Title: Optima
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Publication Date: June 1996
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On October 27, 1995, 1 had the opportunity to interview Ralph Gomory in
New York. Gomory has done fundamental work in integer programming. c_
Among his contributions are the first algorithm for integer programming [1],
[2], now known as Gomory's cutting planes, work with P.C. Gilmore on the --
traveling salesman problem and the cutting stock problem [3], [4], [5], the de-
velopment of corner polyhedra [6], [7], [8], and his work with E.L. Johnson on
group problems [9]. More recently he also developed, together with W.J.
Baumol, an economic theory related to international trade [10], [11], [12].
Gomory received his Ph.D. degree in 1954 from Princeton University. From 1954 to .
1957 he served in the U.S. Navy, and in November 1957, he became a Lecturer in .
Mathematics at Princeton University. In 1959 he joined IBM where in 1970 he became
Director of Research, a post that he held until 1989. He then became President of the .
Alfred P. Sloan Foundation, a non-profit institution supporting various research :
programs in science and technology through grants and fellowships.
During the interview I asked him about his previous and current scientific work, what .
it is like to be president of research of a huge company such as IBM, his view on .. '-
industrial research, and the role of the mathematical programming community. -

j iI


conference notes 7

Journals 9
book reviews 10

gallimaufry 12

PAG 2 N'I~" 50 JUNE 1996~sssl~

' Ralph E. Gomory

OPTIMA: You were in the ii,, I,
physics branch. After a while
you spent some time in the OR
Department. What triggered
your interest in OR at that
RG: I have always been very in-
terested in what people refer to
as the "applications of math-
ematics," but I never thought
about it in that way. I primarily
got interested in mathematics as
a way to understand how things
work rather than as a "thing" in
itself. The OR people were just
40 yards down the hall, and I
knew that they were working on
practical systems. At that time
that meant weapons systems,
and that was the sort of things
that I had in mind to do myself,
so I wanted to learn how it was
done. I was not ,. In Illj I I ,
ning on doing weapons systems,
but I thought that when I was
getting out of the navy then I
might do OR as a career. That
struck me as about right,
given my interest.
OPTIMA: At that time you
started to think about integer
programming. In particular you
developed "the fractional cut
method" as you called it your-
self. Was an algorithm for inte-
ger t},'.;, la mni a big open
question at that time?
RG: No, I did not even know
that there was such a question.
By that time I had gotten out of
the navy and I was at Princeton,
but the navy people had asked
me to stay on as a consultant. So,
every few weeks I would go
down there and consult with
them on problems. One of these
problems was actually an integer
programming problem, but it
was formulated as a linear pro-

gramming problem, and the
variables came out noninteger.
This was very awkward since
they represented the number of
aircraft carriers. The answer 2.2
would not actually be so bad,
but 0.6 is awful because you
then have to ask if you need any
aircraft carriers at all. As soon as
I saw that I thought that in solv-
ing integer linear equations there
is a diophantine way, so prob-
ably in solving integer linear in-
equalities there should be a
diophantine way as well. Of
course I was wrong, but that is
how I started my work. We
needed the answer. Of course
we could round it pretty well,
but it is not the same.
OPTIMA: It is interesting to see
today that both you, and
Dantzig, Fulkerson and Johnson
in their work on the traveling
salesman problem [13], used the
Simplex method as the basis for
an integer programming algo-
RG: That was a method we were
all using at that time. It was a
very natural choice. If you are
using the Simplex method and
you have any contact with prac-
tical situations, you are likely to
run into situations where you
want integral answers. Dantzig
et al., ran into the traveling sales-
man problem, and I ran into the
navy problem. The navy was al-
ready using the Simplex method
to solve a lot of problems, so it
was very natural to ask whether
or not you could adapt the
method such that it produced in-
teger answers. Actually, things
are often much more straightfor-
ward than they appear!

OPTIMA: You also developed
another algorithm for integer
i .... ii iiinwi "the all-integer
ii, tir.i, [14]. You commented
once that it was
computationally inferior to the
' jp,hil...,..I cut method." Why
was that?
RG: At that time it was inferior,
but at some point I would like to
get back to it and take a look to
improve my understanding of it.
I wrote all my programs myself
at that time, and the computers
then were very primitive. If you
did the computations using the
fractional cut method, what you
saw was (assuming you started
out with a problem with integer
coefficients) that the constraint
matrix first stopped being inte-
ger. The determinant of the ma-
trix appears in the denominator
when inverting the matrix, and
this determinant could be very
big. However, once you start to
use fractional cuts, there is a ten-
dency for the determinant to be-
come small again, because if you
pivot on a fraction, then the new
determinant is the old determi-
nant times the fraction, so it gets
smaller. So, very often it hap-
pened that if you did a sequence
of pivots, then the matrix would
again become integral, simply
by reducing the determinant,
through a series of fractional
cuts, down to one again. So ev-
erything turned "magically"
back to integers. So, of course,
this suggested to me that, "We
want integers answers, and we
start with integer coefficients.
Why are we going around
through all these non-integers?"
So I looked around and tried to
come up with a method in which
it was possible to do a Simplex
series of steps but in which ev-
erything would remain integer.

If you think about it a little bit,
this means that the pivot ele-
ment always needs to be equal
to one. So, the problem was how
to arrange that. To my surprise I
did solve it, and I was really
thrilled! I thought, "Wow, this is
a dream come true; this is like
diophantine equations, which I
originally thought to use here.
We can work in the domain of
integers and get integer answers.
This is wonderful! I know it is
going to be very good
computationally!" But it was
not... The reason it did not work
was that the integers you got be-
came absurdly large. So, it just
did i ~i- i .. iii, I am not sure
this has to be the case, but I did
not know anything about nu-
merical analysis. None of us
knew what was happening at all
in these integer programming
problems. One of the things I
want to do eventually is to go
back and look at this using mod-
ern computers. It would be so
easy to make three-dimensional
pictures and see what actually
happens. The fact that the
method was not
computationally successful then
was a great source of regret to
me because I always felt that, in-
herently, it really ought to be
like this. But sometimes those
feelings can be wrong.
OPTIMA: You have also men-
tioned that you find it a bit sad
that corner polyhedra have not
been more explored for use in
computations such as, for in-
stance, to generate a natural se-
quence of test problems. What
do you mean exactly by such
test problems?
RG: I certainly do not mean test
problems for big programs but
test problems for understanding


N- 50


JUNE 1996

PAGI 3 Nj~ 50 JUNE 1996

what is happening. What is the
simplest integer programming
problem really? I assert, but this
is just an intuitive assertion, that
the simplest integer program-
ming problem is defined by a
cone. But, then you still have a
lot of cones in the world that
you can make! So, how would
you order all possible cones in
order of "difficulty"? A corner
polyhedron is basically the con-
vex '-. II of the lattice points in-
side a cone, but these polyhedra
are :- r.,L ii arranged in an or-
der of increasing complexity, be-
cause to each such cone (I am
leaving out one step here, so
what I say is only "approxi-
mately" true) there corresponds
an Abelian group. The size of
the group is given by the deter-
minant of the constraint coeffi-
cients. So, if we were very lucky
and the determinant was one,
everything would come out in
integers automatically. But if the
determinant was equal to 10,
then the group would have 10
elements in it, including zero.
So, you can arrange, so to speak,
all cones in an ascending se-
quence of "complexity," first, the
ones corresponding to groups
with two elements, then the ones
with three elements and so on.
To each of these corresponds a
single polyhedron. As you get
more and more elements, the
polyhedra get more and more
complicated very rapidly. To me
this suggests that there is a very
useful thing to be understood
here, which is: I have got this
very simple cone at all times, but
if I were doing integer program-
ming, the "thing" I have to ex-
plore inside the cone rapidly
gets more complex. In some
sense, corner polyhedra form the

natural link between linear pro-
gramming and integer program-
m ing in the -. 1 a.. t ...- 1-.
OPTIMA: What happened with
that line of research?
RG: I really do not know be-
cause it was the last thing I did
before I became the Director of
Research at IBM. I i- particular
piece of work suffered from the
di.- Ii-I of being harder
than most things were then in
integer programming. It was
kind of an empirical subject at
that time, and the theory of cor-
ner polyhedra was not. The pa-
pers were hard to read, and it
probably was hard to see the
point of it. Every now and then I
do, however, run into people
who are aware of this work.
As far as computing is con-
cerned, to the extent that you
would like to use cutting planes,
these cutting planes can all be
obtained using corner polyhe-
dra. Suppose you are solving a
linear programming problem,
and you come to some vertex,
and you would like to have all
cutting planes that would reduce
it, and suppose you "magically"
had a list of the corner polyhe-
dra for that vertex. It has got all
the cutting planes that are the
faces you need to give you an in-
teger body. Clearly, most of the
work would have to be done be-
cause, first of all, you cannot
generate all those polyhedra to
begin with, i.e., you cannot have
such a list as I was referring to,
but a very promising line of re-
search in my opinion would be
to start to look at this sequence
of polyhedra and then simplify
them in a way that makes them
valid inequalities. It is a lot to be
examined! For example, which

of the facets of a certain polyhe-
dron matter? It is quite ..- -1 --1.
that a reason the number of fac-
ets may be tremendous is that it
has a few "big" ones in front and
then, in addition, it has all kinds
of "fancy stuff." I do not know if
this is true or false.
OPTIMA: That would agree
with the experience from the
study of the traveling salesman
polytope done by Alan Hoffman
and Harold Kuhn, who simu-
lated that someone was sitting
at the center of gravity of the
traveling salesman polytope
with a pistol firing at random.
The result was that only the
trivial facets x,=0 were hit (see
[15], page 118).
RG: Yes, exactly. You see, if you
had to make a guess, you would
say that this is the way it prob-
ably is. With modern methods
you could, in my opinion, just
look at a few classes of facets
and start to see whether they
matter. If only a few mattered,
then you would try and develop
ways to get those few. I think the
corner polyhedra are the sim-
plest possible problems ar-
ranged in order of difficulty, and
I think they would be very inter-
esting to explore. If no one else
does and if I am through with
what I am now doing, then I
OPTIMA: At the end of your
1966 paper on the traveling
salesman problem [15], there is a
discussion. One thing that sur-
prised me with this discussion
was the awareness of the exist-
ence of "difficult" and "easy"
problems. In particular, Jack
Edmonds posed the question of
how problems can be character-
ized such that they get more
tractable. The atmosphere

seemed i, optimistic, from
what I could read out of the dis-
cussion, in the sense that people
were fairly convinced that even-
tually such nice characteriza-
tions would be available.
RG: I think it was the way
people felt, and I guess that I still
feel that way in spite of the com-
plexity theory, at least from a
"practical" viewpoint. Clearly,
you cannot solve all problems in
a systematic way, but you never
have to. You just have to solve
some class of problems that you
are interested in, and very often
you do not have to solve prob-
lems exactly. There may be some
practical problems that are terri-
bly difficult, and where we want
the exact answer, but I think
they form a tiny minority. So,
depending on what you are try-
ing to do, you should be very
pessimistic or very optimistic.
Even if we cannot compute cer-
tain things, there is very often a
saving grace. One obvious ex-
ample is to determine where the
ball will end up in a roulette
wheel. You cannot compute this
well enough to tell between
which pins the ball will end up,
so this problem is, for all practi-
cal purposes, unsolvable. In this
case, however, there is a related
solvable problem, namely a "sta-
tistical" version. Typically, if
you cannot solve a problem,
there is a related class of prob-
lems that you can solve. Maybe
being unable to solve a problem
means that it is too delicate, so
now we can employ a different
technique that we, in the case of
the roulette wheel, happen to
call probability. Perhaps because
I did not live through this evolu-
tion of all things you cannot do,
I remain extremely optimistic!





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_ ~ I~I~ ~

N05 UE19

E. Gomory

OPTIMA: How did you get in-
volved in your present work in
economics [!11 [11], [12])?
RG: When 1 was Director of Re-
search at IBM I used to travel a
lot. In particular, I often went to
Japan, which at that time was
developing very rapidly. I was
impressed by the rapidity with
which they learned to do things,
and they did things extremely
well. People sometimes credit
the Japanese government for be-
ing a major force behind this, but
in my opinion mostly it was ter-
rific work by the companies and
the people. They just did things
better. One question at that time
was whether this was a good or
a bad thing for the United States.
It became clear at a certain point
that the U.S. was losing its semi-
conductor industry to the Japa-
nese competition, and it was a
lot of fuss in the U.S. about that.
I was on a government commit-
tee that was asked to look into
the question whether something
should be done. To help an in-
dustry like that is actually very
"un-American." I went to a
meeting with this committee in
which there were some CEOs of
semiconductor companies and
some economists. The econo-
mists basically said, "Look, it
does not matter if you lose this
industry because something else
will make up for it." There is an
economic theory along those
lines, and I now believe that
what they said was a misinter-
pretation of that theory. The
CEOs 'd ,-1t .iI said that once
you get out of this business,
there is no way to get back in.
So, the two groups talked right
past each other. I was familiar
with the economic theory to
some extent, just from my gen-

eral education, and I certainly
knew what the CEOs were talk-
ing about because I understood
something about that industry,
and I said to myself, "Oh, boy,
there is a real disconnect here." I
was two years away from retire-
ment at that time, so I thought,
"One thing I am going to do is to
look at this problem," because I
was convinced that both parties
had something to say, but both
were wrong. As a matter of fact,
when I left IBM and started my
present job, I developed hepati-
tis, and Herb Scarf came to see
me. He is a very good friend of
mine since the time when we
were both at Princeton. He has
an interest in economies of scale,
and I felt that economies of scale
was the key to this situation. So,
we decided to take a look at it,
and Herb taught me about eco-
nomic equilibria, and then I was
able to go on from there and de-
velop the outlines of a theory of
international trade based on
economies of scale. It turned out
that you, miraculously, had to
do a little integer programming
to understand this because
economies of scale has that qual-
ity you either have a plant or
you do not. Of course, this gave
very different results compared
to the ordinary theory, and it re-
ally said that if another country
gets better and better it could
harm you. Since then, and also
in collaboration with Will
Baumol, we developed a theory
which does not require econo-
mies of scale. It is really the old
theory all over again, but it al-
lows for the fact, which people
had not worked on, that capa-
bilities of a country really do
change over time, and then an-
swers the following question:

Suppose you are country one
and you are dealing with coun-
try two, and country two is
rather undeveloped, but then it
starts to develop. Is that a good
or a bad thing from your "self-
ish" point of view as country
one? The answer is quite inter-
.. i, .. which is, in the beginning
it is better for both countries that
country two develops because
they do better as they develop
more goods, and you benefit be-
cause their labor is so cheap. But,
after they developed to about a
point where their real wage is
about a third of yours, then their
further development is negative
for country one. So, we devel-
oped this theory, and I think it is
going to have a lot of effect.
OPTIMA: What was it like to
become the Director of Research
for such a large and relatively
diverse company as IBM?
RG: You have the very impor-
tant challenge of making the
things that your people do, and
we had really wonderful people,
useful to IBM. We had a research
group varying in size between
two and three thousand people
doing very different things. The
problem was, how can you re-
late what they are doing and
make it useful to IBM? It was
clear to me from the very begin-
ning that if something did not
turn out to be useful to IBM, it
would not survive. There were
some people who felt that IBM
would just pay for science, but I
never believed that for a second.
OPTIMA: Nowadays you read
much about how various indus-
trial research labs have serious
problems. What is your view on
the future of industrial research?

RG: I think that industrial re-
search, if properly done, can be
very productive. And I think
that the Bell Labs experience and
the IBM experience both show
that. Within IBM, research has
been cut less. IBM suffered a
great deal; all parts of the com-
pany were cut down, but re-
search perhaps the least. So I do
not think that IBM has walked
away from research. But, my
general belief is that a mixture of
theory and exposure to practice
is good for both, and therefore I
would say that the "case" for in-
dustrial research is that it is the
milieu in which that can take
place. If it is a milieu in which
this cannot take place, then why
bother? If it is just another group
of people simply doing research
out of contact with applications,
you are not taking advantage of
practical problems, or if it is just
making the next product, you
are not taking advantage of the
I'..... I lir -, that science gives
you. So, I think these research
labs can be very useful if they
can manage, which is not easy.
OPTIMA: Can we afford it?
RG: Sure we can!
OPTIMA: A controversial ques-
tion related to the previous one
is whether research should be
done at all universities, or if we
need a partition into research
and teaching universities?
RG: Why not? This is not a sub-
ject that I know a great deal
about, but there is an interesting
related question, namely, how
much research is "enough"? Re-
searchers think it should always
be more, but it cannot be. So,
there is a question of what is the
correct level. I am currently
quite active in pursuing that


JUNE 1996

N- 50


SRalph E. Gomory


question [16], but I tend to agree
with you. If we set the right level
not every university will be do-
ing research.
For instance, the place where I
went to college was a very good
college, but it was not a univer-
sity, there was only a little re-
search. Faculty did research in
their spare time, but they were
often wonderful teachers. And it
worked fine. A lot of scientists
came out of schools like that, be-
cause they got to know the pro-
fessors, and they got to see how
they thought about things, and it
was a good atmosphere. I cer-
tainly do not believe that the
only people that can teach are
people who develop themselves
very heavily in research.
OPTIMA: How does the math-
ematical programming commu-
nity look to you at present? Do
you think that we consider a
good im 'of theoretical and
practical questions?
RG: I will not comment on that
simply for lack of knowing. My
only exposure to the community
lately was the International Sym-
posium in Ann Arbor in 1994,
which I enjoyed very much.
There I saw a mixture of things.
Rather than commenting on the
community, it might be more
sensible for me to say what I
think is good. It is also quite pos-
sible that my prejudices are out
of date, because this has not
been my primary focus for a
long time.
It happens in many subjects that
they develop an abstract wing
and an applied wing, as for ex-
ample, computer science. There
is quite a gap between people
who develop the theory of algo-
rithms, and the people who de-

velop software systems. It is
very hard to get the theory to
deal with such a tangled mess.
This is, in my opinion, the real
distinction between engineering
and science. In engineering you
have to make the things work,
even if you do not understand
them. When you make disk
drives for instance, a lot of that
is just: "We will do what we
did last year, and see if we can
make these layers a bit thinner
and get the bits packed a little
closer". You could only do cal-
culations about little pieces of
this complex problem, but the
whole thing is too complicated.
So, in real life you almost al-
ways deal with things that are
too complicated to scientifically
analyze. The scientists deal
more with what is understood.
My view is that all these things
are i. II worth d..;,..-. includ-
ing the most theoretical, and
the most applied, and it is too
bad that they tend to look
down on each other, because
the people who do the theoreti-
cal work every now and then
come up with something very
helpful. Every year you can
compute and understand a
little more than last year, so in
the long run they will be helpful
to the people with the applied
problems. The applied people
are going to approach their
problems somehow, and they
keep running into things they
cannot do, which form a fantas-
tic source of interest and stimu-
lation for the more theoretical
people to work on. So, this
whole chain is what I like to see
in a society, and I do not know
to which extent the mathemati-
cal programming society re-
sembles the picture I have just

[1] R.E. Gomory (1958). Outline of an algorithm for
integer solutions to linear programs. Bull. Amer.
Soc. 64, 275-278.
[2] R.E. Gomory (1963). An algorithm for integer
solutions to linear programs. R.L. Graves, P. Wolfe
(eds.). Recent Advances in Mathematical Program-
..ii,. McGraw-Hill, New York, 269-302.
[3] P.C. Gilmore, R.E. Gomory (1961). A linear
programming approach to the cutting stock
problem. Oper. Res. 9, 849-859.
[4] P.C. Gilmore, R.E. Gomory (1963). A linear
programming approach to the cutting stock
problem Part II. Oper. Res. 11, 863-888.
[5] P.C. Gilmore, R.E. Gomory (1964). Sequencing a
one state-variable machine: a solvable case of the
traveling salesman problem. Oper. Res. 12, 655-679.
[6] R.E. Gomory (1965). On the relation between
integer and noninteger solutions to linear pro-
grams. Proc. Nat. Acad. Sci. 53, 260-265.
[7] R.E. Gomory (1967). Faces of an integer polyhedron.
Proc. Nat. Acad. Sci. 57, 16-18.
[8] R.E. Gomory (1969). Some polyhedra related to
combinatorial problems. Linear Algebra Appl. 2,
[9] R.E. Gomory, E.L. Johnson (1973). The group
problem and subadditive functions. T.C. Hu, S.M.
Robinson (eds.) Mathematical Programming,
Academic Press, New York, 157-184.
[10] R.E. Gomory (1994). A Ricardo model with
economies of scale. Journal of Economic Theory 62,
[11] R.E. Gomory, W.J. Baumol (1995). A linear Ricardo
model with varying parameters. Proc. Nat. Acad.
Sci. 592, 1205-1207.
[12] R.E. Gomory, W.J. Baumol (1995). Regions of linear
trade-model equilibria and the conflicting interest
of trading partners. Research Report #95-17, New
York University, Faculty of Arts and Science,
Department of Economics.
[13] G.B. Dantzig, D.R. Fulkerson, S.M. Johnson (1954).
Solution of a large- scale traveling salesman
problem. Oper. Res. 2, 393-410.
[14] R.E. Gomory (1963). An all-integer integer program-
ming algorithm. J.F. Muth, G.L. Thompson (eds.)
Industrial Scheduling, Prentice-Hall, Englewood
Cliffs, 193-206.
[15] R.E. Gomory (1966). The traveling salesman
problem, Proc. IBM Scientific Computing Sympo-
sium on Combinatorial Problems, IBM, 93-121.
[16] R.E. Gomory, H. Cohen (1993). Science: How much
is enough? Scientific American, July, 1993.

I _J X A - A;UFI


N 50

JUNE 1996


50 JUNE 1996

onfere nce

SIFORS 96 14th Triennial
Conference, Vancouver
British Columbia, Canada
July 8-12, 1996
Santa Barbara, California
Aug. 19-23, 1996
SInternational Conference on
Nonlinear Programming
.eijing, China
Sept. 2-5, 1996
> Symposium on Operations
Research (SOR96)
Technical University
Braunschweig, Germany
Sept. 4-6, 1996
> Second International Sympo-
sium on Operations Research
and its Applications
(ISORA '96)
3uilin, China
Dec. 11-13, 1996
; Optimal Control: theory,
algorithms, and applications
Center for Applied
University of Florida
Feb. 27 March 1, 1997
) XVI International
Symposium on Mathematical
Programming, Lausanne
Switzerland, Aug. 1997

-;- --
_. .% ,- cv:_ :

Optimal Control: Theory,

Algorithms, and Applications
Center for Applied Optimization
University of Florida
Feb. 27-March 1, 1997
The conference is to be held at the Center for Applied Optimization, Univer-
sity of Florida, Gainesville, FL. This meeting will provide a unique oppor-
tunity for researchers working on theory, algorithms, and applications of
optimal control to exchange recent research advances, to establish a foun-
dation for joint research cooperation, andto stimulate future research. Pub-
lication of a conference proceedings is planned. For more information, con-
tact Bill Hager: hager@math.ufl.edu.

Report on the DIMACS Workshop on the
Satisfiability (SAT) Problem
March 11-13, 1996
Rutgers University
This conference was held at the NSF National
Center for Discrete Mathematics and Theoreti-
cal Computer Science (DIMACS), Rutgers Uni-
versity. The workshop was organized by Ding- '-' '
Zhu Du, Jun Gu, and Panos Pardalos. The advi- .
sorycommittee members included BobJohnson,
David Johnson, Christos Papadimitriou, Paul t
Purdom, and Benjamin Wah. DavidJohnson and
MosheVardi, Chairman oftheSpecialYear Com-
mittee, provided suggestions and comments on
the early planning of the workshop.
More than 65 researchers from universities, gov-
ernmental agencies, and industrial companies
from 10 countries around the world attended the
workshop which began with a welcome by
DIMACS director, Fred Roberts. Following an
introduction by David Johnson, Professor Steve
Cook (pictured), who was the 1982 Turing Award winner, then delivered a distin-
guished lecture. A total of 34 technical talks were presented. The major topics covered
included practical and industrial SAT problems and benchmarks, significant case
studies and practical applications of SAT problems and SAT algorithms, new algo-
rithms and improved techniques for satisfiability testing, specific data structures and
implementation details of the SAT algorithms, and the theoretical study of the SAT
Problem and SAT algorithms.

The satisfiability problem is central in mathematical logic, computing theory, and
many industrial application problems. There has been a strong relationship between
the theory, the algorithms, and the applications of the SAT problem. This work-
shop brought together a group of distinguished theorists, algorithmists, and prac-
titioners working on the SAT problem and on its industrial applications, which en-
hanced the interaction between the three research groups. As an important activity
of the workshop, a set of SAT problem benchmarks derived from the practical in-
dustrial engineering applications has been provided for SAT algorithm
benchmarking. Overall, the workshop was a great success. Proceedings of the work-
shop will be published later this year by the American Mathematical Society in the
DIMACS Series.


JUNE 1996


i'sii 8 N" 50 JUNE 199


Call for Nominations:
Nominations are being sought
for the Mathematical Program-
ming Society Beale-Orchard-
Hays Prize for Excellence in
SComputational Mathematical

This award is dedicated to the
memory of Martin Beale and
I .... Orchard-Hays, pio-
neers in computational math-
ematical programming. To be
eligible, a paper or a book must
meet the .11 ,i require-
1) It must be on computational
mathematical programming.
The topics to be considered
a) experimental evaluations
of one or more math-
ematical algorithms,
b) the development of quality
mathematical program-
ming software (i.e. well-
documented code capable
of obtaining solutions to
some important class of
MP problems) coupled
with documentation of the
applications of the soft-
ware to this class of prob-
lems (note: the award
would be presented for the
paper which describes this
work and not for the soft-
ware itself),
c) the development of a new
computational method
that improves the state-of-
the art in computer imple-
mentations of MP algo-
rithms coupled with docu-
mentation of the experi-
ment which showed the
improvement, or

d) the development of new
methods for empirical
testing of mathematical
.. techniques
(e.g., development of a
new design for computa-
tional experiments, identi-
fication of new perfor-
mance measures, methods
for reducing the cost of
empirical testing).
2) It must have appeared in the
open literature.
3) If the paper or book is writ-
ten in a language other than
English, then an English
translation must also be in-
4) Papers eligible for the 1997
award must have been pub-
lished within the years 1993
through 1996.
These requirements are in-
tended as guidelines to the
screening committee but are
not to be viewed as binding
when work of exceptional merit
comes close to satisfying them.
Frequency and Amount of
the Award:
The prize is awarded every
three years. The 1997 prize of
$1500 and a plaque will be pre-
sented in August 1997, at the
Swiss Federal Institute of Tech-
nology (EPFL), Lausanne Swit-
zerland, at the Awards Session
of the International Sympo-
sium on Mathematical Pro-
gramming sponsored by the
Mathematical Programming

Judgement criteria:
Nominations will be judged on
the : .1.. .. criteria:
1) Magnitude of the contribu-
tion to the advancement of
computational and experi-
mental mathematical pro-
2) Originality of ideas and

3) Clarity and

II .. ofex-

Nominations must be in writ-
ing and include the titles) of
the papers) or book, the
authorss, the place and date of
publication and four copies of
the material. Supporting justifi-
cation and any supplementary
materials are welcome but not
mandatory. The awards com-
mittee reserves the right to re-
quest further supporting mate-
rials from the nominees.
Nominations should be
mailed to:
Professor Robert J. Vanderbei
Dept. of Civ. Eng. and Opera-
tions Research
ACE-42 r,.:P...... ; Quad
Princeton University
Princeton, NJ 08544
Tel: 609-258-0876
Fax: 609-258-3796
rvdb@princeton.edu http://
The deadline for submis-
sion of nominations is
January 1, 1997.
This call-for-nomination can be
viewed online by visiting:

Stephen M. Robinson

Receives Honorary


i I-.. Faculty of Economics of the
University of Zurich awarded the
Honorary Doctor's degree to
Stephen M. Robinson, University
of Wisconsin-Madison in recognition
of his fundamental -. ,I i l-,. it ......-, to
the theory of nonlinear optimization,
in particular to the stability behaviour
of optimization problems, and to the
development of efficient and robust
solution methods for nonlinear and
stochastic optimization problems,
as well as in appreciation of his
permanent efforts for international
cooperation in the OR community.
-lIi i! 1: KALL
Institute fuer Operations Research
der Universitaet Zuerich
Moussonstr. 15
e-mail: kall@ior.unizh.ch (Internet)
phone: +41 1 257 3771
fax: +41 1 252 1162 UNI ZH IOR

N 50


JUNE 1996

\,I 9 50 JUNE 1996

Volume 72, Number 1

M. Fukushima, The primal
Douglas-Rachford splitting
algorithmfor a class of mono-
tone mapping oith application
to the traffic equilibrium
W. Li, A conjugate gradient
method for the unconstrained
minimization of strictly convex
quadratic splines.
M.S. Gowda, On the extended
linear .il,' i, n il iit problem.
A. Ben-Tal and M. Teboulle,
Hidden convexity in some
nonconvex quadratically
constrained quadratic program-
R.D.C. Monteiro and S.
Mehrotra, A general parametric
analysis approach and its
implication to ., iaili; ill,
analysis in interior point
M. G6the-Lundgren, K. Jornsten
and P. Varbrand, On the
nucleolus of the basic vehicle
routing game.

Volume 72, Number 2

M. Gr6tschel, A. Martin and R.
Weismantel, Packing Steiner
trees: polyhedral investigations.
M. Grotschel, A. Martin and R.
Weismantel, Packing Steiner
trees: a cutting plane i i.-," Iin
and computational results.
A.I. Barros, J.B.G. Frenk, S.
Schaible and S. Zhang, A new
algorithm for generalized
fractional programs.

J.F. Sturm and S. Zhang, An
O(VnL) iteration bound primal-
dual cone ,tii', scaling algorithm
for linear programming.
U. Faigle and W. Kern,
Submodular linear programs on

Volume 72, Number 3

Y. Crama and F.C.R. Spieksma,
Scheduling jobs of equal length:
-, ,,ll, I it facets and computa-
tional results.
H. Tuy, S. Ghannadan, A.
Migdalas and P. Virbrand, A
SI-..,i., it polynomial algorithm for
a concave production-transporta-
tion problem with a fixed number
of nonlinear variables.
K.T. Au, A primal-dual approach
to inexact subgradient methods.
A. Nemirovskii and K.
Scheinberg, Extension of
Karmarkar's algorithm onto
convex quadratically constrained
quadratic problems.
M. Muramatsu and T. Tsuchiya,
Convergence analysis of the
projective scaling algorithm based
on a long-step homogeneous .(i',
scaling algorithm.

Volume 73, Number 1

U. Faigle, W. Kern and M. Streng,
Note on the computational
complexity of j-radii of polytopes
in R".
S. Chopra and J.H. Owen,
Extended formulations for the A-
cut problem.
R. Miiller, On the partial order
polytope of a digraph.

R. Mifflin, A quasi-second-order
proximal bundle algorithm.
A.R. Conn, N. Gould and Ph.L.
Toint, Numerical experiments
with the LANCELOT package
(Release A) for large-scale
nonlinear optimization.
S. Kapoor and P.M. Vaidya,
Speeding up Karmarkar's
algorithm for multicommodity

Volume 73, Number 2

B.V. Cherkassky, A.V. Goldberg
and T. Radzik, Shortest paths
algorithms: Theory and experi-
mental evaluation.
A. Nemirovskii, On
polynomiality of the method of
analytic centers for fractional
J.-S. Pang and J.C. Trinkle,
C'""11.m'.. aii,. i i, formulations
and existence of solutions of
lim,,,i. multi-rigid-body contact
problems with Coulomb friction.

Volume 73, Number 3

S.S. Nielsen and S.A. Zenios,
Solving multistage stochastic
network programs on massively
parallel computers.
L.U. Uko, Generalized equations
and the generalized Newton
S. Wright and Y. Zhang, A
superquadratic infeasible-interior-
point method for linear
., u .hc ,iit 1,11it ll, problems.
P.M. Vaidya, A new algorithm for
minimizing convex functions over
convex sets.


JUNE 1996


P~~d 10 N' 50JUNE199



% I

,_ :-.

I 1

The Traveling Salesman,
Computational Solutions for
TSP Applications By G. Reinelt
Lecture Notes in Computer Science
840 Springer-Verlag, Berlin, 1995
ISBN 3-540-58334-3

According to the author, "The aim of
this monograph is to give acomprehen-
sive survey on heuristic approaches to
solving the traveling salesman problem
and to motivate the development and
implementation of possibly better algorithms."
The book begins with some chapters on basic material,
treating basic concepts, including such geometric con-
cepts as Voronoi Diagrams, Delaunay Triangulations and
Convex Hulls. Although not intended as a text, I think
these chapters make the book very usable in a course for
graduate students.
The core of the book is given in chapters such as "Con-
struction Heuristics," "Heuristics for Large Geometric
Problems," and i' I.... ... I ,, ."Manycomputa-
tional results obtained by the author illustrate how the
considered and analyzed heuristics are characterized as
specific to the TSP.
In the chapter, "Further Heuristic Approaches," some
non-specific 1. ..;il... are briefly described, without
further computational results. A chapter, "Lower
Bounds," gives results for a large number of problem
instances and bounds. The last chapters give a case study
ofa production process and treat some issues on practical
TSP solving.
The book is completed by an index and seven pages of
references, .l. ,i;. II pages where the references are
used, thereby adding to the value of the book. The lay-
out, both the text and tables with results, is straightfor-
ward and adds to the readability. Typing errors are very

This book can be seen as an up-to-date and valuable
extension ofsome chapters of the classical book on theTSP
byLawleretal. Without any doubt, I can recommend this
book, both for practitioners as well as researchers, since
it gives a concise and crisp overview of the heuristical
approach to solving the TSP.
Reference Lawler, E.L.,J.K. Lenstra,A.H.G. Rinnooy
Kan and D.B. Shmoys (Eds.), The Traveling Salesman
Problem, Wiley, Chichester, 1985.

Global Optimization in
Engineering Design
Nonconvex Optimization and Its
Applications, Vol. 9
Edited by Ignasio E. Grossmann
Kluwer Academic Publishers,
Dordrecht, 1996
ISBN 0-7923-3881-2

lobal Optimization is concerned with
the problem of computing I ..I .11.
optimal solutions of nonlinear func-
tions. Although the application of
local optimization methods is very
popular in the field of engineering, the use of these tech-
niques becomes inadequate in certain problem cases where
the local solution either produces a significant cost pen-
alty or results in an incorrect solution to a physical prob-
lem. Therefore, the need for global scope optimization
methods becomes critical in the context of engineering
The chapters of this volume describe some of the most
recent developments of deterministic approaches in glo-
bal optimization and their use in engineering applications.
The first two chapters by Epperly and Swaney, present an
LP-based branch and bound algorithm for finding the
i, ,i ,i .? ,, .. r. t ... . "i .. , ,, .. . .. .. C- ... ,i 1 I. ...

: 7


N' 50


JUNE 1996

i i1Sit8astnmg8lIlbl1 I asm180 tiauI nB titi3ilUBIIU8

useful for a variety of engineering applications such as
phase and chemical equilibrium problems and flowsheet
optimization problems.
The next two chapters, by Visweswaran and Floudas,
describe the improvement of an original cutting plane
algorithm GOP designed by the same authors, now pre-
sented in a branch and bound framework. Additionally,
an implementation of the new GOP algorithm and com-
putational results related to various problems in chemi-
cal engineering design and control and mathematical
programming are presented in detail.
Chapter five describes the implementation of interval
analysis optimization methods for solving global optimi-
zation problems, with emphasis on issues related to pro-
cessdesign. Ci ,. I ,I ,,,. . I. .,, i. 1 .,
interval analysis that employs procedures for search ac-
celeration and fast elimination ofinfeasible search spaces.
The algorithm is applied for solving nonconvex mixed
integer nonlinear programs.
The last six chapters emphasize applications in engineer-
ing design, such as planning ofprocess networks, stochas-
tic planning and scheduling models, heat exchanger
networks, layout design, design of truss structures, batch
design, water distribution systems, and general process
Chapter titles and authors are listed below:

Epperly, T.G.W. and R.E. Swaney, "Branch and Bound
for Global NIP: New Bounding LP"
Epperly, T.G.W. and R.E. Swaney, "Branch and Bound
for Global NLP: Iterative LP Algorithm and Results"
Visweswaran, V. and C.A. Floudas, "New Formulations
and ...i;, Strategies for the GOP Algorithm"
Visweswaran, V. and C.A. Floudas, "Computational
Results for an Efficient Implementation of the GOP
Algorithm and Its Variants"
Byrne, R.P. and I.D.L. Bogle, "Solving Nonconvex Pro-
cess Optimisation Problems Using Interval Subdivision

'ACE 11

Vaidyanathan, R. and M. El-Ialwagi, "Global Optimi-
zation of Nonconvex MINLP's by Interval Analysis"
Liu, M.-L. N.V. Sahinidis and J.Parker Shectman, "Plan-
ning of Chemical Process Networks via Global Concave
lerapctritou, M.G. and E.N. Pistikopoulos, "Global
Optimization for Stochastic Planning, Scheduling and
Design Problems"
lyer, R.R. and I.E. Grossmann, "Global Optimization of
F. .. F 1. .'... T 1 .with Fixed Configuration for
Multiperiod Design"
Quesada, I. and I.E. Grossmann, "Alternative '
Approximations for the Global Optimization ofVarious
I .. .. Design Problems"
Sherali, H.D., E.P. Smith and S. Kim, "A Pipe Reliabil-
ity and Cost Model for an Integrated Approach Toward
Designing Water Distribution Systems"
Smith, E.M.B. and C.C. Pantelides, "Global
Optimisation of General Process Models"
I i, I ... I I,1, 1. .. ... ...n iti graduate students,

cal programming and operations research.

( .

N050JUE 99

Iterative Methods for Linear and
Nonlinear Equations

By C. T. Kelley
Frontiers in Applied Mathematics 16
SIAM, Philadelphia, 1996
ISBN 0-89871-352-8

his book is devoted to giving a modern
view of iterative methods for solving lin-
ear and nonlinear equations. The reader
should have a basic knowledge of analy-
sis and numerical linear algebra because
most of the results are proved with mathematical rigor
The chapters are concluded with sets of examples and
exercises. Throughout the text one can find motivating
examples mainly from boundary value problems with
partial differential equations. Many of the chapters con-
tain links to MATLAB code which is provided per anony-
mous ftp from the author.
The first three chapters are devoted to the iterative solu-
tion of systems of linear equations. The first chapter
contains a description of basic concepts for stationary
iterative methods. Conjugate gradient methods are pre-
sented in the second Ih ..1 .... i,, 1.'. : ... ... i ....1 ...
and the variants for solving normal equations. In the

;7 :. ,... i '

I wish to enroll as a member of the Society.

'. ii2 ,I ..., r i; for my personal use and not for the benefit of any library or institution.

L I willpay my membership dues on receipt ofyour invoice.

0 I wish to pay by creditcard (Master/Euro or Visa).

ii 1 .. i GMRES is discussed for nonsymmetric
systems of equations. Other methods presented are Bi-
CG, CGS, Bi-CGSTAB, and TFQMR. All these meth-
ods have been widely used for the solution of large scale
linear systems. A chapter on fixed-point iterations intro-
duces concepts for the solution of nonlinear equations.
Newton's method is discussed in depth including conver-
gence rate analysis, implementation hints, and variants
such as the chord method, finite difference method, and
Shamanski's method.
Chapter 6 is a novelty in a textbook presentation. The
concept of inexact Newton methods, where each New-
ton step is only computed up to a certain accuracy, is
intertwined with iteration solvers from the earlier chap-
ters. The question of how to control the residual error in
the linear solver while retaining the fast convergence of
Newton's method is raised, and it is shown in Chapter 7
that Broyden updates can be used in this context. They
are applied to linear and nonlinear problems. All the
previous convergence issues deal with local convergence
results.C l. I . ', : 1" 1- ". -. 1,. . ,. ; .FI,
last chapter. Various implementations ofArmijo's rule are
This book is an excellent textbook for mathematicians and
engineers who want an insight into modern iterative
methods for solving nonlinear equations.

Mail to:
The Mathematical Programming Society, Inc.
c/o International Statistical Institute
428 Prinses Beatrixlaan
2270 AZ Voorburg
The Netherlands








Cheques or money orders should be made payable to
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one of the currencies listed below.
Dues for 1996, including subscription to the journal
Mathematical Programming, are Dfl.105.00 (or
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Student applications: Dues are one-half the above
rates. Have a faculty member verify your student sta-
tus and send application with dues to above address.
Faculty verifying status




JUNE 1996


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Programming, such as table of
contents, policy, and subscription
can be found via the homepage
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Follow these instructions:
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Donald W. Hearn, EDITOR
Utrecht University
Department of Computer Science
P.O. Box 80089
3508 TB Utrecht
The Netherlands
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Industrial and Systems Engineering
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Journal contents are subject to change by the publish-,.


JUNE 10)96

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ismp97@masg 1.epFl.ch
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fon hr the Do,..iwi1
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INFOP-\! \lescunz.
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donna llewellynkisye.gatech.edu
I)D,.sdlinc for cdi
nc-c (c I i-\1A
'n.-pc~rr 5icr i ]I ''Ii.



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