[16] "Congestion Toll Pricing of Traffic Networks," Network Optimiza-
tion, Springer-Verlag series Lecture Notes in Economics and
Mathematical Systems, 51-71, 1997 (with P. Bergendorff and M.




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 three-year 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 branch-and-
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-
and-project method," a disjunctive
programming-based algorithm for
tackling general mixed-integer 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

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

Robert Weismantel, Book
Review Editor

I was born in Miinchen, Germany
in 1965. After studying mathemat-
ics in the years 1984-1988 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 1989-1991, 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
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

~- - -

10SP I MWA 56

t! rr., i Fr. 01.tiro-c no )d

Karen Aardal, Editor

high-quality articles. At the
Lausanne Symposium I became
Council Member-at-Large 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
"lift-and-project method."
In 1994 I implemented a
branch-and-cut code that uses
Gomory cuts to solve general inte-
ger programs. I have also been
working on crew-scheduling for
the railways, cutting plane algo-
rithms for general integer programs
and a semi-definite programming
approach for the clique problem in

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 real-world 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 semi-definite
cluster, with talks by Michel Goemans,
Franz Rendl and Christoph Helmberg.
The natural question, "Is semi-definite
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 bc-opt, a branch-and-cut
system being developed jointly between
CORE and XPRESS. He once again
pleaded for more mixed-integer pro-
gramming data, but in a model-form 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 mixed-integer (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), andJean-Francois Puget,
from ILOG, described to us the world
of Constrained Logic Programming and
its relation to general integer program-
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 lift-and-project 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.

in Isis DECEMBER *'1'








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SAlgorithms and Experiments (ALEX98) Building Bridges
Between Theory and Applications
Trento, Italy
February 9-11, 1998
) Symposium on Combinatorial Optimization, CO98
April 15-17,1998
Brussels, Belgium
E-mail: bfortz@ulb.ac.be
) Internation Conference on Interval Methods and Their Application
in Global Optimization (INTERVAL'98)
April 20-23, 1998
Nanjing, China
URL: http://cs.utep.edu/interval-comp/china.html
) INFORMS National Meeting
April 26-29, 1998
Montr6al, Quebec, Canada
U RL: http://www.informs.org/Conf/Montreal98/
) Sixth Conference on Integer Programming and Combinatorial Op-
timization, IPCO '98
June 22-24, 1998
Houston, TX
URL: http://www.hpc.uh.edu/~ipco98
) INFORMS International Meeting
June 28-July 1, 1998
Tel Aviv, Israel
U RL: http://www.informs.org/Conf/TelAviv98/
) Fourth International Conference on Optimization
July 1-3, 1998
Perth, Australia
URL: http://www.cs.curtin.edu.au/maths/icota98
) Optimization 98
July 20-22, 1998
Coimbra, Portugal
URL: http://www.it.uc.pt/~opti98
) ICM98
Berlin, Germany
August 18-27, 1998
URL: http://elib.zib.de/ICM98
August 19-21, 1998
Saarbruecken, Germany
URL: http://www.mpi-sb.mpg.de/~wae98/
) INFORMS National Meeting
October 25-28, 1998
Seattle, WA
) International Conference on Nonlinear Programming and Varia-
tional Inequalities
Hong Kong
December 15-18, 1998
) Sixth SIAM Conference on Optimization
May 10-12, 1999
Atlanta, GA
) 19th IFIP TC7 Conference on System Modelling and Optimization
July 12-16, 1999
Cambridge, England
E-mail: tc7con@amtp.cam.ac.uk

Ninth Annual
ACM-SIAM Symposium on
Discrete Algorithms
January 25-27, 1998
San Francisco, California
URL: http://www.siam.org/

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 9-10, 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://
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://


International Congress of Mathematicians (ICM'98)
August 18-27, 1998
Berlin, Germany
URL: http://elib.zib.de/ICM98

Plenary talks will be given by:
Jean-Michel Bismut: Differential Geometry and Global Analysis
Christopher Deninger: Arithmetic Algebraic Geometry, L-Functions 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


Marcelo Viana: Dynamical Systems, Ergodic Theory
Vladimir Voevodsky: Algebraic Cycles and Motives

Mathematical Software Session
International Congress of Math-
ematicians 1998 (ICM'98)
Berlin, Germany
August 18-27, 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 IMU-ap-
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.
Winfried Neun
Konrad-Zuse-Zentrum fir
Berlin, Germany
E-mail: 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
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 (e-mail:
moehring@math.tu-berlin.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 should be sent, prefer-
ably by electronic mail, to:
ICM'98 Session on Math. Soft-
ware, c/o W. Neun, Konrad-Zuse-
Zentrum (ZIB), Takustr. 7,
D-14195 Berlin, Germany
E-MAIL: 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
4. The availability of the software
and the terms and conditions for
distribution should be easily acces-
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-
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.
Submission of Presentations:
March 1, 1998
Notification of Acceptance:
April 1, 1998

International Conference on
Operations Research (OR98)
31 August 3 September 1998
ETH Zurich, Switzerland
Up-to-date 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,
B) Discrete (Burkard, Hertz,
2. Stochastic Modelling, Optimiza-
tion and Simulation (Rieder,
3. Econometrics and Statistics
(Deistler, Garbers, Schmitz)
4. Mathematical Economics, Game
Theory and Decision Theory
(Brachinger, Ulrike Leopold-
5. Banking and Finance (Buehler,
Frauendorfer, Zechner)
6. Operations and Production
Management (Guenther,
Jammernegg, Tempelmeier)
7. Energy and Ecology (Haurie,
8. Telecommunication (Martine
Labbe, Mechthild Stoer)
9. Logistics and Transportation
(Domschke, Fleischmann, Staehly)
10. Fuzzy Systems and Neural
Networks (Rommelfanger, Brigitte

Conference Languages:
English and German
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 e-mail) of the
b) an extended abstract of two
pages (indicating intended section).
The extended abstract should be
submitted either as hard copy (four
copies) or by e-mail as ASCII/TeX/
LaTeX-file to:
Institute fuer Operations Research
der Universitaet Zuerich
OR 98
Moussonstrasse 15
CH-8044 Zuerich
E-mail: 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




- W 4 .. 'r .g w

4 =- o ;' MMF

Linear Programming:
Foundations and Extensions

by Robert J. Vanderbei
Kluwer Academic Publishers
Boston, 1996
ISBN 0-7923-9804-1

This book presents a thoroughly modern treatment
of linear programming that achieves a healthybalance
between theory, implementation, computation, and
between the simplex method and interior-point 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-
The Need for New Linear Programming
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 in-depth treatment of the most impor-
tant interior point methods.
Vanderbei's Book Is Thoroughly Modern
Vanderbei's book is completely up-to-date. Aside
from a nice treatment of the simplex method, it also
contains a very up-to-date treatment of interior point
methods, including the homogeneous self-dual 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
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
self-dual 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 self-dual formulation and algorithm, qua-
dratic programming via interior point methods, and
general convex optimization via interior point meth-
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 much-simplified variant of the
Klee-Minty 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 hard-core 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 fill-in 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.

Geometry of Cuts and Metrics

by M. M. Deza and M. Laurent
Berlin, 1997
ISBN 3-540-61611-X

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-
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 so-called 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 non-negative multiple of
its path metric is hypercube-embeddable. 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 hypercube-embeddable. There are
some distance functions for which this problem is easy
to solve. For others, the decision about hypercube-
embeddability is NP-hard. For various other classes of
metrics, there are conditions available that can be tested
in polynomial time and ensure hypercube-
Part 5 deals with the geometry of the cut cone and
the cut polytope. It surveys extensively polyhedral
material, including the fundamental facet-manipulat-
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 n-dimensional 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 self-contained. I
found it helpful to read Chapter 1, the outline of the



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 up-to-date, 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

Lectures on Polytopes

by G. Ziegler
Berlin, 1995
ISBN 0-387-94329-3

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 example-oriented
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 LP-algorithms. 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 edge-graphs of 3-dimen-
sional polytopes are characterized by planarity and 3-
connectedness. This is the famous theorem ofSteinitz
which is the basis for many further results about 3-di-
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 higher-dimensional polytopes. In
analogy to the theorem of Steinitz, the question is
whether cell-complexes with given geometric or com-
binatorial properties are isomorphic to the face-lattice
ofpolytopes. For such problems, oriented matroids and
Gale-diagrams have proven very useful. As an appli-
cation of this theory, the reader is presented with a 5-
dimensional polytope which has a 2-dimensional face
whose shape cannot be arbitrarily preassigned. Mean-
while, Richter-Gebert have constructed a 4-polytope
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 "Upper-Bound-
Theorem" and the "g-Theorem." The concept of
shellability and the related h-vectors, 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 fiber-polytopes which are
important for Grobner-bases. 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 "warm-ups" 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
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.







Deadline for the next


February 28, 1998

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