Title: Optima
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Permanent Link: http://ufdc.ufl.edu/UF00090046/00045
 Material Information
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: March 1995
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Bibliographic ID: UF00090046
Volume ID: VID00045
Source Institution: University of Florida
Holding Location: University of Florida
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III I'ii fMi 111111 iLll h1I

1T Features Department is to
U provide high-quality feature ar-

R ematical programming. Our goal is to
E publish at least one article in every issue
S of OPTIMA. Feature articles have always
beenwelcome and several have appeared.
They have, however, appeared less frequently dur-
ing the last few years. Given the very rapid devel-
opment in many areas of mathematical program-
ming, the increased interaction with other areas,
and the vast number of new successful applica-
tions, it is felt strongly among members that fea-
ture articles should again constitute a prominent
part of OPTIMA. In order to stimulate this ac-
tivity, I have solicited articles by leading experts
in various fields of mathematical ...., i 1.1..1 i" !
In -,. .. ,,l, year articles on combinatorial op-
timization used in railway planning, Grabner
bases;,,,l ',r .'1 I ,;. ii,,ii:... random ized al-
gorithms, and structural optimization are ex-
pected to appear. It is our hope that many people
will be inspired by these contributions and sub-
mit articles for forthcoming issues.
point out that a feature article should be written
in away that is understandable to the general MPS
member. It could typically present a new tech-
nique, the state-of-the-art ofan area, provide his-
torical perspective, or describe an application.
Formulae are allowed but should be used spar-
i,,-i ill r ,r..,, most welcome. Belowis a
complete list of feature articles that have appeared
so far in OPTIMA. Prospective authors can re-
quest any of those from me. The recommended
way ofsubmitting an article is by e-mail using the
address given below. The format should be a plain
text file with TeX commands wherever needed.
I ,o.. or photos can be sent by ordinary mail.
Articleswillbesubjectto 1;I _., ..l; i. 1 1..r 11
not be refereed.


t a t :i ,

Interviews 2-6

MORE ON PAGE NINE more feature articles 7-9

conference notes

book reviews





I II II II I 10 IFJ9111hl III, 11r, oil

- ----- ~s

Department of OPTIMA is the new home T
for readers interested in computational ac- W

...I ..' ,I I ., I. I, 1 . y . l, R

t . .. , ,l ,t .. -, -, - ,. I .
II , .., ,
,,, I. ,, r,, ,, , ,11, 1 ,

i., 1 ,

tain information on test problems and problem gen-
erators (including evaluating same), compile and
maintain a list of recent technical reports related to
computation, and compile and maintain a list ofavail-
able commercial quality software (including evaluat-
ing same). Obviously, some of the more n ,".. in-
formation is best transmitted through a directory
accessible via anonymous ftp, with OPTIMA guiding
readers to what is available on the Internet.
In order for this column to be successful and respon-
sive to reader views, I herein announce a general call
for contributions to this Department. This is to include
n o, ,, I... .. q l [I .. I , .. ,,u rl ,I .... . II
as related news items about conferences, workshops,
appointments, awards, and personal achievements that
you would like to communicate to your peers.



Features Department is to collect and
compile news. I am currently building
a network of news reporters to assist me
in this task. We welcome news about
conferences, workshops and other ac-
tivities as well as personal news, such as
appointments, awards and other
achievements. News regarding software
should be sent to the Software and Com-
putation editor, Faiz Al-Khayyal. Any-
.r- ,ll. F ,, .. p .q r._r ,
encouraged to contact me as quickly as
possible. News should preferably be sub-
mitted to me by e-mail using the same
format as the feature articles.
OPTIMA exists to provide service to the
members. Any ideas on topics for feature
articles or any other activities are, of
course, most welcome!


Karen Aardal
Features Editor
Department of Econometrics
Tilburg University
P.O. Box 90153
5000 LE ,il-..,, ,
The Netherlands
e-mail: aardal@kub.nl
Tel.: +31 13663254
FAX: +31 13 663280

Feature articles published in

earlier issues ofOPTIMA:

* The ellipsoid algorithm
(P. Wolfe, Number 1, June 1980)
* Algorithms: The influence offinite precision arithmetic
(W. Murray, Number 2, Oct. 1980)
* Mathematicalprogramming at Oberwolfach
(A. Bachem, Number 3, March 1980)
* Some roads hardly taken
(R. G. Jeroslow, Number 4, July 1981)
* Mathematicalprogramming activities in the USSR
(B. Korte, Number 6, March 1982)
* Atplay in thefields of scheduling theory
(E.L. Lawler et al., Number 7, June 1982)
* Testing the theory of evolution
(L.R. Foulds, Number 10, Aug. 1983)
* Theprogramming of(some) intelligence: opportunities
at the OR/AI interface
(R.G. Jeroslow, Number 14, Jan. 1985)
* Are you all salesmen here?
(E.L. Lawler et al., Number 19, Sept. 1986)
* Highlights ofMike Todd's research
(L.E. Trotter, Jr., Number 26, April 1989)
* Some comments on notation for quasi-Newton methods
(W.C. Davidon et al., Number 32, March 1991).

laudk Lemarifch(l r Ic ii'L I
Ill, 19 4PDaii : i.- I' :I I malillill tll
.,1 s L','- I l ,i ; /I'/,l l a d a ,-lll ll i
lyzing numerical methods in
nonsmooth optimization. The
prize was jointly awarded to
RogerJ.-B. Wets, who will be inter-
viewed in the next issue of OP-
TIMA. The Dantzig Prize is
awarded once every three years by
the Mathematical Programming
Society and the Society for Indus-
trial and Applied Mathematics to
recognize original, broad and deep
research making a major impact
on the field. Among Lemarechal's
many important contributions to
nonsmooth optimization is the so-
called bundle method. Next to his
theoretical work, he has been in-
volved in numerous collaborative
projects on applied optimization
problems. He recently published,
with J.-B. Hiriart-Urruty, the two-
volume book, Convex Analysis
and Minimization Algorithms
(Springer Verlag, 1993).
Lemarechal graduated from the
engineering school in Toulouse
(part of the French "GrandeEcole"
system) and received his Ph.D.
degree from Universite de Paris IX,
Dauphine. In 1971 he received an
appointment at INRIA (Institut
National de Recherche en
Informatique et Automatique) le
Chesnay, France, where he is still

I 1' f11.1. ,_ l .;Iu 1hit ii n I I t i
liI I "r ii l 1 il0 110 1 Ihlllth
I/ (l lllll11: h 'll )
CL: I went to INRIA just as I had
graduated. It was an institute created
in 1967 for work in applied math-
ematics, and more specifically, on
problems coming from industry. At
the start it was a very small institute.
We were about 10 people, including
the housekeepers and the concierge!
I first worked on partial differential
equations and ordinary differential
equations. Then I moved to optimi-
zation, and I was lucky.enough to get
an application from the glass com-
pany, Saint Gobain. It concerned the
management of a glass oven and
could be interpreted as a multi-prod-
uct, multi-machine lot-sizing prob-
lem. It was very well formulated. All
the equations of the model were
present, so for me it was "only" a
matter of finding an appropriate al-
gorithm. However, it was a nonlin-
I. 1 r -, i-'. ] i ,. .bhIn. fr- hiI.,
classical methods were not really
suitable. It took me awhile to analyze
it since Iwas quite fresh in optimiza-
tion. The problem was decompos-
able, so Lagrangian relaxation was,
in principle, suitable, but it was not
convex, so the Lagrangian dual was
nondifferentiable. At that time I
didn't know that nonsmooth optimi-
zationexisted. Iwas,however, lucky
to find Lasdon's book, Optimization
Theory for Large Systems,
1.i..'.1l!., ll 1970), which has got a
very good introduction to elemen-
tary convex analysis in an appendix.
This appendix contained all the
material necessary to understand
what can be done in this subject.

--- -- ---- --- --I

N? 45

MARCH 1995


I'AGL 3 N~45 MARCH 199

The glass company project was the
origin of the development of bundle
methods. It took me about one year
to find the appropriate solution
scheme, and when I had finished, I
went back to Saint Gobain and told
them very proudly "OK, I have got
the solution," but they told me,
"What? We don't care about this
p ,.,l 1 .1 ` !. 1, ,. I i -t i r i.- t ...... J
and second, there was a mistake in
the formulation anyway." If the mis-
take had not been present, the prob-
lem would have become perfectly
convex with a nice dual, so all this
research would not have been car-
ried out. So, it was a "happy mis-
OPTIMA: One of your most
important contributions to
nonsmooth optimization is
the bundle method. Could you
describe the principle behind
this technique?
CL: There are two ways of describ-
ing bundle methods. The way to
explain it that appeared at that time,
i.e., about 20 years ago, is as follows.
If you minimize a smooth function,
you first need to find a descent direc-
tion. If the function is smooth, i.e., if
the gradient exists, it suffices to move
in the direction opposite to the gra-
dient, or even anywhere in the half-
space opposite to the gradient. If,
however, the function is nonsmooth,
instead of having one gradient, you
have a bunch of generalized gradi-
ents-or subgradients-which form
the so-called subdifferential. If you
generate a subgradient (by an
oracle), you are not sure that the
direction opposite to this
subgradient is a descent direction.
Whatyou need to do to find a descent
direction is to consider the set that
forms an obtuse angle to all
subgradients, which is a set much
smaller than the half-space opposite
to the subgradient. In the bundle
method you start by computing one
subgradient. If the opposite direction
is a descent direction, it is all right,

but if it is not, there is a way of forc-
ing the oracle to compute a new, dif-
ferent subgradient that forms an
acute angle with your direction. You
then pivot your direction so as to
have an obtuse angle with both
subgradients, and so forth. You can
prove that either your starting point
is optimal, or eventually, after
enough such pivots, you really get a
descent direction.
One more comment: this approach
was "invented" simultaneously by
Phil Wolfe and myself, but Phil be-
haved wonderfully towards me.
While I was very young and totally
unknown, he immediately moved
aside, and then he started to put my
own work continuously to advan-
tage, advertising it at every occasion.
I am also very grateful to Michel
Balinski, who had a decisive influ-
ence on my career, helping me and
encouraging me from the very begin-
ning. Then Bob Mifflin was the
"worker of the first hour." Later on,
K.C. Kiwiel, J.J. Strodiot, J. Zowe
joined us (in alphabetical order!), as
well as several others. It is my plea-
sure to acknowledge the important
role they all played in the develop-
ment of the field.
OPTIMA: In which direction
did your work proceed after the
Saint Gobain application?
CL: What happened then, in the mid-
'70s, was that we quite rapidly found
appropriate algorithms which were
rather satisfactory for optimization
of convex functions. We were also
able to generalize these algorithms to
nonconvex functions. On the other
hand, my point was, and this I feel
very strongly about, that work in
nonsmooth optimization should
keep close to smooth optimization.
You should not become an expert in
a difficult subject before you are an
expert in the easy subject in which the
difficult one is contained. Therefore,
I did work on smooth optimization.
Furthermore, I considered the re-
search in nonsmooth optimization to

have reached a kind of equilibrium,
by the development of appropriate
software that can solve significant
problems, and on top of this, appli-
cations were coming that involved
"ordinary" optimization. However,
I did not totally abandon my spe-
cialty; I kept an eye on the
nonsmooth world.
OPTIMA: Which are currently
the main issues in nonsmooth
CL: On the purely theoretical side, a
S i , l -,I l ,.t..i, i'- L ,1 l- 1' .lw to
properly generalize the concept of
second derivative to the case of func-
tions which are convex but which
have no first derivative. The motiva-
tion is to design algorithms that are
really extremely fast (those existing
at present may be slow; there is a
reason for that). I have been asking
myself this question for about 20
years, and I must confess that no
substantial progress has been made.
On the application side there are
several issues. It would, for instance,
be a good thing to develop compu-
tational codes especially tailored to
Lagrangian relaxation, enabling the
resolution of large-scale real-life
problems (Lagrangian relaxation is
one of the main applications of
nonsmooth optimization). This is,
however, fairly difficult since large-
scale problems involve broad exper-
tise, almost by definition. Develop-
ing such codes involves joint work,
not only between applied mathema-
ticians but also with people from the
application world as well. This is
why there has been very little devel-
opment of software for decompos-
able systems so far. For instance, I
have for several years been in contact
with EDF (the French Electricity
Board), and progress is very slow.
OPTIMA: If you look at the
real-world problems that you
have studied, would you say
that there are tools available
for solving them?

CL: I would say that the tools are not
too bad. What I would like to have is
excellent tools, which is much more
OPTIMA: What is the major
deficiency of the present tools?
CL: Speed of convergence. From the
theoretical point of view it is very
difficult to develop excellent tools. It
has actually been proved that it is im-
possible! The complexity theory of
Nemirovskii (Nemirovskii and
Yudin, Problem Complexity and
Method Efficiency in Optimization,
W iley,19- ,,. ...J i l! ... . .. ,
not exist any fast algorithm to mini-
mize a general convex function.
Therefore, if we really want "excel-
lent tools," we are forced to use heu-
ristic methods. This negative theo-
retical result is extremely important
as a safeguard to prevent people
from trying to prove theorems that
have got to be wrong. Unlike the tra-
ditional theory of computational
complexity where it is still not
known whether P is equal to NP,
Nemirovskii's result tells you thatfor
sure such algorithms do not exist!
The same theory tells you also that
one particular variant of a bundle
method is optimal in a certain sense:
its rate of convergence cannot be
made faster independently of the
number of variables. These results
are true only if you consider the gen-
eral class of convex functions; for
some particular subclasses, the situ-
ation may change.
OPTIMA: Does the community
aim at developing a general-
purpose code for solving generic
nonsmooth problems?
CL: As far as solving generic
nonsmooth problems, the existing
codes are all right, at least theoreti-
cally. On the other hand, it is ex-
tremely difficult to develop codes for
specific problems. The reason is sim-
ply that we do not know how to clas-
sify convex functions. Let's consider
the generic problem of minimizing
convex functions given the oracle


N? 45

MARCH 1995



that computes the subgradient. We
don't know right now which sub-
classes of convex functions are easy
in the sense that they can be mini-
mized quickly with the help of this
oracle. Maybe some day we will
know what kind of functions are
easy, but right now it is impossible to
OPTIMA: In this way it is hard
to know exactly what you are
developing for.
CL: Right! We have got to develop
general codes, but we have no idea
what particular problem will be
more amenable to a specific method.
Therefore, we cannot develop the
specific method.
The big open question is the one that
I mentioned earlier.
OPTIMA: You mentioned
earlier that a very important
open question is how to general-
ize the second-order concepts
from classical calculus. If you
had such a generalization, what
would the consequences be?
CL: The aim would be to have
algorithms that, due to Nemirov-
skii's result, would not be
superlinear, but at least, based on
some serious grounds, likely to be
very fast. Alternatively, this generali-
zation might help us to say which
problems are easy, i.e., amenable to
fast convergence.
OPTIMA: Which areas provide
input to, or interact with,
nonsmooth optimization?
CL: The introduction of convex
analysis into the field, more deeply
than has been the case before, is one
of the new inputs that can be useful.
Here I think more of the introduction
of convex analysis in numerical op-
timization, rather than on the theo-
retical side. Convex analysis is, of
course, well-known on the theoreti-

cal side for optimality and stability
conditions, but for algorithmic de-
velopment it is relatively new. I also
feel strongly about the harmony be-
tween smooth and nonsmooth opti-
OPTIMA: What is your feeling
about possible future develop-
ments in nonsmooth
CL: For the future, there is another
field of application of nonsmooth
optimization, which has just been
born but is growing very fast,
namely, eigenvalue optimization.
You have a symmetric matrix that
depends on certain parameters and
you want to adjust these parameters
such that the maximal eigenvalue
becomes as small as possible. This is
a nonsmooth problem occurring in
both so-called robust control, which
is an extremely expanding subject in
optimal control, as well as in combi-
natorial optimization (see, e.g., C.
Delorme and S. Poljak, "Laplacian
Eigenvalues and the Maximum Cut
Problem," Mathematical Program-
ming62 (1993) 557-574, or L. LovAsz,
"On the Shannon capacity of a
graph," IEEE Trans. Information
Theory 25, (1979) 1-7). There are al-
ready some approaches available for
this problem, such as a very recent
one by Nemirovskii based on interior
point methods; see also the work by
the team of S. Boyd at Stanford (In-
formation Systems Laboratory). This
technique is extremely powerful but
rather heavy. Convex optimization
would, hopefully, result in a more
flexible and versatile method even
though the speed of convergence
would be more modest.

I also think that there will be an ex-
treme need for development of algo-
rithm s for q1i .i I ., [. .ti rtn, i ,
and this leads me to a second way of
describing bundle methods. Think of
the Dantzig-Wolfe algorithm where
you solve a sequence of linear pro-
gramming problems. The bundle
principle can be viewed along the
lines of just adding a quadratic term
to the Dantzig-Wolfe master pro-
grams. This has the beneficial effect
of stabilizing the Dantzig-Wolfe
mechanism, which is highly un-
stable. Each master program will
now be a quadratic problem instead
of a linear one. So, the more we want
to have software for large problems,
the more we will need efficient soft-
ware for solving quadratic problems.
OPTIMA: What is, in your
opinion, the situation in the
field of general nonlinear
pI 11,o1n' in..1?
CL: I feel that nonlinear program-
ming in general needs fresh blood.
One proof to me that this is the case
was, for instance, that when the ellip-
soid method was developed, it cre-
ated a lotofnoise. Next, it was the de-
velopment of interior point methods,
which now very much dominate the
literature in the field. This was to me
a sign that the community was in
need of a new idea and that new fresh
blood is needed which, in a sense, is
a bad sign for the nonlinear commu-

Moreover, I think that the general
work that has been done in nonlin-
ear programming the last 40-50 years
has basically come to an end. There
are few important theoretical devel-
opments to foresee. I feel that the
only really important developments
are going to be at the computational
side, like the improvement of the
simplex method for linear program-
ming over the last decades. I also
believe that for the field to revive,
new ideas are needed; one is the use
of convex analysis, which is why I am
so excited about it.
I agree with John Dennis who says
that we should work closer to the
applications because research in this
field has to be motivated by applica-
tions. I am extremely proud that my
own research has started with a spe-
cific application. I consider myself
very lucky for this reason. In general,
it L, ..1 !-ui t u Ii because-these-applica-
tions, of course, come from people
who are totally outside our world,
and we have no common language.
They express themselves in a way
that is extremely hard for us to un-
derstand and conversely, of course.
Therefore, it requires a lot of gener-
osity from both sides-and, in par-
ticular, from our side!


N 45

MARCH 1995


'AGE 5 N~45 MARCH 1995

ick den Hertog and
Jiming Liu were two of
the three finalists for the
A.W. Tucker Prize which
is awarded by the Math-
ematical Programming Society
for an outstanding paper by a
student. The third finalist, and
winner, was David P. William-
son who was interviewed in the
previous issue of OPTIMA.
Dick den Hertog received both
his undergraduate and his Ph.D.
in computer science and applied
mathematics from Delft Univer-
sity of Technology, Delft, The
Netherlands. His thesis "Interior
point approach to linear, qua-
dratic and convex programming-

OPTIMA: How did you choose
your Ph.D. topics?
DDH: I also worked on interior point
methods for my Engineering degree,
and I really liked it, so Icontinued the
work for my Ph.D.
JL: After I got my bachelor's degree,
I went to another institute for three
years as a computer programmer.
After a while I got tired of comput-
ers, so when I decided to take a
master's degree, I wanted to choose
a topic between computer science
and applied mathematics. That is
why I chose computational math-
ematics. In 1990 I tried to apply to
some schools in the U.S., and even-
tually I got financial support from
George Washington University. I


Algorithms and complexity"
was supervised by Kees Roos.
He currently has an appoint-
ment with the consultancy firm
Center for Quantitative Meth-
ods (CQM), Eindhoven, The
Jiming Liu received his under-
graduate degree in combinato-
rial optimization from Beijing
Institute of Technology. He is
completing his Ph.D. thesis at
George Washington University,
Washington D.C., supervised by
Anthony Fiacco. His submission
to the Tucker Prize competition
consisted of five papers on sta-
bility and sensitivity analysis of
generalized equations and varia-
tional inequalities.

already knew of Anthony Fiacco
since I had studied his book with
McCormick (A.V. Fiacco and G.P.
McCormick, Nonlinear Program-
ming, Sequential Unconstrained
Minimization Techniques, Wiley,
1968) already in China. When I came
to V .I l..-i hin. in ProfessorFiaccowas
interested in stability and sensitivity,
so he asked me to write some papers
with him. So I started very early, just
after beginning my Ph.D., to write
papers. Later on, I started to write on
my own. Some of this work is follow-
ing new directions and some is ex-
tending my work with Fiacco.
OPTIMA: Within your field of
research, what is the most cen-
tral result?

DDH: Of course, Karmarkar's work
is the most important one. It was the
root of all developments in the field
of interior point methods. Besides
that, I think the book of Fiacco and
McCormick was very stimulating for
me. My research mainly concen-
trates on the path-following meth-
ods, which forms a subclass of the
interior point methods, and the path-
following techniques go back to the
book of Fiacco and McCormick.
Also, the book of Nesterov and
Nemirovskii (Y.E. Nesterov and A.S.
Nemirovskii, Self-concordantFunc-
tions andPolynomialTimeMethods
in Convex Programming, Central
Economical and Mathematical Insti-
tute, USSR Academy of Science,
Moscow, USSR, 1989), published
later,i. :a- .m im l ..It n I I l.- .,1, I,. r
me. Here I also want to mention that
the work by, for instance,
Anstreicher, Gonzaga, Ye and, last
but not least, my supervisor Kees
Roos has all been very important to
me. When I started to work on inte-
rior point methods, Kees Roos gave
me his articles, and they stimulated
me to start working in this field. For
me, Karmarkar is the "father" of the
interior point methods, but I should
be careful here! Some of the interior
point methods go back to the'60s, but
at least he gave the impetus that
started an important development.
JL: Can I saysomethinghere?As I see
this whole development, I think a
main contribution of Karmarkar is
that he in some sense pointed out that
the simplex method is not the only
DDH: But Khachian had already
pointed that out, so I don't think it is
as simple as that.
JL: But Karmarkar was the first one
to really challenge the simplex

DDH: If you look at the implemen-
tation of interior point methods to-
day, they are very different from
Karmarkar's. So, it' -., 1 difficult to
check if the claim that Karmarkar
made at the time is really true, at least
for the variant that he proposed.
OPTIMA: So, let me ask you,
Jiming, what the central results
are in your area.
JL: I think that Fiacco and Steve
Robinson pioneered this field. At the
beginning they made very important
contributions. Fiacco proved some
very basic sensitivity results, but
these results do not take care of the
nonsmooth case. I believe that Steve
Robinson was the first one to come
up with a very nice idea about how
to deal with the nonsmooth charac-
teristics of the Karush-Kuhn-Tucker
OPTIMA: What would you con-
sider to be your own biggest
DDH: I developed a machinery to
prove complexity results for path-
following methods, also for long-
step methods, which could easily be
extended to the nonlinear convex
case. To obtain these results I used
many results of others.
JL: I think I wrote a couple of papers.
In three years there were about 20,
and these papers cover a lot of areas,
but most concern stability and sensi-
tivity analysis.
OPTIMA: Can you give an
example of such a result?
JL: Actually, it's kind of technical!
For instance, Steve Robinson proved
a very beautiful result which says
that if we have the MFCQ and gen-
eral second-order sufficient condi-
tions, then the perturbed solution is
up-Lipschitz continuous, which is
the basic property we want. Since my
background is half computer science
and half applied mathematics, I al-

N? 45


MARCH 1995

PAG 6h N045MRC 19

ways want to deal with something
we can compute, so I developed a
theory trying to estimate the
Lipschitz coefficient. I believe this
work is important in practical situa-
tions. Another important contribu-
tion is that I proved a very nice result
for variational inequalities. I provide
a necessary and sufficient condition
for strong stability and I think that
result is very cute.
OPTIMA: Which are the re-
search questions that you would
like to study next?
DDH: Concerning interior point
methods, I would like to try to get
some further complexity results. For
linear programming problems, the
most efficient methods are the




primal-dual ones. If you look at the
nonlinear programming problems,
then we can get good complexity re-
sults only for the path-following
methods. So, I think it is a nice area
here to prove some complexity re-
sults for the primal-dual methods for
nonlinear problems. In practice these
methods have proved to be very ef-
ficient, but the theory is not yet there.
JL: I am trying to integrate different
algorithms in optimization. My gen-
eral point of view on optimization is
that here, I talk about nonlinear pro-
gramming problems since I wrote
most of my papers on nonlinear pro-
gramming; this subject has reached
maturity. But one development
would be the following: Suppose we

Associate Professor


Department of Operations Research


are dealing with some kind of gen-
eral nonlinear programming prob-
lem-because my feeling is that users
usually don't want to provide special
structure. Then assume that for a
given algorithm this problem has got
a good complexity result. Maybe for
another class of problems this
.Il...rir I ..i:..m n .' i ,'is veryhigh.
For a second algorithm we probably
have another distribution of com-
plexity results, and so on. So, the
point here is first of all: How can we
identify which type of algorithms are
suitable for which class of problems?
Of course, this is difficult. I have been
doing a lot of programming, and I
studied a lot of algorithms, so I have
coded many algorithms to see which
algorithm works best for a given

The Department of Operations Research, Stanford Uni-
versity, is seeking applications for the position of Associate
Professor (Research) in connection with modeling and algo-
rithm development work in the Systems Optimization
Laboratory. The successful candidate will collaborate with a
. ,11, l group conducting research on planning under uncer-
tainty. The appointment may be held for a period of up to
six years, depending on performance and the i iil I1 .h of
research funding. This is a non-tenure track, externally-
funded position, which is not renewable. Initial external
funding to support this position is in place at the moment.
Applicants should display excellence in research through
publications in the stochastic mathematical programming
area. In addition, they should possess extensive experience
in modeling and the implementation of large-scale algo-
Letters of application and resumes should be addressed to:
SOL Search Committee
Department of Operations Research
Stanford University
Stanford, CA 94305-4022
The deadline for receipt of applications is May 1, 1995.
Stanford University is an affirmative action/equal oppor-
tunity employer and especially welcomes applications
from women and minorities.

DDH: Best for your implementation
JL: Right... it's not mature yet, but my
current results are very promising,
especially for unconstrained optimi-
OPTIMA: Is there some particu-
lar event from your time as a
Ph.D. student that you remem-
ber particularly well?
DDH: I remember several occasions
when the whole interior point
method group at Delft was very ex-
cited about new results which we
obtained. I also remember many
phone calls with my supervisor,
Kees Roos, late in the evenings, when
one of us suddenly discovered some-
JL: It's a happy occasion to obtain a
new result. I also remember once I
wrote a paper, and I discovered later
that the basic assumption was
wrong! It was very embarrassing.
But mostly I remember the happy
OPTIMA: What are your plans
for the near future?
DDH: I like it very much to work in
the consultancy firm. It is very enjoy-
able to talk to people about their
problems and to solve them by OR
techniques. I am actually very aston-
ished that these techniques can be so
useful in practice, because I heard so
many negative stories. I don't think
they are true. But I still want to stay
in touch with the academic world
and to publish papers, mainly on
practical work. I was also surprised
that these two years at CQM gave me
a lot of very good ideas on what I
would like to work on in the future.
What I would like to do is to work
about four years at CQM, then two or
three years at a university, four years
CQM and so on. That would be ideal!
JL: I prefer to get an academic job, but
the market now is very tough, and I
don't think it will improve in the next
two or three years. I just have to see
what happens!

I -


N 45

MARCH 1995



In their paper [1], the group mod-
estly describes the technique as fol-
lows: "In 1954 Dantzig, ,,II ..I..
and Johnson [2] showed a way to
solve large instances of the TSP; all
that came afterward is just icing on
the cake. The purpose of the present
paper is to describe some of the icing
we have added on top of the previous
layers. Our icing comes in five fla-

a. l -\ppkpiic, Bob Bixby,
\ .L ( .l u .1 Hill Cook have
...I i, ... i 97 in the
i l'i ii h. -.... i .r instance is
. t, l,.. I -. r .... .. .r w which an
. .I i"." I,! .h .. .. I .... 'n. T he pre-
. 1-- 4 I . .. (instance
tn!, ,. I ,.'. i, ,r I I B)-was held
[. i ... ,. ,. ,.-
. pl.b .. ' ,. ~r .. I.. Igues started
their first runs on a 3,038-city in-
stance in January 1992, using about
sixty workstations and a computer
program based on known techniques.
After monitoring the growth of the
branch-and-cut tree, they realized
that new tricks were needed. By April
1992, they had come up with suffi-
ciently good tools to solve the 3,038-
city instance. Since then they have
improved their techniques further
and solved several previously un-
solved instances from the TSPLIB.
The 7,397-city instance was actually
solved twice. The first time was on a
network of Sun Sparcations at
Bellcore, taking the idle cycles from
machines at night and weekends. The
second time, to write a certificate of
the optimal solution, was with the ar-
ray of Silicon Graphics machines at
Silicon Graphics' Center in Houston.
The array consisted of 10 Challenge
XL machines, each having 20 proces-
sors and 2 gigabytes of memory.

(i) new ways of finding cuts,
(ii) new ways of handling the
LP relaxations,
(iii) new ways of selecting an
edge on which to branch,
(iv) new ways of finding an
incumbent tour,
(v) solving the problem in
parallel on a network of
UNIX workstations."
Along with a comprehensive descrip-
trc.n .-f r-h "Li-re-r lI r h nFr ing trhe"
present a nice historical account ot
the computational development since
Dantzig, Fulkerson and Johnson.
The paper is available by anonymous
ftp from netlib.att.com in the direc-
tory netlib/att/math/applegate/TSP.
The certificate of optimality of in-
stance pla7397 can be found in the
same directory.




SO -I /

The "LaserLogic"
process was de-
vised at the
AT&T Bell
Laboratories fa-
cility in Cedar




Ul-L, : .' J- J 1
sey, in 1986. At
the time, it seemed as if it might well
be competitive with other methods
for producing custom chips. For the
technique to work, it was necessary to
be able to get good solutions to the
TSP in under 20 minutes on their
workstations. Given the speed and
memory capacity of the workstations
at the time and the size of the largest
instances, it was not possible to use
very sophisticated TSP heuristics.
However, Jon Bentley (of AT&T Bell
Labs' Computing Science Research
Center) and David Johnson did pro-
vide a fast implementation of the
simple nearest neighbor heuristic,
which easily met the time bounds
and produced solutions that were two
to four times shorter than the ones
initially used, which were simply to
cut the interconnections in the order
in which they occurred in the listings
produced by the CAD tools used to
design the circuits. The LaserLogic
process eventually proved


MARCH 1995

N? 45


The instance arose in an AT&T Bell
Laboratories application involving
programmable logic arrays (PLA's),
and was contributed, together with a
33,810 and an 85,900-city instance,
by David Johnson. Programmable
logic arrays are VLSI chips that are
:., i,, I; ,I-.i r ..! as a regular array
of interconnected components which
can be converted to custom inte-
grated circuits by using a laser to
vaporize specified interconnections.
( 'I" i i. the order in which the in-
terconnections are to be vaporized is
a more-or-less canonical traveling
salesman problem.

-TIM""U"N"09mali"m m

uncompetitive but, as David Johnson puts it,
"Nevertheless, it is nice to know, even after the
fact, precisely how much room for improvement
there was. For larger instances, we must settle for
slightly weaker estimates, such as those deter-
mined by the Held-Karp lower bound on optimal
tour length, which is typically between 0.5 and
1% below the optimal length. Had the
LaserLogic application arisen today, we probably
could have provided much better solutions within
the specified 20 minute time bound, although
,il not optimal ones. On a modern workstation,
nearest neighbor takes just 3 seconds to handle
the 85,900-city instance. Thus we might try
something like the ( I ''. --Wright savings heuris-
tic, which gets within 10.1% of the Held-Karp
lower bound for this instance in about 17 sec-
onds, or a fast implementation of 3-opt, which
gets within 3.7% in about 90 seconds, or even the
famous Lin-Kernighan algorithm, which gets
within 1.6% in about 7 minutes." For further de
tails see [3].

[11 D. Applegate, R.E. Bixby, V. ChvAtal and
W.J. Cook, "Finding cuts in the TSP," Prelimi-
nary report, August, 1994.
[2] G.B. Dantzig, R. Fulkerson and S.M.
Johnson, Solution of a large-scale traveling
salesman problem," Operations Research 2
(1954), 393-410.
[3] J. Bentley, L. McGeoch, D.S. Johnson a d E.
Rothberg Near-optimal Solutions to Veryarge
Traveling Salesman Problems (forthco ng).



The Committee

on Stochastic





COSP is an official committee of MPS, currently with 12 members:
Aharon Ben-Tal (Israel), John R. Birge (U.S.A.), Michael Dempster
(England), Jitka Dupaovi (Czech Republic), Yuri Ermoliev (Austria/
Ukraine), Kurt Marti (Germany), John Mulvey (U.S.A.), Andris
Prekopa (Hungary), Secretary Andrzej Ruszczyiski (Austria/Poland),
Tamas Szintai (Hungary), Chair: Stein W. Wallace (Norway), Will-
iam Ziemba (Canada).
The major activity of COSP is to organize an International Confer-
ence in Stochastic Programming every three years. So far, there have
been six such conferences: Oxford, England (Dempster, 1974),
Koszeg, Hungary (Prikopa, 1981), Laxenburg, Austria (Wets, 1983),
Prague, Czechoslovakia (Dupacovi, 1986), Ann Arbor, U.S.A. 'P.;...:
1989), Udine, Italy (Andreatta and Salinetti, 1992). The next meet-
ing will be in Israel in 1995. It is described separately in this issue of
OPTIMA. During its history, COSP has had two previous chairs,
namely Andris Prekopa and Roger J.-B. Wets. Until 1992, the
secretary was Jitka Dupacovi.
About one year before a conference takes place, the chair of COSP
appoints two committees, one to suggest the site of the next meeting,
the other to nominate a new chair and secretary of COSP. This com-
mittee may also nominate new ordinary members. During the confer-
ence a business meeting is held, and those present vote on the choice
of the next site and on new members. The list of names is sent by the
chair to the Council -.I Ii' t.., approval. The current committees
are chaired by Sen (committee members) and King (site).
COSP operates an e-mail list of scientists interested in stochastic pro-
gramming. Anything sent to cosp@iiasa.ac.at will be forwarded to
all addresses on the list. To be included in this list, please contact
Ruszczy'nski at rusz@iiasa.ac.at. This rather extensive list has been
used to communicate new results, report on computational experi-
ments, and gives news about positions, conferences, workshops, etc.
Another concern of COSP is the establishment of a database for sto-
chastic programming problems and the design (modification) of
standard input formats.
Karl Frauendorfer (frauendorfer@sgcll.unisg.ch) and David Gay
(dmg@research.att.com) are working on these issues. Frauendorfer
takes the main responsibility for the contents of the database, whereas
G i; '!1 run it; Gay already administers netlib's Ip/data and I .',.
erators collections (linear programming test problems).
Together they also consider the possibility of adding features to the
standard input format. Anyone with ideas on the input format or
with problems they think fit for the problem base, should contact
Frauendorfer or Gay. There will be a special session on these issues at
the next International Conference on Stochastic i',... I ,..'i'' in
Haifa, Israel.
Stein W. Wallace, Department of Managerial Economics and Opera-
tions Research, The Norwegian Institute of Technology, University
of Trondheim, N-7034, Trondheim.
Eithan Schweitzer, Faculty of Industrial Engineering and Manage-
ment Technion-Israel Institute of Technology, Haifa 32000, Israel.

-- ----


N? 45

MARCH 1995

PAGE 9 N0 45 MARCH 1995s





The Third National Conference on Optimiza-
tion, with about 100 participants, was held in
Xi'an October 5-10, 1994. At the conference it
was unanimously decided that a permanent
organization for mathematical programming
shouldbe created, which ledto the establishment
of the Chinese Mathematical Programming
Society. The councilwasformedwith YueMinyi
as president, Yu Wenci andZhang Xiansun as
vice presidents, and Han Jiye as secretary.
The purpose of the society is to promote the de-
velopment of the theory and application of
mathematicalprogrammingin China. To coor-
dinate with the activities ofMPS, it was decided
to hold triennialnationalsymposium thesame
years as the international symposia. Hence, the
next Chinese symposium willbeheldin 1997in
Wuhan, which is the capital ofthe Hubeiprov-
ince. In addition to the triennialsymposia, con-
ferences on special topics in mathematicalpro-
gramming will occasionally be organized. A
biannual newsletter will be issued in Chinese.

S Now for a few words about
COAL and the decision to
dissolve it: The Committee on
I Algorithms (COAL) was formed in
1973 by a handful of members of
Sthe Mathematical Programming
Society to develop guidelines for
the reporting of computational re-
sults and for the comparison of
optimization software. The guide-
lines were subsequently published
and adopted by Mathematical Pro-
gramming and Operations Research,
as well as other journals.
In order to keep the growing com-
munity of researchers informed of
new developments in computa-
tional mathematical programming
and to promote the goals of COAL,
a newsletter (later called the COAL
Bulletin) was started with the first
issue appearing in September 1978.
Initially, the newsletter was only
distributed to a -mall croup of r -
searcher- ii naed in so tn, are Jd -
velopm'ent bt %a-t n e entuail l enCr
to all mctr.,cL ei- %i lien the -!oci>t\
Agreed to i. C r thie pubbiaLatin n. i.
mailing co.t-.
In addition Io th. nei -letter,
COAL acil el\ prrnimoted increa ed
research on compuitatioial i-iue~,
and world to open more
channel- tr the publiic.-
tion of -ikch researh. (. O
This wai a.ionimliihed
by spon-oring as.e.-oln- at MIl'S
SORSA/TIMS, and SIAM meetings,
organizing small focused meetings,
Sand maintaining a library of test
problems and problem generators.
As the quality and significance of
Sthe contributions improved, the
Beale/Orchard-Hayes Prize was
created to recognize outstanding
developments in computational

optimization. As computational
issues became more and more an
integral part of the optimization
community and the optimization
literature, the members of COAL
reduced their activities as a group.
Then, at the MPS meeting in Ann
Arbor last August, COAL voted
unanimously to dissolve itself,
having succeeded in accomplishing
all of its goals. There will be a final
farewell issue of the Bulletin to
include contributions that were
in-hand at the time, an article on
the history of COAL and the news-
letter, and a master in-
S dex of all issues.
O Simultaneously, the
MPS Council suggested
F that an expanded
S OPTIMA can serve as
the focal point for com-
W municating with inter-
A ested members. There
are still many issues that
R need to be addressed
E and certainly the speed
of progress in comput-
inlg. both hardware and
ort% ire., call- for a forum to keep
abrieat ol dei. elopments. The Soft-
i\.ire ind Computation Department
%ili till lhi. \ oid.

mpt tat ion

Fa. I- khayyal, Software and
Computation Editor
School of Industrial and Systems
Georgia Institute of Technology
Atlanta, Georgia 30332-0205 USA
e-mail: faiz@isye.gatech.edu
phone: +1 404 8943037
fax: +1404 8942301



N? 45

MARCH 1995

PAC 10 N"l45 MARCH 1995~

c F. FF r[ I r

E ; : ,





N? 45

MARCH 1995

Montr6al, May 10-12
Seventeenth Symposium on
Mathematical Programming
with Data Perturbations
Washington, D.C.,
May 25-26, 1995
Ettre Majorana Centre for
Scientific Culture International
School of Mathematics "G.
Stampacchia", Erice, Sicily,
Italy, June 13-21, 1995
) 6th Stockholm
Optimization Days,
Stockholm, Sweden,
June 26-27, 1995
VII International Confer-
ence on Stochastic Program-
ming, Nahariya, Israel,
June 26-29, 1995
SConference on Optimization
'95, Braga, Portugal,
July 17-19
International Symposium
on Operations Research with
Applications in Engineering,
Technology, and Management
(ISORA), Beijing, Aug. 19-22,
International Workshop on
Parallel Algorithms for Irregu-
larly Structured Problems,
Lyon, France, Sept. 4-6, 1995
Symposium on Operations
Research 1995, University of
Passau, Germany, Sept. 13-15
AIRO '95 Annual Confer-
ence, Operational Research
Society of Italy, Ancona, Italy
Sept. 20-22, 1995
ICCP-95-lnternational Con-
ference on Complementarity
Problems: Engineering &
Economic Applications, and
Computational Methods,
Baltimore, Maryland, U.S.A.
Nov. 1-4, 1995
) Conference on Network
Optimization, Feb. 12-14,
1996, Center for Applied Opti-
mization, Gainesville, Florida
1 XVI International
Symposium on Mathematical
Programming, Lausanne,
Switzerland, Aug. 1997

AGE 11 No45 MARCH 1995

Montr6al, May 10-12
GERAD (Groupe d'itudes et de Recherche en Analyse des Decisions)
5255, avenue Decelles, Montreal, CANADA, H3T 1V6
Tel: (514) 340-6043; e-mail: jopt95@crt.umontreal.ca
FAX: (514) 340-5665

Seventeenth Symposium on
Mathematical Programming
with Data Perturbations
Washington, D.C.
May 25-26, 1995
This symposium is designed to
bring together practitioners who use
mathematical programming optimi-
zation models and deal with ques-
tions of sensitivity analysis with re-
searchers who are developing tech-
niques applicable to these problems.
Contributed papers in mathemati-
cal programming are solicited in
the following areas:
(1) Sensitivity and stability analysis
results and their applications. (2)
Solution methods for problems in-
volving implicitly defined problem
functions. (3) Solution methods for
problems involving deterministic or
stochastic parameter changes. (4)
Solution approximation techniques
and error analysis.

Clinical presentations that describe
problems in sensitivity or stability
analysis encountered in applica-
tions are also invited.
Abstracts of papers for presentation
should be sent in triplicate to Pro-
fessor Anthony V. Fiacco. Ab-
stracts should provide a good tech-
nical summary of key results, avoid
the use of mathematical symbols
and references, not exceed 500
words, and include a title and the
name and .,II mailing address of
each author. The deadline for sub-
mitting abstracts is 17 March
Approximately 30 minutes will be
allocated for presenting each paper.
Anthony V. Fiacco, Organizer
Sponsored by the Department of
Operations Research and the Insti-
tute for Management Science and
Engineering, School of Engineer-
ing and Applied Science, The
George Washington University,
I,.1 l ..i. D.C. 20052. Tel.
(202) 994-7511.

Ettre Majorana Centre for
Scientific Culture
International School of Math-
ematics "G. Stampacchia"
Erice, Sicily, Italy
June 13-21, 1995
This workshop aims to review and dis-
cuss recent advances and promising re-
search trends in the field of Nonlinear
Optimization concerning theory, algo-
rithms and innovative applications.
Both the finite and the infinite dimen-
sional cases will be of interest.
As usual, the course ,il be structured to
include invited lectures and i....- I -r... I
lectures. Proceedings including the in-
vited lectures and a selection of contrib-
uted lectures will be published.
Invited lecturers are:
J. Abadie, V. Demyanov, Y.G.
Evtushenko, M. Fukushima, L. Grippo,
J.J. Judice, O.L. Mangasarian, J.J.
More, J. Nocedal, J.S. Pang, P.M.
Pardalos, E. Polak, L. Qi, S.M.
Robinson, R.T. Rockafellar, R.
Schnabel, E. Spedicato, Ph. Toint
For details contact:
Prof. Gianni Di Pillo, Dipartimento di
Informatica e Sistemistica, Universith di
Roma "La Sapienza", via Buonarroti 12,
00185 Roma, Italy.
e-mail: erice@peano.dis.uniromal .it
FAX: +39-6-48299218

_____ ____ ____ I ____ ____ ____ ____ ___

VII International Conference
on Stochastic Programming
Nahariya, Israel, June 26-29,
The VII Conference on Stochastic
. be hosted by the
Technion-Israel Institute of Tech-
nology, and held in Nahariya, Israel
(near Haifa) on June 26-29, 1995.
The conference is I. .II. 1... by the
EURO XIV-14th European Confer-
ence on Operational Research, which
will be held at the Hebrew Univer-
sity, Jerusalem, July 3-6, 1995.
The Conference will take place at the
Carlton Hotel, Nahariya, Israel.
Nahariya is a lovely small resort town
on the Mediterrenean Sea, located 30
kilometers north of Haifa and about
140 kilometers from Ben-Gurion Air-
port. The weather in June-July in
Nahariya is about 28-30C by day and
about 20-22C by night. The hotel is
located at the center of the town with
shops and restaurants nearby. It pro-
vides :,ll service, air-conditioned
rooms with telephone, radio, T.V.
and video service, outdoor swimming
pool, sauna, night-club, bar, cafeteria
and a restaurant. The beach is within
walking distance.


6th Stockholm
Optimization Days
Stockholm, Sweden,
June 26-27, 1995
Theoretical, computational
and applied papers are wel-
come for the 6th Stockholm
Optimization Days, a two-
day conference on optimiza-
tion, to be held at KTH
(Royal Institute of Technol-
ogy) in Stockholm, Sweden,
June 26-27, 1995.
Sessions are planned
on various aspects of
optimization, includ-
ing nonsmooth F
optimization, linear
and nonlinear pro-


Invited speakers include:
A. Ben-Tal, Technion, Haifa, Israel
R.E. Bixby, Rice University, Houston, TX, USA
J. Desrosiers, GERAD, Montreal, Canada
P.E. Gill, UC San Diego, CA, USA
C.C. Gonzaga, Federal Univ. of Rio de Janeiro, RJ, Brazil
A. Griewank, Technische Universitat Dresden, Germany
D.W. Hearn, U. Florida, Gainesville, FL, USA
C. Lemarechal, INRIA, Rocquencourt, France
R. Mifflin, Washington State University, Pullman, WA, USA
W. Murray, Stanford University, CA, USA
A. Nemirovskii, Technion, Haifa, Israel
Yu. Nesterov, CORE, Belgium
M.L. Overton, Courant Institute, NY, USA
C. Sagastiztbal, INRIA,
SRocquencourt, France
A. Sartenaer, FUNDP,
Namur, Belgium
T. Steihaug, University
0 R of Bergen, Norway
Ph. Toint, FUNDP,
S Namur, Belgium
R. Vanderbei, Princeton
University, NJ, USA

Abstracts (maximum 200 words) should be sent
by May 1, 1995, (preferably by e-mail) to:
e-mail: optdays@math.kth.se
Optimization Days
Division of Optimization and Systems Theory
S-100 44 Stockholm Sweden
Fax: +46 8 22 53 20.
Further information may be obtained from the
addresses listed above.
The conference is financially supported by the
Goran Gustafsson Foundation and the Swedish
National Board for Industrial and Technical Devel-
opment (NUTEK). Organizers are Ulf Brannlund
(head), Anders Forsgren, Per Olov Lindberg and
Krister Svanberg from the Division of Optimization
and Systems Theory, Department of Mathematics,
Royal Institute of Technology (KTH).


N? 45

MARCH 1995

'AGE 11

PAG 12 ~ ~ ~ IR I No45 MACH 199

The main topics of the conference are:
*Stochastic programming, theory
and application
.Stability and sensitivity analysis
*Stochastic combinatorial
*Statistical approach to stochastic
*Stochastic approximation techniques
*Optimization of discrete event
(simulation) models
-Application of stochastic program-
ming to artificial r,. !,Io .- finance,
production, reliability, etc.
One hour, state-of-the-art tutorial
and review lectures will be presented
by invited speakers: J.R. Birge, A.
Gaivoronsky, G. Pflug, B. Polyak,
A. Shapiro and S. Zenios. The tuto-
rial and review lectures will be held
in special (non parallel) sessions.
Contributed sessions and presenta-
tions are welcome. The contributed
presentations are expected to be 30
minutes (including about 5 minutes
for questions and/or discussion).
Presentations will be grouped into
sessions of 90 minutes each, by topic.
S11... will be two or three simulta-
neous parallel sessions.
The social part of the conference will
consist of a reception, sponsored by
the Mathematical Programming Soci-
ety, a dinner at Rosh-Haniqra after
visiting the famous caverns formed
by the powerful waves of the
Mediterrenean, and a banquet on the
last evening of the conference. An ex-
cursion to Galilee on Friday, June 30,
1995, will be organized for partici-
pants who stay over the weekend.
The members of the international
scientific committee are: Z. Artstein
(Israel), M. Avriel (Israel), H. Ben-
Haim (Israel), A. Ben-Tal (Israel),
J.R. Birge (U.S.A.), M.A.H.
Dempster (Canada), J. Dupaeovd
(Czech Rep.), D. Elmakis (Israel),
Y. Ermoliev (Russia), A. Gaivoronsky
(Italy), G. Infanger (U.S.A.), A.J.
King (U.S.A.), P. L'Ecuyer (Canada),
T.M. Liebling (Switzerland), K.
Marti (Germany), A.S. Nemirovskii
(Israel), G. Pflug (Austria), A.
Prekopa (U.S.A.), B. Polyak

(Russia), S.M. Robinson (U.S.A.),
R.T. .... I ir. I r- (U.S.A.), U.G.
Rothblum (Israel), A. Ruszczyhski
ii'..1 ,,, I, S. Sen (U.S.A.), A. Shapiro
(U.S.A.), M. Teboulle (Israel), S.
Uryasev (U.S.A.), and S.W. .II .
The members of the organizing
committee are: Arkadi Nemirovskii,
Reuven Y. Rubinstein (Chair),
I ,l r..i Schweitzer of the Faculty of
Industrial r '-. i. ; i.. L and Manage-
ment at the Technion, and Marc
T i .. il.: of the Department of
Statistics and Operations Research
at Tel-Aviv University.
The last date for submitting abstracts
was March 1, 1995, but for more in-
formation and for registration please
contact the conference secretariat:
Mrs. Nilly Schnapp, Faculty of In-
dustrial Engineering and Manage-
ment, Technion, Haifa 32000, Israel.
Fax: 972-4-235-194 E-mail:
iernsO 1 @technion.technion.ac.il
The e-mail address of the organizing,
committee is: gosp@ie.technion.ac.il

Conference on
Optimization '95
Braga, Portugal, July 17-19
The main aim of the meeting is to
gather experienced researchers in
Optimization and to invite them to
describe their latest results, experi-
ences or applications in a t ,. ..11I,
atmosphere for an audience with an
expected large number of students.
The conference is organized by the
Optimization Group ofAPDIO -
The Operations Research Society of
Portugal. It will take place at the
University of Minho, in Braga, a city
50 km north of Oporto. The pro-
gram includes contributed communi-
cations and invited lectures in six
different themes by the following
(1) Linear Programming, D.
Shanno, Rutgers University, USA (2)
Nonlinear Programming, J. Dennis,
Rice University, USA (3) Global
Optimization, R. Horst, University

of Trier, Germany (4) Integer
Programming, L. Wolsey, CORE,
Belgium (5) Network i i..1.. ,
T. Magnanti, MIT, USA (6)
Complementarity and Variational
Inequalities, K. Murty, University of
Michigan, USA.
There is a limit of 60 contributed
talks. The conference language is
Prospective authors are requested to
submit an extended abstract (1 page,
1.5 space), by March 30, 1995, with
a cover page including the name of
the author, affiliation and address,
and key words. All the submitted
abstracts will be refereed by the Pro-
gram Committee. Authors will be
notified of acceptance by May 1, 1995.
Submit abstracts to:
Departamento de Produgao e
Escola de Engenharia Universidade
do Minho
4700 I- .., 1 ... .
Tel.: +351-53-604455
FAX: +351-53-604456 e-mail:



International Symposium on
Operations Research with
Applications in Engineering,
Technology, and Management
Beijing, Aug. 19-22, 1995
The symposium is intended to pro-
vide a forum for researchers working
in Operations Research who deal
with theoretical, computational and
applications aspects of optimization.
Optimization is understood in the
widest sense to include linear, non-
linear, stochastic, combinatorial, and
multiobjective systems. Papers pre-

sending original research in these ar-
eas are sought. Typical, but not ex-
clusive, topics of interest include:
Linear and nonlinear .. .......;
Combinatorial and global optimi-
zation Multiobjective optimization
SStochastic ... i,.....i .. Sched-
uling and network flow Queuing
systems Quality technology and re-
liability Simulation Optimiza-
tions in VLSI Neural network Fi-
nancial modeling and analysis *
Manpower planning Production/
Inventory control Flexible manu-
facturing systems Decision analysis
Decision support systems Micro-
computer software of OR methods.
Papers on real-world applications
will be *.- il appreciated.
Authors are requested to submit 5
copies (in F,! ; I!i of an extended
abstract of not more than 10 pages
by April 1, 1995, to one of the fol-
lowing addresses:
Professor Kan ( Il-, n Institute of
A .l ,.,t I ,,.! .,.....- u r 1 .- ,
Academy of Sciences, Beijing
100080, P.R. China; or Dr. Ding-
Zhu Du, Computer Science Depart-
ment, University of Minnesota,
Minneapolis, MN 55455, U.S.A.
The extended abstract should in-
clude the e-mail address of the con-
tact person. Authors will be notified
of acceptance or rejection by April
25, 1995. A camera-ready copy of
each accepted paper is required by
May 30, 1995. The conference wel-
comes any special session on the
above topics. The proposal for a spe-
cial session should also be sent to
one of the above addresses before
April 1, 1995. A formal proceedings
will be published and selected papers
will be put in a special issue of The
Journal of Global Optimization.
One author of each accepted paper
should attend the conference and
present the paper.
The symposium will be held at the
West Suburb Hotel, a three-star ho-
tel in the university area, 15 kilome-
ters from the center of Beijing. The
room rate is about US$35 per day.

-------~ ------~ _L

N 45

MARCH 1995



One day of : l. ....... to the Great
S II11 is included in the 7. _: I ,,;...
fee (US$300).
For information about the program,
registration and local arrangements,
please contact Kan Cheng at FAX
86-1-254-1689 or e-mail
ISORA@amath3.amt.ac.cn or
D.-Z. Du at FAX 1-612-625-0572
or e-mail dzd@cs.umn.edu.
Conference sponsor: The Asian-
Pacific Operations Research Center
within APORS and CAS.
Co-sponsors: The Institute of Ap-
plied Mathematics, Chinese Acad-
emy of Sciences; The Operations
Research Society of ( !,,-c The Na-
tional Natural Science Foundation
of China; The State Science and
Technology Commission of China

International Conference on
Complementarity Problems:
Engineering & Economic Appli-
cations, and Computational
The Johns Hopkins University
(Homewood Campus) Balti-
more, Maryland, U.S.A.
Nov. 1-4, 1995
The conference will bring together
for the first time engineers, econo-
mists, industrialists, and academicians
from the U.S. and abroad who are in-
volved in pure, applied, and/or com-
putational research of
complementarity problems, to
present and discuss the latest results
in this subject and to offer sugges-

Symposium on Operations Research 1995
University of Passau, Germany, Sept. 13-15
Sections at this conference include: Linear Programming; Nonlinear Pro-
gramming; Combinatorial and Discrete Optimization; Stochastic Models
and Optimization; Realtime Optimization; Scheduling; Control Theory;
Statistics, Econometrics; Macroeconomics; Mathematical Economics and
Game Theory; Neural Networks and Fuzzy Control; Simulation; Decision
Support and Information Systems; Banking, Finance, Insurance; Produc-
tion; Logistics; Transportation and Traffic; Inventory; Practical OR
(Application Reports); Decision Theory and Experimental Economics;
and Environmental Aspects.
Conference languages are English and German. The scientific program in-
cludes invited plenary and semiplenary lectures as well as contributed pa-
pers. Presentation of the latter is limited to 30 minutes including discussion.
The I.- ,, i for submission of abstracts is April 1, 1995.
Software i', i.. ,.. Participants are encouraged to present software solu-
tions for their contributions or software systems.
Mailing address for abstracts and further information:
Prof. Dr. P. :.l. I .r i.... Universitdt Passau, Wirtschaftswissenschaftliche
Fakultat D-94030, Passau
Tel. +49-951-509-339 e-mail: sor95@winf.uni-passau.de

tions for collaborative research and
further development of the field.
The conference will last four days
and will consist almost entirely of in-
vited presentations. There will be a
small number of selected contributed
talks, and the conference is limited to
100 participants (including the
speakers). A refereed volume of pro-
ceedings of the conference will be
There are three major themes of the
conference: engineering applications,
economic equilibria, and computa-
tional methods. Each theme will be
represented by experts in the area.
Topics to be covered in the confer-
ence are listed below.

AIRO '95 Annual Conference,
Operational Research Society
of Italy
Ancona, Italy, Sept. 20-22,
CALL 1995
AIRO '95 Prof. Ferdinando
F 0 R Pezzella Istituto di Informatica -
S- Facolth di Ingegneria Universith
S: degli Studi di Ancona Via Brecce
Bianche 60131 Ancona, Italy
Tel: +39-71-2204826
FAX: +39-71-2204474
e-mail: airo@anvaxl.unian.it

Engineering applications: Contact
mechanics problems, structural me-
chanics problems, nonlinear obstacle
problems, elastohydrodynamic lubri-
cation problems, traffic equilibrium
Economic applications: Applied gen-
eral economic equilibrium, game-
theoretic models, NEMS.
Computational methods: Pivotal and
path f.;.1.. methods, smoothing
techniques, quadratic programming
based methods, interior point meth-
ods, and projection/proximal based
methods; software development,
modeling language interfaces.
Contact one of the organizers for fur-
ther details if you are interested in
participating in the conference or in
contributing a paper for possible pre-
Michael C. Ferris, (on leave at) De-
partment of Economics, University of
Colorado Campus, Box 256, Boul-
der, CO 80309
Tel.: (303) 492-2651
E-mail: ferris@cs.wisc.edu
Jong-Shi Pang, Department of Math-
ematical Sciences, The Johns
Hopkins University, Baltimore, MD
21218 Tel.: (410) 516-7216
E-mail: jsp@vicpl.mts.jhu.edu

Conference on
Network Optimization
Feb. 12-14, 1996, Center
for Applied Optimization
University of Florida
Organized by Bill Hager, Don
Hearn and Panos Pardalos
The conference will bring together
researchers working on many differ-
ent aspects of network optimization:
algorithms, applications, and soft-
ware. The conference topics include
diverse applications in fields such as
engineering, computer science, op-
erations research, transportation,
telecommunications, manufacturing,
and airline scheduling.
Since researchers in network optimi-
zation come from many different
areas, the conference will provide
a unique opportunity for cross-
disciplinary exchange of recent
research advances as ..11 as a foun-
dation for joint research cooperation
and a stimulation for future research.
Advances in data structures, com-
puter technology, and development
of new algorithms have made it pos-
sible to solve classes of network
optimization problems that were
recently intractable. For example,
recent advances have been made in
techniques for solving problems re-
lated to airline scheduling, ir..lIr..
communication and transportation,
and VLSI chip design. Computa-
tional algorithms for the solution of
network optimization problems are
of great practical .l, ....
The conference will be held at the
Center for Applied Optimization,
University of Florida, Gainesville,
All presentations are invited. A col-
lection of refereed papers will be
published in book form by Kluwer
Academic Publishers. For further
details, please contact one of the
conference organizers.

- II ~---"- -------~---


N? 45

MARCH 1995



The Linear Complementarity Problem
by R.W. Cottle, J.-S. Pang, and R.E. Stone
Academic Press, Boston, USA, 1992
ISBN 0-112-192350-9
The linear complementarity problem (LCP) originated in the works of Cottle and others
as a unifying model for the study of linear and quadratic programming problems and
bimatrix games. Given an n x n real matrix Mand a real n-vector q, the problem seeks
a nonnegative vector x such that Mx+q is nonnegative and orthogonal to x. LCP and
its various generalizations, namely, the nonlinear complementarity problem, the gen-
eralized complementarity problem (over closed convex cones), the variational inequal-
ity problem, generalized equations, and the recent vertical (horizontal, mixed) linear
complementarity problem, have found many applications in optimization, economic
-, I rr -i',...i-'-i- .i .,,_,,, nn ... .. ,. ,, .i. ThestudyofLCPhasledto new ideas
and techniques for analyzing complex systems. For example, Scarfs computational
scheme for finding fixed points ofcontinuous mappings, Robinson's inverse and implicit
function theorems for nonsmooth functions, and (the somewhat recent) univalence
results forpiecewise affine functions have had their origins in the LCP theory. Although
basic existence, uniqueness, and stability questions have been answered and various
computational schemes have been proposed, there are still many interesting open
The book under reviewcontains an exhaustive account of the LCP. Of the seven chapters
in the book, the first two deal with introduction and background material. Chapter
3 deals with the existence and uniqueness aspects. In this chapter, various matrix classes
associated with the LCP are introduced and studied. Chapter 4 deals with the pivotal
algorithms for solving LCPs. Here, the algorithms due to Lemke, Cottle,
Chandrasekaran, and Van der Hayden are fully discussed. In Chapter 5, splitting and
L TJ ..... pr, ; .. . r; .-.. II. j, ,, .. .. I I. 1 1 1, ... I ,. I J... .I. I, I, I, I .., .... I -.
to the interior point methods. This chapter also deals with residues and error bounds.
Intricate geometric and degree-theoretic analysis ofLCP is covered in Chapter 6. The
final chapter deals with the stability and sensitivity aspects of the LCP.

"This extremely
well-written book
can be used either
as a textbook (at
the senior under-
graduate or at the
graduate level) or
as reference book."

---~ -- -----~


N 45

MARCH 1995
_... ...... .

MARCH 1995

_________ ________

The notes and references given at the end of each chapter outline the historical devel-
opment of the subject. The exercises vary from routine to .1, 1.. ;'.:. There is an
extensive bibliography. This extremely well-written book can be used either as a text-
book (at the senior undergraduate or at the graduate level) or as reference book.
This book is highly recommended to anyone interested in the LCP, linear and qua-
dratic programming, optimization, variational inequalities, and other related topics.

Interior-Point PolynomialAlgorithms in

Convex Programming
by Y. Nesterov and A. Nemirovskii
SIAM, Philadelphia 1994
ISBN 0-89871-319-6
Theappearance, !....... I .. r..i.. I. l' . r ... t ...'..i chapterinthestudy
ofcomplexity in mathematical programming, which has since resulted in the production
of over 1000 papers according to the bibliography [3]. Karmarkar's method had a slight
-I .. -, -I ,.j ,,, .. ... I I ,. ,L ,,,- 1 ~ ., l- r .. 1... I ,,,r .. . r! ....,,l .. .,1, 1
polynomial-time algorithm for linear programming at the time-KI ,. I,; ... ... 11.
,. 1 I ,.l- ..... I, .. I fY u d in and-N r. ,.. I- .1 c,.1 i . ...... . ... !
convex optimization. The computational developments since Karmarkar's paper, both
for interior-point and simplex methods, have l'.. ,,, ., nr ..I *. II- .11 1
in the lead article by Lustig, Marsten, and Shanno with commentaries on it appearing
in [5]. These developments, however, are not the subject of the present book, which
provides a truly comprehensive study of the foundations of interior-point methods for
convex programming.
Karmarkar used projective transformations and an auxiliary potential function in his
algorithm which was pi .. ... T' .. 1 I .... .....i : ... 1- .. .. in a rather restrictive
form. A large amount of effort went into .J..'. i.., -.TL i .J extending these ideas and
removing the restrictive assumptions over the next few years. Also, connections with
classical barrier methods and methods of centers were established, and the first path-
following methods, with a superior theoretical complexity bound, were developed by
Renegar and soon thereafter Gonzaga. These led to primal-dual algorithms and the
explosion of research referred to above.
At the same time as these developments, mainly concerned with complexity issues and
practical computation for linear programming, Nesterov and Nemirovsldi began their
path-breaking (as opposed to; ,r-I .-. 1,. I research into what the key elements of
interior-point methods were, what allowed polynomial complexity bounds to be estab-
lished, and to what general classes ofproblems such analyses could be extended. This book
is the result of five years of their investigations.
The key idea is that of a self-concordant barrier for the constraint set, a closed convex
set or cone in a finite-dimensional space. The notion of self-concordance requires that
the convex barrier function satisfy certain inequalities between its various derivatives;

of its second derivative, which defines a semi-norm at every point of the interior of the
convex set or cone. These conditions ensure, for instance, that Newton's method behaves

N? 45


Na 45

"This is a book that every
mathematical program-
mer should look at, and
every serious student of
complexity issues in opti-
mization should own."

nicely in a reasonably global sense. From such a barrier one ... -' r'. I..I .. :
methods, of either barrier or method-of-centers type; if the barrier satisfies an additional
propertynaturalfor a convexcone, oneobtainspotential-r. r, .....I I .I in,. I i1 .
the number of iterations necessary to obtain an e-optimal solution depends polynomially
on In (1/i) and 0, a parameter associated with the barrier.
Chapter 1 of the book provides a very useful overview of the ideas underlying the work
and the contents of each chapter. Then Chapter 2 contains the basic definitions and
properties of self-concordant functions and barriers, including the beautiful result that
every convex set in R" admits a self-concordant barrier with parameter 0 of order n.
C hapters3 and4areconcerned nl, 0 .1 ,I.t .. II .. n .. .. 111. i ..l, ,. .. i ... ,, ...
respectively and demonstrate that the r ,. I "'. '".- ..... I -..,, I .1 .. L convex
programmingproblem' :rl, ..,r1. .... .. ,;r withalinearobjective function) is the
knowledge of a self-concordant barrier together with its first two derivatives for the
constraint set, with a reasonably small value for its parameter. (The authors also show
concern for the practical efficiency of variants of their methods.)
Chapter 5 provides tools for constructing such barriers and several examples. While the
result quoted above assures the existence of a barrier with parameter of the order of the
dimension, such a barrier may not be easily computable. For example, the usual barrier
for a polyhedral set in R" defined by m inequalities is the standard logarithmic barrier,
with parameter m not n. On the other hand, the cone of positive semidefinite matrices
of order n, a set of dimension n(n+1)12, admits a barrier of parameter n. (This cone arises
frequently in important optimization problems.) ( hi .... I .. ,. I..... ... fthe
toolsdevelopedpreviouslyto .i. .,,,,,,. .. .. 1, ,l-.I ... ..... .I,. .... ii ,
methods for their solution. Chapters 7 and 8 address extensions to variational inequali-
ties and various acceleration techniques respectively.
This is a book that every mathematical programmer should look at, and every serious
student ofcomplexity issues in optimization shouldown. I.. ,..I.!.. .... .. 'II.....' ,
thefirstchapter, theintroductorym ar .1 .I i.. I.' ,I -.ir ,. l.i.l.I.._.,.,,.
cal notes at the end of the book can be read. For a more detailed study, a serious com-
mitment is necessary; this is a technically demanding tour-de-force. The authors provide
m motivation an d e .., i .... . i-,..i-.. ,,,, ,i,.| i. ir.i .. .. r.. i .. I, .
The reader is advised to skim forwards and backwards to help understand some of the
definitions and results. For example, the standard logarithmic barrier function -S In x
for the nonegative orthant is introduced on page 40 (with related barriers on pages 33
t..! .. r I ., r r Ip, I ri n,..r ,, ...,n ,, .! ll, .. I ,., I c concordantfunctions
are first defined on page 12. Likewise, a hint of the barrier-generated family on page 66
would assisting understandingthedefinitionofaself- ..,.....i, i, I .,, ,I ..,. I ,.l.' 58. The
authors' overview in Chapter 1 is also very helpful in showing the direction the argument
will take.
There seem to be very few misprints. One possibly confusing one appears in (2.2.16):
w2 should be wo in the numerator. And the excellent bibliographical notes were prepared
for an earlierversion of the book; the chapter-by-chapter remarks need the chapter numbers
increm ented byone.T . l.I,..: .i. ,r It. ... l, ,r l, in. I .- ..i.t r, r..i.. .r.
that have since appeared in print. Forc .nq ..1. :, ... ... il .i, .i n .i,. .,
unfortunately, this has not been ,r... ri, ...1 ,r.. I ...... the summer of 1993.
Before concluding, let me mention two subsequent reports. Giiler [1] has found a lovely
connection between self-concordant barriers for convex cones and their characteristic
functions, whichwereintroducedby .. ... i... i r.i b. f....rr .r ...l.. n, i, ..r..,. In
homogeneous cones. And Nesterov and the reviewer [4], by further restricting the class

S__ I

MARCH 1995


of cones and their barriers II. I 11 1 ....... important examples), have developed
long-step and symmetric primal-dual interior-point methods for certain convex prob-
lems, extending those for linear programming.
In summary, this is an outstanding book, a landmark in the study of complexity in
mathematical programming. It will be cited frequently for several years and is likely to
become a classic in the field.

1. O. Gdiler, "Barrier functions in inte-
rior-point methods," Department of
Mathematics and Statistics, University of
Maryland, Baltimore County, Baltimore,
MD, 1994.
2. N.K. Karmarkar, "A new polyno-
mial-time algorithm for linear program-
ming," Combinatorica 4 (1984), 373-
3. E. Kranich, "Interior point methods
for mathematical programming: a bibli-
ography," available from NETLIB: send

e-mail to netlib@research.att.com
containing the message "send intbib.tex
from bib" and/or "send intbib.bbl from
4. Yu.E. Nesterov and M.J. Todd, "Self-
scaled cones and interior-point methods
in nonlinear programming," School of
Operations Research and Industrial En-
gineering, Cornell University, Ithaca,
NY, 1994.
5. ORSA Journal on Computing, 6
I ''I 1ff.


Constructive Approximation

by R. A. De Vore and G. G. Lorentz
Springer-Verlag, Berlin, 1993
ISBN 3-540-50627-6
As stated in the preface, the book under review is to be seen as a modern version of G.
G. Lorentz: Approximation ofFunctions, NewYork, 19(.' i .., ..if, i ri, rhI -
same classical questions in approximation theory, a field also known as constructive
theory of functions. Of course, the authors have added many new results obtained since
the publication of that book, and the presentation is in a more general frame; never-
theless the choice of the topics considered has almost been unchanged, and some modern
aspects of approximation theory are only sketched or missing. New are, for example,
the theory of K-functionals and interpolation spaces, and the Ditzian-Totik approach
to algebraic approximation. Furthermore, there are three chapters (5, 12, 13) on splines,
approximation by splines and interpolation bysplines, altogether more than 100 pages
which is about a quarter of the book.
Another major part of about 100 pages is devoted to best approximation, in particular,
best approximation by algebraic and trigonometric polynomials as well as by splines.
Chapter 3 is concerned with the standard material, e.g., the general theory of existence
and uniqueness ofelements of bestapproximation, the Chebyshev theory, and the theory
of Haar systems. The "Central Theorems ofApproximation" .. ,I1.J 1./1 the authors,
namely Jackson-Bernstein-type theorems for best approximation, are given in Chap-
ters 7 and 8. The latter deals in detail with the influence of the endpoints in case of
algebraic approximation. Here the reader can find the classical results ofA.F. Timan
and V.K. Dzjadik as well as the more recent ones of Z. Ditzian and V. Totik.

__ _ _ ___ .1 _____ ___ ___ ___ ____ ___ ___ __

N? 45

MARCH 1995


N 45

"...we can recommend
this book to everyone
who is interested in the
topics dealt with and
who wishes to have a
presentation in modern
language of approxima-
tion theory."

Apart from this there are some minor chapters ,1;,,:.. with Lagrange, Hermite and
I.,, I. I, .," .... ..I ...' i ,, i.'. ,withapproximationbylinearoperators, inparticular
with the norms of projection operators and the Butzer-Scherer theory of commuting
operators (Chapter 9), I. ... ... ..... i i ... .. ..., ..... ...1 (Chapter 10), and
with saturation and Miintz approximation (Chapter 11).
Chapters 2 and 6 provide the necessary material on the function spaces used through-
out and on the moduli of smoothness and K-functionals. In particular, the equivalence
between moduli of smoothness and K-functionals in certain instances, Marchaud-type
inequalities and the extension theorems ofWhitney are given here. There is also a short
exposition of the so-,. .:1. i 0,q-interpolation spaces.
Let us also mention Chapter 1 which may serve as an introduction presenting, among
others, the celebrated theorems of K. Weierstrass (1885), L. Fej&r (1900) and S. Bernstein
(1912) on the approximation of continuous functions by polynomials.
All in all, the book is exactly what the authors state in the preface or, in other words,
"a modern presentation of selected topics in classical approximation theory." Most of
the material is covered by at least one of the textbooks on approximation theory which
were published up to .... d... ri, ....il .I.Achieser(1956), P.J. Davis(1961),
A.F. Timan (1963), I.P. Natanson (1964/65), E.W. Cheney (1966), G.G. Lorentz
(1966), or the book on splines by L.L. Schumaker (1981).
Of course, one cannot give an absolute answer to the question whether the choice of
topics is optimal or whether there was a need to write this book at all. This answer has
to be left to the reader, but we can recommend this book to everyone who is interested
in the topics dealt with and who wishes to have a .... .... ......1. ... I 1,........ : of
approximation theory.
Let us finally mention that the book contains an extensive bibliography, even if some
of the above-mentioned ,I.... 1 .... ;.. On the other hand, there is only a very
poor index, e.g., no entries mentioning the theorems of Bernstein, Weierstrass, Fejer
or Timan can be found, and the entries 'Fej&r operator' or 'Fourier series' only give a
hint to Page 3, although these topics are dealt with in detail in Chapter 11.

MARCH 1995



MARCH 1995
..-. ... ...- ..-- . .--.. -.... 1

MARCH 1995

The Faculty ofManagement, McGill University,
announces a tenure track position in Operations
Management beginning Sept. 1995. Apply in
writing to: ProfessorAlbertTeitlebaum, Faculty
of Management, McGill University, 1001
Sherbrooke St. West, Montreal, Quebec, Canada
H3A 1G5.iScholars interested in conducting
research in Germany may contact the Alexander
von Humboldt Foundation, Suite 903, 1350
ConnecticutAve. NW, Washington, DC 20036,
Phone: 202-296-2990, Fax: 202-833-8514.
#Deadline for the next OPTIMA is June 1, 1995.

I Don.Ild l i \ h e 'ni I 1 1 i
I d earn, i\ ull Edu

.lll L i l ,1 1.

The Netherlands
e-mail: aardal@kub.nl
Georgia Tech
Industrial and Systems Engineering
Atlanta, GA 30332-0205
e-mail: faiz@isye.gatech.edu
Department of Econometrics
Tilburg University
P.O. Box 90153
5000 LE Tilburg
The Netherlands
e-mail: talman@kub.nl
Elsa Drake, DESIGNER

Journal contents are subject to change
by the publisher.

Mail to:
The Mathematical Programming
c/o International Statistical Institut
428 Prinses Beatrixlaan
2270 AZ Voorburg
The Netherlands

Cheques or money orders should be
payable to The Mathematical Progr
Society, Inc., in one of the currency
below. Dues for 1995, including sul
to the journal Mathematical Progra
are HFL100.00 (or USD55.00 or D
or GBP32.50 or FRF300.00 or CHI
Student applications: Dues are /2 th
Have a faculty member verify your s
and send application with dues to ab


I wish to enroll as a member of the Society. My subscription is for
Society, Inc. my personal use and notfor the benefit of any library or institution.
teI enclose payment as follows:

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es listed
he above rates.
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