MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER
A return of the MPS Symposium to
Europe after nine years brought a
record 799 registrants and 656
scheduled presentations during five
busy days at the University of Amsterdam,
August 5-9. The meeting, headed by an organiz-.
ing committee of J. K. Lenstra, A. H. G. Rinnooy
Kan and A. Schrijver, featured an outstanding
scientific program, awarding of the Society
prizes, and a lively social program against the
backdrop of the infinitely varied life of the city.
Opening ceremonies included plenary talks by J.
Tindbergen,firstrecipientof theNobelPrize in Econom-
ics, and W. R. Pulleyblank who summarized progress
and trends in mathematical programming. The George
B. Dantzig Prize for original research with a major im-
pact on mathematical programming went to Martin
Gr6tschel and Arkady Nemirovsky. Three Delbert Ray
Fulkerson Prizes for outstanding papers in discrete
mathematics were awarded to Alfred Lehman; Martin
Dyer, Alan Frieze and Ravi Kannan; and Nikolai Mnev.
The Beale/Orchard-Hays Prize for excellence in compu-
was given to Irvin
Lustig, Roy Marsten
and David Shanno.
Reports by the vari-
ous prize committees
are in this issue of
PAGE IWO D
1991 Dantzig Prize
Martin Gratschel, Technische
Universitat Berlin and Zuse Institut,
Berlin and Arkady S. Nemirovsky,
CEMI, USSR Academy of Sciences,
Moscow were awarded the 1991
Dantzig Prize by the MPS and
SIAM during opening ceremonies
at the MPS Symposium. PAGE TWO o
BOOK REVIEWS 16-18
TOP: WR. PULLEYBLANK
BOrOM: J. TINDBERGEN [LEFL
CHAIRMAN GEORGE NEMHAUSER.
I __ __ -
PAG 2 umer hity-iv NOEMER 99
The MPS business meeting on
Wednesday featured the
awarding of the A. W. Tucker
Prize for outstanding student
paper to Michel X. Goemans of
MIT. Leslie Hall of Princeton
and Mark Hartman of the
University of North Carolina
were Tucker prize finalists.
Details on the work of all
Also at the business meeting,
MPS Chairman George
that the 15th Symposium
will be held at the Univer-
sity of Michigan in Ann
Arbor in August, 1994.
John R. Birge of the
Department of Industrial
and Operations Engineer-
ing will be program
TOP: DANTZIG PRIZE WINNER MARTIN GROTSCHEL
[RIGHT] JOINED BY A. SCHRIJVER AND L. LOVASZ.
BOTTOM LEFT: BEALE/ORCHARD-HAYS PRIZE WINNERS
DAVID SHANNO, ROY MARSION, AND IRVIN LUSTIG WITH
PRIZE COMMITTEE CHAIRMAN ROBERT MEYER [SECOND
FROM RIGHT]. BOTTOM RIGHT: ARKADY NEMIROVSKY
RECEIVES DANTZIG PRIZE FROM PRIZE COMMITTEE
CHAIRMAN T.L. MAGNANTL OPPOSITE PAGE: THE
SECOND FULKERSON PRIZE IS GIVEN BY COMMITTEE
MEMBER LOUIS BILLERA [LEFT) TO MARTIN DYER,
ALAN FRIEZE, AND RAVI KANNAN.
The awarding of two prize winners and the
selection of these specific individuals was
particularly attractive to the prize committee
for several reasons. First, by giving two
awards, two outstanding researchers whose
contributions go far beyond their own
individual contributions could be recog-
nized; they have been role models to other
researchers and have precipitated entire
streams of research within the optimization
community. Second, with these two
selections, contributions from both the
discrete and continuous domains of
mathematical programming could be
recognized. Third, these awards would be
the first time that the Dantzig Prize has
recognized work conducted by a researcher
from continental Europe and from the
The individual prize citations are
"Martin Gr6tschel has made numerous and
substantive contributions to the theoretical,
algorithmic, and computational aspects of
combinatorial optimization. As one of the
principal architects of the field of polyhedral
combinatorics, he has played an important
intellectual and leadership role in
S the emergence of polyhedral
combinatorics as a field with both
deep intellectual content and
P considerable practical utility.
Through a broad and impressive
range of investigations that touch
upon many aspects of theory,
computation, and application,
including such topics as the
traveling salesman problem, the
S linear ordering problem, statistical
physics, large-scale circuit layout
design, network design, and
combinatorial implications of the
ellipsoid algorithm, Martin
Gr6tschel has been one of the few
members of the mathematical
programming community who
rightly earns the title of 'man for
-- I -- I------~-----~
PAGE 3 number thike/llrt-Ive NOVEBER 199
1991 PRIZE RECIPIENTS
"Arkady S. Nemirovsky has been a pioneer
in the study of the complexity of continuous
optimization problems. His numerous
contributions have both defined an entire
new field of scientific investigation within
mathematical optimization and produced
innovative and significant algorithms and
methods of analysis. His contributions
include the conceptualization and analysis
of informational complexity for smooth,
convex and stochastic programming
problems, the development of new algo-
rithms such as the ellipsoid algorithm and
the assessment of the basic arithmetic
complexity for broad classes of convex
problems. Arkady S. Nemirovsky has been
an innovator, conceptualizer, and a brilliant
analyst, and as such has been leading
contributor to the field of mathematical
optimization in the past two decades."
In the process of reaching its decision, the
Committee also recommended that the
future Dantzig Prize Committees adopt the
view that the Prize be awarded for a body of
research that has had a significant impact
upon the field of mathematical program-
ming, and not for a single piece of work or a
singular contribution. This interpretation is
consistent with the credentials held by all
the previous recipients of the award.
T. L. MAGNANTI (CHAIR)
After lively discussions about the relative
merits of the many excellent nominees, the
Beale/Orchard-Hays Prize Committee is
pleased to announce that the 1991 prize has
been awarded to Irvin J. Lustig, Princeton
University, Roy E. Marsten, Georgia
Institute of Technology, and David F.
Shanno, Rutgers University, for their paper
"Computational experience with a primal-
dual interior point method for linear
programming," first published in 1989 as
Technical Report SOR 89-17 of the Depart-
ment of Civil Engineering and Operations
Research of Princeton University and
subsequently presented at the NY ORSA/
TIMS Meeting in October 1989 and pub-
lished in Linear Algebra and Its Applications,
Volume 152, pages 191-222, 1991.
This paper is the culmination of a body of
research on computational aspects of
primal-dual interior point methods for
linear programming, aspects of which were
also reported in the following earlier papers:
K.A. McShane, C. Monma and D. Shanno,
"An Implementation of a Primal-Dual Interior
Method for LP," ORSA JoC 1, 70-83 (1989).
I.C Choi, C. Monma and D. Shanno, "Further
Development of a Primal-Dual Interior Point
Method," ORSA JoC 2, 304-311 (1990).
I. Lustig, "Feasibility Issues in a Primal-Dual
Interior Point Method for LP," Math. Prog. 49,
Since its re-introduction in 1984, the interior
point method for linear programming has
been the subject of lively debate with respect
to its performance relative to the simplex
method. While this competition, which has
been an invigorating force in the mathemati-
cal programming area, fortunately still
continues, the thorough description in the
prize paper of its algorithmic approaches
and the free distribution to research
universities of the OBI code that imple-
ments these methods has succeeded in
firmly establishing the competitiveness of
the interior point method. In addition to
summarizing and re-interpreting some of
the techniques introduced in the earlier
papers for handling feasibility issues and
simple bounds, the prize paper presents a
method for handling free variables and
discusses the role of the barrier parameter
and the effects of preprocessing, scaling, and
removing variables. Finally, it presents a
convincing computational comparison with
MINOS 5.3, a widely distributed, state-of-
the-art simplex code. In order to limit
further stimulation of the interior-point/
simplex controversy, we conclude simply
with the observation that this comparison
certainly presents very good evidence that,
for large-scale linear programs, these
methods are worthy adversaries.
The Beale/Orchard-Hays Prize Committee
ROBERT R. MEYER (CHAIR)
PAGE FOUR M
PAGE 4 numbecl~r r tityfv NVMER'9
1991 PRIZE RECIPIENTS
Prizes in Discrete
Up to three Fulkerson Prizes are jointly
awarded by the Mathematical
Programming Society and the American
Mathematical Society every three years.
Both Societies appoint a committee that has
the task of making recommendations to the
Societies. The prizes are given for single
papers, not series of papers or books. The
term "discrete mathematics" is intended to
include graph theory, networks, mathemati-
cal programming, applied combinatorics,
and related subjects.
The 1991 Fulkerson Prize Committee
consisted of Louis J. Billera, Martin
Gr6tschel (chairman) and Paul Seymour.
The committee recommended three papers
for the award. MPS and AMS accepted the
recommendations. The laudations that were
submitted to MPS and AMS and that
describe the scientific achievements of the
prize-winning papers are listed below.
The Selection Committee recommends that
one Fulkerson Prize be given to Alfred
Lehman for his paper "The width-length
inequality and degenerate projective
planes" published in W. Cook and P.
Seymour (Eds.), Polyhedral Combinatorics,
DIMACS Series in Discrete Mathematics
and Theoretical Computer Science, Vol. 1,
1990, American Mathematical Society,
Alfred Lehman is Professor of Mathematics
at the University of Toronto, Toronto,
Among the prize-winning papers, Lehman's
paper is undoubtedly closest to the heart of
Ray Fulkerson's research. It is a fundamen-
tal contribution to combinatorial optimiza-
tion, one that Fulkerson would have greatly
admired. It brings our understanding of
clutters with the width-length inequality
almost to the same level as that of
Let us put Lehman's result in perspective.
One of the core problems in combinatorial
optimization is to characterize the linear
programs that have an integer optimal
solution. This problem seems to be out of
reach, at least for the time being. A more
specialized question is: What are the
matrices A such that the linear program max
cTx s.t. Ax < b has an integral optimum
solution for all vectors c and b? This leads to
the theory of totally unimodular matrices,
that was developed by Hoffman and
Kruskal (1956) and many others. These
matrices are now fully understood through
the work of Seymour (1980).
Another line of attack on the general
problem seeks for a characterization of those
0/1-matrices A for which the so-called
packing problem max cTx s.t. Ax < 1, x > 0
has an integral optimum solution for all
vectors c > 0. This area of research has
resulted in the theory of antiblocking
polyhedra developed by Fulkerson (1971,
1972) and the theory of perfect graphs and
perfect matrices. Alfred Lehman in a
seminal preprint of 1965, which was
published in Lehman (1979), characterized
the antiblocking pairs of polyhedra by a so-
called width-length inequality. Further
outstanding results are due to L. LovAsz
(1972) who solved the famous weak perfect
graph conjecture and to M. Padberg (1974)
who proved beautiful regularity properties
of the so-called almost perfect matrices or
equivalently the minimally
The related covering problem is to charac-
terize those 0/1-matrices A for which min
cTx s.t. Ax > 1 x > 0 has an integral optimum
solution for all c > 0. This is the problem the
paper we talk about addresses.
For the study of this problem, Fulkerson
(1970, 1971) introduced the theory of
blocking polyhedra. Cornu6jols and Novik
(1989) call the matrices for which this
integrality property holds ideal, and
Seymour (1990) refers to the clutters that can
be derived from the rows of ideal matrices
as clutters with the max-flow
As in the antiblocking case, Alfred Lehman
provided in his preprint of 1965 a width-
length inequality that characterizes ideal
matrices and clutters with the max-flow
min-cut property, respectively. It is natural
to ask for an excluded minor characteriza-
tion of ideal matrices, i. e., a characterization
of almost ideal matrices. This is the blocking
analogue of the "antiblocking question" of
determining minimal imperfect graphs or
almost perfect matrices. This blocking
analogue seems just as hard as the open
antiblocking problem, or even harder, for
there is not even a conjecture as to what the
answer might be. Almost no progress was
made for a long time despite the fact that
many prominent researchers of the field
worked hard on it. Even proving an
analogue of Padberg's theorem had eluded
researchers until Lehman's breakthrough.
In the paper for which the Fulkerson Prize is
awarded, Alfred Lehman made a funda-
mental and unexpected step towards a
decent characterization of almost ideal
matrices. Quite a variety of different almost
ideal matrices are known. For instance, in
the language of clutters, there is the set of
edge sets of all odd circuits of K,, the set of
all consecutive triples from eight vertices
arranged in a circle and several others. Yet
Lehman proved that all these almost ideal
matrices have certain very remarkable
regularities, except for one infinite family,
the so-called degenerate projective plane
clutters. Lehman's theorem is a deep
blocking analogue of Padberg's result on
almost perfect matrices.
Many people have wondered why
Lehman's 1965 paper was only published in
1979. The committee members recently
learned that Lehman was reluctant to
publish this paper despite the fact that it
contains really outstanding results. He felt
that the final touch to the theory was
missing. This is what the prize-winning
PAGE 5 number thirty-five NOVEMBER 1991
1991 PRIZE RECIPIENTS
paper is adding to the theory of
A number of papers have meanwhile
appeared that restate Lehman's result in
other mathematical languages, polish the
approach, provide different proof tech-
niques and view it from different angles.
Among these are papers by Cornudjols and
Novik (1989), Padberg (1990), and
G. Cormujols and B. Novik, "On the weak
max flow min cut property," Working Paper,
GSIA, Carnegie-MelIon University, Pitts-
burgh, September 1989.
D.R. Fulkerson, "Blocking polyhedra," in: B.
Harris (ed.), Graph Theory and Its Applica-
tions, Academic Press, New York, 1970,
D.R. Fulkerson, "Blocking and anti-blocking
pairs of polyhedra," Mathematical Program-
ming 1 (1971) 168-194.
D.R. Fulkerson, "Anti-blocking polyhedra,"
Journal of Combinatorial Theory B 12 (1972)
A.J. Hoffman and J.B. Kruskal, "Integral
boundary points of convex polyhedra," in: H.
W. Kuhn and A. W. Tucker (eds.), Linear
Inequalities and Related Systems, Princeton
University Press, Princeton, New Jersey, 1956,
L. Lovisz, "Normal hypergraphs and the
perfect graph conjecture," Discrete Math-
ematics 2 (1972) 253-267.
M. Padberg, "Perfect zero-one matrices,"
Mathematical Programming 6 (1974) 180-196.
M. Padberg, "Lehman's forbidden minor
characterization of ideal 0-1 matrices,"
Working Paper No. 334, cole Polytechnique,
Laboratoire d'Econometrie, Paris, France,
P. Seymour, "Decomposition of regular
matroids," Journal of Combinatorial Theory B
P. Seymour, "On Lehman's width-length
characterization," in: W. Cook and P.
Seymour (eds.), Polyhedral Combinatorics,
DIMACS Series in Discrete Mathematics and
Theoretical Computer Science, VoL 1,1990,
American Mathematical Society, 107-117.
The selection committee recommends that
one Fulkerson Prize be given to Martin
Dyer, Alan Frieze and Ravi Kannan for
their paper "A random polynomial time
algorithm for approximating the volume of
convex bodies" published in Proceedings of
the Twenty-first Annual ACM Symposium on
Theory of Computing, Seattle, Washington,
May 15-17,1989, Association of Computing
Machinery, pp. 375-381 and in a more
detailed version in Journal of the Association
of Computing Machinery 38 (1991) 1-17.
Martin Dyer is from the School of Computer
Studies of the University of Leeds, U. K.;
Alan Frieze is from the Department of
Mathematics, and Ravi Kannan is from the
Department of Computer Science, of the
Carnegie-Mellon University, Pittsburgh,
The paper is concerned with the following
problem: we are given a convex body in n-
dimensional Euclidean space, and we wish
to find its volume. Earlier work of Grbtschel,
Lovasz and Schrijver gave an algorithm,
with running time polynomial in n, that
estimates the volume to within a factor that
is exponential in n; and results of Elekes and
of Barany and Fiiredi showed that if the
body is presented by means of a "member-
ship oracle", which is usual, then no
algorithm with polynomial running time
can estimate volume within a polynomial
factor. However, these results are concerned
with algorithms in the course of which no
random decisions are made. The prize-
winning paper of Dyer, Frieze and Kannan
shows that, in remarkable contrast, if we
permit the algorithm to incorporate flipping
coins, then one can estimate volume with
high precision; more precisely, for any
arbitrarily small d > 0, one can generate a
random variable, with the property that its
ratio to the true volume is between 1 -d and
1 + d with probability at least 4. The
algorithm has running time bounded by a
polynomial in n and 1/d. This is the first
instance of the use of a random algorithm
providing a super-polynomial
improvement in accuracy.
The method is, roughly, to surround the
body with a nicer (that is, algorithmically
more tractable) convex body, to generate a
uniformly distributed random point inside
the larger body and to count how often it
falls into the smaller one. This provides an
estimate of the ratio of the two volumes.
There are a number of difficulties with this
approach, arising from the fact that if we ask
the larger body to be "really" nice, the ratio
of the volumes becomes exponential, and
we cannot estimate it in polynomial time by
Monte Carlo methods. The paper overcomes
this by using a series of bodies, each inside
the next; but the intermediate bodies are no
nicer than the original, and this produces
the second difficulty-how can one generate
a random point, uniformly distributed over
a convex body? Solving this in a satisfactory
way is the main achievement of the paper.
The technique used is to approximate the
body by its intersection with a sufficiently
dense cubic lattice and to take a random
walk on these lattice points. This converges
to a uniform distribution, but it is important
that it does so in polynomial time, and
proving the latter is complicated. However,
it would suffice to prove that the second
eigenvalue of the "Laplacean" of the walk is
well separated from the first eigenvalue, and
for this the paper uses a recent result of
Sinclair and Jerrum, which relates the
separation of these eigenvalues to connectiv-
ity and expansion properties of the underly-
ing graph. (The result of Sinclair and Jerrum
is itself an extension of a result of Alon.) So
it suffices to establish that the underlying
graph indeed has these expansion proper-
ties, but unfortunately that is not true. To
make it true the paper adjoins to the domain
of the random walk a few more
neighboring points from outside the body,
to smooth it off, and then the expansion
property is shown to hold for the enlarged
domain by applying isoperimetric inequali-
ties from geometry.
This paper has had a major impact on
numerical integration, statistical sampling
and other fields. Also, there has been a great
deal of subsequent work on modifying the
PAGE SIX >
PAGE 6 number th~1_ irtyfie OVMBR 99
1991 PRIZE RECIPIENTS
techniques of Dyer,
Frieze and Kannan to
reduce the running time of
the algorithm. In particular, papers by
Lov&sz-Simonovits, Applegate-Kannan, and
Dyer-Frieze have brought down the running
time to roughly n*, which promises practi-
cal applicability in the near future.
N. Alon, "Eigenvalues and expanders,"
Combinatorica 6 (1986) 83-96.
D. Applegate and R. Kannan, "Sampling and
integration of near log-concave functions,"
preprint, November 1990.
I. BWrAny and Z. Ffiredi, "Computing the
volume is difficult," Proc. 18th Annual ACM
Symposium on the Theory of Computing,
M. Dyer and A. Frieze, "Computing the
volume of a polytope: A case where random-
ness provably helps," preprint, January 1991.
G. Elekes, "A geometric inequality and the
complexity of computing volume," Discrete
and Computational Geometry 1 (1986)
M. Gr6tschel, L. Lovisz and A. Schrijver,
"Geometric Algorithms and Combinatorial
Optimization," Algorithms and Combinato-
rics 2, Springer-Verlag, 1988.
L. Lovisz and M. Simonovits, "Mixing rate of
Markov chains, an isoperimetric inequality,
and computing the volume," Proc. 31st
Annual Symposium on the Foundations of
Computer Science, 1990, IEEE Computer Soc.,
A. J. Sinclair and M. R. Jerrum, "Approximate
counting, uniform generation and rapidly
mixing Markov chains," Information and
Computation 82 (1989) 93-133.
The selection committee recommends that
one Fulkerson Prize be given to Nikolai E.
Mnev for his paper "The universality
theorems on the classification problem of
configuration varieties and convex polytope
varieties" published in O. Ya. Vim (Ed.),
Topology and Geometry-Rohlin Seminar,
Lecture Notes in Mathematics 1346,
Springer-Verlag, Berlin, 1988,527-544.
Nikolai Mnev is a Senior Researcher at the
Institute of Socio-Economic Problems of the
Academy of Sciences of the USSR in
Leningrad. The work in this paper consti-
tutes Mnev's 1986 Ph.D. thesis, written
under A. M. Vershik in Leningrad [Mn86]
and was first published as a research
announcement in [Mn85].
The results of this paper represent a
landmark in discrete geometry. They relate
directly to the theory of oriented matroids, a
subject partly invented in the Ph.D. thesis
[B174] of R. G. Bland, a student of Ray
Fulkerson. One of Bland's aims was to
provide a combinatorial framework for the
treatment of linear programming. This led,
for example, to his discovery of what has
come to be known as Bland's pivot rule to
avoid cycling in the simplex method. While
Bland's work showed how far the combina-
torial approach to linear programming
might be carried, this paper of Mnev points
to the inherent limitations of any approach
that ignores the data to any essential degree.
A precursor to the result of Mnev is a long-
standing conjecture in discrete geometry,
first formulated by Ringel [Ri56]. Though
Ringel stated it in a form concerning lines
in the plane, we give the version in
terms of points.
MPS VICE-CHAIRMAN JAN f
KAREL LENSTRA MEETS WITHIN
TREASURER LES TROTTER
AND NEW COUNCIL
MEMBERS CLOVIS CONZAGA
AND STEVE ROBINSON
DURING THE AMSTERDAM
SYMPOSIUM LEFT TO RIGHT]. -4
MASAKAZU KOJIMA AND
BERNARD KORTE, ALSO
RECENTLY ELECTED TO
COUNCIL, WERE NOT "'il
PAE ume tiryfieNOEME 19
1991 PRIZE RECIPIENTS
Isotopy Conjecture: Given two labeled sets of n
points in general position in the plane (i. e., no
three on a line), such that corresponding triples
of points have the same orientation (clockwise or
counterclockwise), then it is possible to continu-
ously move one set of points into the other so
that all the intermediate configurations are in
general position (and so also maintain
The Isotopy Conjecture was considered
almost obvious by many, although serious
attempts to prove it went on for many years
without much success. It can be put in a
form that appears more relevant to math-
ematical programming. Suppose Ax < b and
A'x < b' describe combinatorially equivalent
convex polytopes (i. e., having the same
facial structure). Then these two data sets
are indistinguishable from the point of view
of any purely combinatorial approach to
linear programming. The conjecture would
imply that one could continuously change
the data Ab into the data A',b' preserving
the combinatorial type of the resulting
The results of Mnev show that this conjec-
ture is as false as it could be. To understand
what is true, consider the case of a configu-
ration of n points in the plane p, = (x. ,y.),
i=l,...,n, which we consider to be repre-
sented by a single point (xi,...,x ,...,y ) in
R2" The combinatorial type, or oriented
matroid, of the configuration is the set of
orientations of the various triples pi,p and
the representation space of a given combina-
torial type is the set of all points in R2"
having this combinatorial type. The Isotopy
Conjecture thus states that the representa-
tion space of any combinatorial type is
connected. Since a representation space is
determined by certain determinants being
positive, negative or zero, it is a
semialgebraic set (i. e., the solution set of a
system of polynomial equations and strict
inequalities). What Mnev showed was that
given any semialgebraic set M, one can
construct, for some n, a configuration of n
points in the plane whose combinatorial
type has a representation space that is in a
sense equivalent to M (so that, for example,
they will have the same homotopy type).
This result is called the Universality Theorem;
it says, roughly, that arbitrary semialgebraic
sets are no more general than those encoded
by combinatorial types of point configura-
tions (oriented matroids) or convex
The problem of determining the topological
type of the set of polytopes having a given
combinatorial type was proposed by
Vershik (himself a student of L.V.
Kantorovich) to his seminar in Leningrad as
early as the mid-1970's. Here again the
expectation was that these spaces would be
connected. For something as broad as
universality to be true was completely
unexpected. Technically, the proof is quite
an achievement, combining combinatorial
geometry, classical projective geometry and
algebraic geometry in a very clever way.
The idea is to mimic the computation of a
polynomial function by means of construc-
tions, originally due to von Staudt (a
student of Gauss), used to coordinatize
projective planes. These constructions are
based on the usual ruler-and-compass
constructions of Euclid. The key is to show
that the combinatorial type so defined by a
system of polynomials has a representation
space which is equivalent (in a well-defined
sense) to the semialgebraic set cut out by
Thus representation spaces of point configu-
rations--and, by a straightforward equiva-
lence by means of Gale duality, representa-
tion spaces of convex polytopes-can be
almost arbitrarily complicated topologically,
having any number of connected compo-
nents, holes, etc. This indicates, for example,
that algorithms for solving linear program-
ming problems may ignore the actual data
at their own risk-two combinatorially
equivalent polytopes can be represented by
data coming from totally different regions in
the representation space. It is possible that
this result contains the seeds of a proof that
there can be no strongly polynomial
algorithm for linear programming.
Given a semialgebraic set M, the corre-
sponding combinatorial type can be
constructed in time (and has size) that is
linear in the complexity of M, so it follows
that the problem of deciding if a given
system of polynomial equations and
inequalities has a solution (the so-called
decision problem for the Existential Theory
of the Reals, referred to as ETR) is linearly
reducible to deciding whether a given
oriented matroid arises from an actual
configuration. Since ETR is known to be NP-
hard, it follows that the problem of deciding
whether an oriented matroid is realizable by
a configuration of points is also NP-hard.
(An independent proof of this is given by
Shor in [Sh91], where one can also find a
clear exposition of the proof using Mnev's
methods.) Finally, it follows that the
algorithmic Steinitz problem-determining
whether a given poset is actually the face
lattice of some convex polytope-
is also NP-hard.
Incidentally, after the proof of the Universal-
ity Theorem, but before the extent of Mnev's
work was fully understood in the West,
there was a flurry of activity in the U. S. and
Europe leading to a counterexample to
isotopy by Jaggi, Mani-Levitska, Sturmfels
and White [JMSW89]. See [BSt89] for a
discussion of this and related work as well
as a description of the work of Mnev.
PAGE8 nmbe thity-iveNOVEBER'99
1991 PRIZE RECIPIENTS
18' 174] R.G. Bland,
subspaces of R" and orientability of
matroids," Ph. D. thesis, Cornell University,
Ithaca, NY, 1974.
[BSti9] J. Bokowski and B. Sturmfels,
"Computational Synthetic Geometry,"
Lecture Notes in Mathematics 1355, Springer-
Verlag, Heidelberg, 1989.
[JMSW89] B. Jaggi, P. Mani-Levitska, N.
White and B. Sturmfels, "Uniform oriented
matroids without the isotopy property,"
Discrete Comput. Geom. 4 (1989) 97-100.
iMn85] N.E. Mnev, On manifolds of
combinatorial types of projective configura-
tions and convex polyhedra," Soviet Math.
Dokl. 32 (1985) 335-337.
[Mn861 N.E. Mnev, Thesis, Leningrad, 1986.
[Mn88] N.E. Mnev, "The universality
theorems on the classification problem of
configuration varieties and convex polytope
varieties," in: O. Ya. Viro (Ed.), Topology and
Geometry-Rohlin Seminar, Lecture Notes in
Mathematics 1346, Springer-Verlag, Berlin,
[Ri56l G. Ringel, "Teilungen der projectiven
Ebene durch Geraden oder topologische
Geraden," Math. Z., 64 (1956) 79-102.
[Sh91] P.W. Shor, "Stretchability of pseudo-
line arrangements is NP-hard," In Applied
Geometry and Discrete Mathematics -The
Victor Klee Festschrift, American Math. Soc.,
Providence, 1991, to appear. a
Tucker Prize to Goemans;
Hall and Hartman
The A. W. Tucker Prize honors a paper
written by a student on any aspect of
mathematical programming. The 1991
prize committee, which consisted of
Richard W. Cottle (chair), Thomas M.
Liebling, Richard A. Tapia and Alan C.
Tucker, selected Michel Goemans as the
winner with Leslie Hall and Mark
Hartman being finalists. All three
presented their work in a special session
at the 14th Symposium.
A summary of the background and work
of the finalists follows:
a Class of
Born and raised
MICoELcoMANS Diploma in
Mathematics from l'Universitd Catholique
de Louvain in Louvain-la-Neuve. In 1987, he
commenced doctoral studies in Operations
Research at the Massachusetts Institute of
Technology. During the summer of 1989,
Mr. Goemans was a member of Dr. David
Johnson's group at AT&T Bell Laboratories
in Murray Hill, New Jersey. He received his
Ph.D. in 1990 under the supervision of
Professor Dimitris J. Bertsimas. Upon
graduation, he joined the teaching staff in
the Department of Mathematics at MIT. A
paper by Mr. Goemans describing the
"parsimonious property" was awarded
second prize in the 1990 George E.
Nicholson student paper competition
sponsored by the Operations Research
Society of America.
Mr. Goemans' paper belongs to the body of
literature concerning worst-case and
probabilistic analysis of (heuristics for)
combinatorial optimization problems. The
goal of this thesis is to evaluate analytically
linear programming relaxations for a class
of connectivity problems including the
survivable network design problem, the k-
edge-connected problem, the TSP, the k-TSP
and the Steiner tree problem. The central
theme in this study is the aforementioned
"parsimonious property" which in rough
terms says that if the cost function satisfies
the triangle inequality, there exists an
optimal solution to a classical LP relaxation
for which the degree of each vertex is the
smallest it can possibly be. The work has
three major sections, corresponding to the
investigation of structural properties, worst-
case analysis, and probabilistic analysis,
In the first part, Mr. Goemans specifies the
combinatorial optimization problems to be
discussed and the corresponding linear
programming relaxation. He derives his
result on the parsimonious property and
develops a number of its consequences. He
obtains the Held-Karp lower bound for the
traveling salesman problem. He shows that
the LP relaxation bounds corresponding to
the Steiner tree problem, the k-edge-
connected network problem, and the Steiner
k-edge-connected network problem can be
computed in the manner of Held and Karp
(using Lagrangian relaxation and solving
minimum spanning tree problems as
In the second part, Mr. Goemans uses the
parsimonious property to carry out worst-
case analyses of the duality gap correspond-
ing to the LP relaxations. He introduces two
heuristics for the survivable network design
problem and gives bounds that depend on
the actual connectivity requirements. He
shows that the value of the LP relaxation of
PAGE 9~ ~~ iube hirtaieNVEBR'
1991 PRIZE RECIPIENTS
the Steiner tree problem is within twice the
value of the minimum spanning tree
heuristic and that analogous results for
several generalizations of the Steiner tree
problem can be obtained. Other contribu-
tions in this part of the work include a new
relaxation (of the Held-Karp type) for the k-
person traveling salesman problem and a
demonstration that an existing heuristic for
this problem gives a value that is within 3/
times the value of the new relaxation.
The third part contains probabilistic analysis
of the duality gap for the LP relaxations.
Concentrating on the Euclidean model, he
generalizes a theorem of Steele on the
asymptotic behavior of Euclidean
functionals. The generalization is particu-
larly convenient for the analysis at hand.
Mr. Goemans shows that the duality gap
(for the various problem types) is almost
surely a constant; he provides theoretical
and empirical bounds on these constants.
The analysis leads to the conclusion that the
undirected LP relaxation for the Steiner tree
problem is "fairly loose."
Mr. Goemans' parsimonious property has
far-reaching consequences in the complexity
analysis of heuristics in the worst case and
probabilistic sense. His very well written
paper combines new ideas with others that
had been around for some time and thereby
unifies problems whose kinship was
suspected but never spelled out.
Leslie Ann Hall, "Two Topics in
Discrete Optimization: The Polyhedral
Structure of Capacitated Trees and
Leslie Ann Hall attended college at Yale
University where she earned a B.S. in
Mathematics in 1982. After a year working
in New York City and a year on a Fulbright
Scholarship studying mathematics at the
Technische Hochshule Darmstadt in
Germany, she entered the doctoral program
at M.I.T. Operations Research Center. In
1989, she received her Ph.D. and since has
been an Assistant Professor in the Depart-
ment of Civil Engineering and Operations
Research at Princeton University. Ms. Hall's
thesis consists of two distinct parts. Profes-
sor Thomas Magnanti was the advisor for
the first part and Professor David Schmoys
was the advisor for the second part.
The first part presents a description of the
polyhedral structure of the capacitated
minimal spanning tree problem. This
variant of the minimal spanning tree
problem has a designated root and the
requirement that each subtree of the root (a
subtree with only one edge incident to the
root) contains at most k nodes. This capaci-
tated problem is essentially the same as the
identical-customer vehicle routing problem.
As part of her investigations, Ms. Hall
showed that the intersection of a well-
known and much-studied integer polyhedra
(the spanning tree polyhedron and the
matching polyhedron) is itself an integer
polyhedron. Demonstrating great insight,
she also found a number of novel types of
facets for describing the capacitated minimal
spanning tree polyhedron.
The second part of the thesis concerns a
general framework for developing polyno-
mial-time approximation schemes for a
variety of important NP-hard scheduling
problems. Heretofore, such approximation
schemes have been ad hoc and generally
applied to only the simplest (least realistic)
scheduling models. She called her method
the "outline-scheme" approach. Her
"outline" captures critical information about
an instance of a scheduling problem and
allows an optimal or near-optimal solution
to be computed in polynomial time.
In this well-written doctoral dissertation,
Ms. Hall shows versatility, treating two
disjoint subjects. In the first, she effectively
uses simple proof techniques to characterize
a non-trivial polyhedron from combinatorial
optimization and also finds non-trivial
facets of others. In the second part, she
develops interesting heuristics for schedul-
ing problems. The manuscript pairs charm-
ing modesty with scientific rigor.
Mark E. Hartmann, "Cutting Planes and
the Complexity of the Integer Hull."
Mark E. Hartmann was born in Redwood
City, California, and grew up in Salt Lake
City, Utah. In 1985, he received his under-
graduate and Master's degrees in Math-
ematical Sciences at The Johns Hopkins
University. His doctoral studies were
undertaken at Cornell University from
which he received the Ph.D. in 1989. Mr.
Hartmann's Ph.D. thesis was supervised by
William Cook with whom he studied for a
year at the Institut fiur Okonometrie und
Operations Research in Bonn. He completed
the writing of the dissertation while
teaching probability and statistics at Johns
Hopkins. He is currently a Postdoctoral
Fellow and an Assistant Professor at the
University of North Carolina at Chapel Hill.
This work belongs to the area of polyhedral
combinatorics. A significant part of it is
concerned with establishing lower bounds
on the Chvatal ranks of polyhedra. (The
Chvatal rank of a polytope is the number of
times a certain "closure" operator needs to
be applied to obtain its integer hull, i.e., the
convex hull of the integer points in the
polytope.) The Chvatal rank gives a
measure of the "tightness" of the
In the first part of this paper, Mr. Hartmann
describes two general methods for proving
lower bounds on the ChvAtal rank of a
polytope. The first is a type of integral
transformation that enables one to transfer
lower bounds from an imbedded polytope
for which it can be easier to prove a lower
bound. This method gives lower bounds for
common relaxations of the set covering, set
partitioning and knapsack problems-all
based on Chvatal's bound for the stable set
problem. The second is a geometric method
of "defending" the integer hull by finding a
sequence of point sets such that if all the
points in a given set are contained in a
polytope, then all points in the next set
survive the closure operation applied to the
polytope. This method generalizes the one
used by ChvAtal in obtaining lower bounds
PAGE TEN ~.
PAE1 ubrtityfv OEBR19
for the stable set problem. It is used for
obtaining lower bounds on the Chvatal rank
of various combinatorial optimization
problems, thereby settling, in a unified
manner, conjectures of Chvatal on the
traveling salesman problem, Barahona,
Gr6tschel and Mahjoub on the bipartite
subgraph problem, and Jiinger on the
acyclic subdigraph problem.
The second part of the paper addresses the
study of integer points in an arbitrary
rational polytope. It gives an upper bound
on the number of vertices of the integer hull
of the polytope in terms of the dimension,
the number of inequalities and maximum
size of an inequality in a system of inequali-
ties used to describe the polytope. The proof
involves dissecting the polytope into smaller
polytopes, called reflecting sets, and then
showing that the number of reflecting sets
which contain vertices of the integer hull is
at most the number of reflecting sets "near
the surface" of the polytope.
The decomposition of the polytope into
reflecting sets can be made constructive,
thereby yielding a generalization of
Lenstra's celebrated result that the integer
linear programming problem can be solved
in polynomial time when the dimension is
fixed. Indeed, it is even possible to obtain a
list of the vertices of the integer hull in
Mr. Hartmann's writing style is remarkably
full of beauty, power and precision. The
results are not entirely unexpected, since
many had been conjectured before, but they
are treated in a unified manner. The paper is
notable for its brilliant, transparent-and
-RICHARD W. COTILE
8c.VI4 'Beale Trust
THIS Trust commemorates Evelyn Martin Lansdowne Beale, FRS
(1928-1985) who was the secondchairman of the Mathematical
Programming Society. During his career as a Scientific Officer at the Ad-
miralty Research Laboratory, a Group Leader at CEIR (Corporation for
Economic and Industrial Research), the Scientific Adviser at Scicon
(Scientific Control Systems Ltd.), and a Professor of Mathematics at Imperial College,
London, he was motivated by the needs of commerce and industry, he was highly
proficient at building mathematical models of real-world problems for investigation
by computer calculations, heled thedevelopment of the UMPIRE and SCICONIC math-
ematical programming systems, he invented many of the algorithms for linear, quadratic
and integer programming that are fundamental to such systems, and he presented and
published many learned papers that are major contributions to basic research. This career
is a brilliant example of sustained work that not only served industry but also was of
great importance to the academic development of statistics, operational research and
mathematical programming. He showed clearly that experience in the business world
can be of immense value to academic studies and that participation and profound un-
derstanding at the frontiers of academic research are often vital to the successful solu-
tion of real commercial problems. His professional knowledge was of particular ben-
efit to his colleagues and to students at Imperial College, where as a Visiting Professor
he taught actively and guided research over a period of 18 years until shortly before
The purpose of the Trust reflects these achievements. In order to make an impact, it aims
at a relatively narrow range of activities that would benefit substantially from moder-
ate financial support. Indeed the Trust Funds will be devoted to the assistance of stu-
dents and their studies in the Mathematics Department of Imperial College. Martin
attached great importance to his duties in this department, and he gave inspiration, en-
couragement and guidance to many of its students, so the Trust is a very suitable me-
morial to the academic side of his career.
The Trust is registered with Charity Commission. Section 5 of the Trust Deed gives the
following details of the activities that may be supported:
''AE Uu me r hryfv OEBR19
The Trustees may pay or apply the income of the Trust Fund for the purposes of the
advancement of education and learning in the Department of Mathematics ("the Depart-
ment") of the Imperial College of Science and Technology of Prince Consort Road London
SW7 ("the College") and in particular for the assistance and support of students and their
studies in the Department in such manner as the Trustees shall in their absolute discretion
think fit and in particular (and without prejudice to the generality of foregoing) the
Trustees may pay or apply such income:
(a) for the provision of bursaries to allow or
facilitate the admission to the Department of
students who in the opinion of the Trustees
deserve or are in need of financial support
(b) for the provision of funding for the
continuation by students of the Department
of their studies beyond the period for which
funding is normally provided for them by
local or central government or by other
bodies or authorities (whether public or
(c) for the payment of travel and living
expense incurred or to be incurred by
students of the Department in attending
(d) for the payment of travel and living
expenses incurred or to be incurred by
students of the Department in
(i) visits to academic, industrial,
scientific or other establishments
(ii) visits enabling students of the
Department to collaborate with
researchers outside the College
(iii) visits enabling students of the
Department to accompany their
supervisors while such supervisors
may be on sabbatical leave
(e) for the payment of travel and living and
other expenses incurred or to be incurred by
students of other universities or educational
or other establishments while visiting
(f) towards the improvement of facilities for
students within the Department
(g) for the avoidance of doubt in this clause
the expression "students" shall mean both
undergraduate students and graduate
students of the Department.
It is hoped that many members of the
Mathematical Programming Society will
show their appreciation of Martin's career
and friendliness by making a donation to
this memorial. TheTrustees are Mrs. E. M.
L. Beale (Windhover House, Treyarnon
Bay, Padstow PL28 8JS, England), Prof. M.
J. D. Powell (DAMTP, Silver Street, Cam-
bridge CB3 9EW, England) and Prof J. T.
Stuart (Department of Mathematics, Impe-
rial College, London SW7 2BZ, England).
If you wish to make a contribution, please
send a cheque (made payable to the
"E.M.L. Beale Trust") to one of the Trust-
ees. All contributions will be acknowl-
edged. We would be very grateful for
your support. a
-M. J. D. POWELL
The interest in parallel processing is rapidly in-
creasing, but information on the research
groups working in the field with respect to
subjects and parallel equipment makes up a
distributed database with no well-defined
As a first step towards changing this situation,
COAL intends to compile a list of sitesworking
with parallel processing inoptimization. Hence
if you work with parallel optimization in prac-
tice, please send a short description containing
information on the hardware available, the
projects being carried out (project titles only)
and the names (and if possible e-mail ad-
dresses) of the people involved in the research.
The description can be sent by mail or e-mail
using the following addresses. News on the
project including how to get access to the com-
piled information will appear in the next issue
of OPTIMA and the COAL Newsletter.
2100 COPENHAGEN O
r N B
OPTIMIZATION DAYS 1991
May 4,5 and 6, 1992
Center de recherche sur les transports
University de Montrial
cole des Hautes Ftudes Commerciales
cole Polytechnique de Montreal
Groupe d'Etudes et de Recherche en
Analyse des Decisions (GERAD)
Ecole des Hautes Etudes Commerciales
Cole Polytechnique de Montreal
College Militaire Royal de Saint-Jean
The scientific meeting "Optimization Days" is
organized each year jointly by the above institu-
tions of the Montr6al region. The aim of the meet-
ing is to survey current trends of research in opti-
mization methods and their applications and to
provide a good opportunity for interaction be-
tween various research groups.
All those interested in optimization methods and
their present or potential applications are kindly
invited to participate. We appeal especially to
those who can give talks on new methods of op-
timization and their applications.
Sessions will consist of invited and contributed
talks. Papers presenting original developments as
well as those of expository nature will be consid-
ered. The languages of the conference will be
French and English. Plenary speakers will be:
L. Devroye (Montr6al)
B. Gavish (U.S.A.)
P. Jaillet (U.S.A.)
C. Lemar6chal (FRANCE)
C. Revelle (U.S.A.)
In 1992, Optimization Days will be held at cole des
Hautes ttudes Commerciales, 5255, avenue
Two copies of a 100-200 word summary defining
clearly the content of the paper, together with a
registration form, should be forwarded before
December 15, 1991, to:
Michel Gendreau and Patrice Marcotte
Center de recherche sur les transports
University de Montreal
Case postal 6128, Succ. "A"
Montr6al, CANADA, H3C 3J7
Telephone: (514) 343-7575
Fax: (514) 343-7121
Registration fee:($CDN) Before April 15:
$100; After: $125 (Students $20)
($US) Before April 15: $90; After $100
Authors should also send a copy of their
summary via e-mail, if this is possible.
Authors will be notified of the acceptance of
their talks by March 15, 1992. Summaries of
the talks will be distributed at the confer-
ence. For more information, please contact
SIAM Activity Group on Optimization
May 11-13, 1992
Hyatt Regency Hotel
The Fourth SIAM Conference on Optimization will
address the most important recent developments
in linear, nonlinear,and discreteoptimization. The
conference will feature recent advances in algo-
rithms and software for the solution of optimiza-
tion problems. It will also featureimportantappli-
cations of optimization in control, networks,
manufacturing, chemicalengineering, operations
research, and other areas of science and engineer-
The conference will bring together mathemati-
cians, operations researchers, computer scientists,
engineers, and software developers. It will provide
an excellent opportunity for sharing ideas and
problems among specialists and users of optimi-
zation in academia, government, and industry.
-s~ II" II- ---
PAGE 13 number thirty-five NOVEMBER '99'
Interior point methods
Algorithms for optimization problems in
Network optimization methods
Parallel algorithms for optimization
Contributed presentations in lecture or
poster format are invited in all areas of
optimization research and applications.
A minisymposium may focus on any topic consis-
tent with the conference themes. It consists of four
15-minute presentations, with an additional five
minutes for discussion after each presentation.
Prospective minisymposium organizers are asked
to provide a title, a description (not exceeding 100
words), and a list of speakers and titles of their
For more information contact:
3600 University City Science Center
Philadelphia, PA 19104-2688 U.S.A.
Telephone: (215) 382-9800;
Fax: (215) 386-7999
SYMPOSIUM ON DYNAMIC
University of Geneva and International
Academy of the Environment
Organizer: Dipartement d'dconomie
commercial et industrielle, University
Sponsor: The International Society of
Dynamic Games (ISDG)
The Symposium is the fifth in a series of meetings
dedicated to the area of dynamic games and is the
official, biannual scientific meeting of the ISDG.
The aim of the meeting is to bring together re-
searchers from various disciplines wheredynamic
game settings are studied and to report the latest
developments both in theory and application.
This year, the Symposium is putting particular
emphasis on the theme Dynamic Games and En-
vironmental Management Modeling.
The deadline for receipt of title and three
copies of a 500-word extended abstract is
February 1, 1992.
Early registration fee is US$200 before
May 1, 1992.
A collection of papers presented at the
Symposium will be published.
For further information, please contact:
A. Haurie, Chairman,
D6partement d'6conomie commercial et
University de Geneve
2, rue de Candolle
Telephone: (int'l) 41 22 705 72 44
Fax: (int'l) 41 22 28 5213
CONFERENCE OF THE
EUROPEAN CHAPTER ON
Organized by: R. E. Burkard, F. Rendl
and G. Rote
University of Technology
April 13-15, 1992
The 5th Meeting of the European Chapter on
Combinatorial Optimization (ECCO) will take
place in Graz, Austria, April 13-15,1992. The con-
ference willbe hosted by the Institute of Mathemat-
ics, University of Technology, Graz. For further in-
formation, please use the address given below.
There will be no conference fee,but participants are
expected to cover their own travel, accommoda-
tion and living expenses.
October 1991: Second announcement and
January 31, 1992: Deadline for abstracts
March 1992: Last announcement with
Professor Dr. Rainer E. Burkard
Institute fir Mathematik B, Technische
Kopernikusgasse 24, A-8010 Graz
Phone: +++0316 873 7125
The Royal Institute of Technology
A second, and much expanded, Stockholm Opti-
mization Days was held on Monday and Tuesday
following the MPS Symposium. Some 25 speakers
from 11 countries presented papers on a variety of
optimization topics including dual methods, de-
composition, subgradient optimization, global
optimization, interior methods, large-scale optimi-
zation, as well as a variety of applications. Copies
of the program, including abstracts, may be ob-
tained from the conference organizer, P. O.
Lindberg, Division of Optimization and Systems
Theory, Department of Mathematics. The confer-
ence was sponsored by the G6ran Gustafsson
Foundation and the Swedish National Board for
--- - --- ~---~ --
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M.C. Golumbic and R. Shamir, "The Interval
Sandwich Problem is NP-Complete,"
Y. Crama, A.W.J. Kolen, A.G. Oerlemans
and F.C.R. Spieksma, "Minimizing the
Number of Tool Switches on a Flexible Ma-
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J.-P. Barthdlemy, O. Hudry, G. Isaak, F.S.
Roberts and B. Tesman, "The Reversing
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M.H. Rothkopf, "Models of Auctions and
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E. Boros, P.L. Hammer and J.N. Hooker,
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U.G. Rothblum and S.A. Zenios, "Scalings of
Matrices Satisfying Line-Product Constraints
and Generalizations," RRR 15-91.
R. Shamir, "Probabilistic Simplex Analysis,"
E. Boros, P.L. Hammer and X. Sun, "Network
Flows and Minimization of Quadratic
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M. Yue, "A Simple Proof of the Inequality: M F
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F.S. Roberts, "Quo Vadis, Graph Theory?,"
M.C. Golumbic and R. Shamir, "Complexity
and Algorithms for Reasoning about Time: A
Graph-Theoretic Approach," RRR 22-91.
A. Kogan, "On Statistical Properties of Boolean
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A. Kogan, "On the Essential Test Sets of
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Their Minors," RRR 31-91.
B. Avi-Itzhak and S. Halfin, "Priorities in
Simple Fork-Join Queues," RRR 32-91.
H. Jiirgen, Y. Crama and F.C.R. Spieksma,
"Approximation Algorithm for the Multidimen-
sional Assignment Problems with Decomposable
Costs," RRR 33-91.
R.E. Bixby, J.W. Gregory, I.J. Lustig, R.E.
Marsten and D.F. Shanno, "Very Large-Scale
Linear Programming: A Case Study in
Combining Interior Point and Simplex Meth-
ods," RRR 34-91.
F.K. Hwang and U.G. Rothblum, "Generali-
zation of an Engineering Principle," RRR 35-91.
- -- -- --- ---~ --- ----- ---- --s
i--~~ ------** ;>, A >_ -
Edited by E H. Clark, V. F.
Dem'yanov and F.
This comprehensive book forms the proceedings
to Nonsmooth Optimization held in Erice from
June 20 to July 1,1988. It reflects the variety of the
field and its wide scope through 27 contributions,
briefly noted as follows. The classifications given
are for the convenience of the reader; it is the re-
sponsibility of the reviewer to introduce such an
artificial arrangement. The main themes of the
book are: 1. Analysis of nonsmoothness; 2.
Optimality conditions; 3. Calculus of Variations;
4. Stability questions; 5. Algorithms; 6. Applica-
1. Analysis of Nonsmoothness
The contribution of F. H. Clarke exemplifies how
the classification we adopt is artificial. Motivated
by a question of regularity of the solutions of a
problem in the calculus of variations, he introduces
toolsof nonsmooth analysis centered around what
is called proximal analysis. The elegant concepts
of this theory apply to calculus of normals to an
intersection and pertain to the realm of perturba-
tion analysis so that four of the announced sections
could host this chapter.
The study of A. D. Ioffe centers around the con-
cept of subdifferential. Again it is motivated by
crucial problems in optimization theory (here an
example arising from optimal control is treated).
It clearly shows the difficulties and the achieve-
ments of the concept of subdifferential, especially
in the infinite-dimensional case.
The provocative and amusing titleof the lectureby
V. F. Dem'yanov "Smoothness of nonsmooth
functions" canalsobe takenseriously. In fact most
of the efforts of researchers in nonsmooth analy-
sis consist of devising tools which are not too far
from the smooth case orobey rules which are close
I E W
enough to the
usual laws. H e r e
Dem'yanov en- large his pre-
ceding views in accepting functions whose direc-
tional derivatives are differences of convex func-
Generalized derivatives obtained from a tangent
cone notion are considered by K. H. Elster and J.
Thierfelder. Their approach is axiomatic as were
those of A. D. Ioffe (Proc. Fermat's Days 1985), D.
Ward and others. A key pointof their contribution
lies in their requirement that the recession cone to
the tangentconeto aset must contain the recession
cone of the set. This ensures that a closed tangent
cone to an epigraph is an epigraph.
A. Marino and C. Saccon introduce a class of
functionals for which the nonconvexity is con-
trolled. They apply the properties they get to ex-
istence and multiplicity results of solutions to
variational inequalities ofVon Karman type asso-
ciated with the plate problem with obstacle.
2. Optimality Conditions
The links of nonsmoothness with optimality con-
ditionsarestrongandwellknown. Itisalso thecase
for generalized convexity conditions: this justifies
the refinements of convexity considered in the
Chapter treated by E. Castagnoli and P.
F. Giannessi, M. Pappalardo and L. Pellegrini
choseto attack thequestionof necessary conditions
in mathematical programming problems by con-
sidering the image of the mapping (j,k) gathering
the objective j and the constraint mapping k.
J.-B. Hiriart-Urruty addresses nonconvex optimi-
zation problems, in particular the maximization of
a convex function on a convex set and the minimi-
nation of a difference of two convex
functions. His aims are local and glo-
bal optimality conditions, which he
reaches through the techniques of
Convex Analysis, especially using the
B. N. Pshenichny deals with implicit
multifunction theorems and approxi-
nation of sets.
The problem of controllability studied byJ. Warga
is a weakening of an openness property. Given a
set X,x0 E X, a convex subset C with nonempty
interior intC of a topological vector space Z, the
problem consists of finding conditions in order to
have for some v E Rm, r > 0 and each t e(0,r)
f(xd) + tv E int f(X n g -'(int C)).
This problem is intimately linked with higher
order optimality conditions in optimal control.
3. Calculus of Variations and Optimal
The contributionof E. De Giorgiand L. Ambrosio
reflects the vitality of the calculus of variations.
Motivated by the study of energy functionals cor-
responding to mixtures of different fluids as liq-
uid crystals, they introduce new classes of func-
tions of bounded variation type with which clas-
sical techniques of calculus of variations can be
used. One has to notethatamong thenew features
of their approach lies the fact that the functional
are defined in terms of Hausdorff measures.
R. B. Vinter provides a unified treatment of nec-
essary conditions for a class of nonstandard prob-
lems in dynamics optimization, in particular op-
timal multiprocess problems. Again the power of
proximal analysis is illustrated.
4. Perturbation and Stability Questions
The central part of the contributionofP.D. Loewen
deals with a formula for the generalized gradient
of the value function of a perturbed differential
J. Gauvin gives a syntheticalsimplified versionof
the works of Gauvin-Janin about the performance
function of parametrized mathematical program-
R.T.Rockafellarstudies themultifunctions whose
values are the Kuhn-Tucker points of a general
class of parametrized minimization problems,and
- --- -~--
PAGE 17 sl"Ll"r~nubr tiryfieNOEME '9
he gives conditions to ensure their differentiabilty.
Among the appealing features of his treatment lies
the symmetry of the roles played by primal and
Saddle functions are considered in the contribu-
tions of E. Cavazzutti and N. Pacchiarotti who
present closure and convergence concepts and a
The analogy with fuzzy mathematics in which sets
blur to become functions is exploited in the work
of S. Dolecki who constructs operators akin to F-
funtionals and conjugations.
J. Morgan deals with stability questions for two-
level optimization problems, focusing her atten-
tion to well-posedness properties.
R. Orlandoni, O. Petrucciand M. Toseques study
the perturbation of curves of maximum slope for
a class of nonsmooth and nonconvex functions;
they obtain a compactness theorem.
C. Zalinescu opens a chapter for systematic as-
ymptoticconvex analysis. In particular, he obtains
a formula for the recession function of a marginal
penalization for mathematical programming
problems which are described by Lipschitzian
functions and give conditions for exactness.
The material presented byM. Gaudioso and MJ.
Monaco is centered on the idea of bundle algo-
rithm. However, quadratic approximations are
considered also in viewoffindingsearchdirections
for descent methods.
It is a combination of the bundle idea and of the
trust region method that J. Zowe undertakes.
Besides a study of the trajectory of optimal solu-
tions to the model problem as a function of the
radius of the trust region (or rather of a penalty
coefficient), he presents convergence results and
N.Z. Shore reports on subgradient type methods
with space dilatation inconnection with estimates
of the value of the dual problem of a mathematical
programming problem, in particular quadratic
E. Polak gives an elementary exposition of the
principles governingdescent algorithms for mini-
max problems; in particular, he describes a two-
phases method of centers related to Huard's
method of centers.
Applications are given in several contributions of
the volume. Two chapters are more obviously
oriented towards applications. One is the treat-
ment of quasi-variational inequalities givenby M.
De Luca and A. Maugeri in view of transportation
networks. Another one is certainly the overview
of the optimization problems pertaining to the
flightof anaircraft ina windshear presented by A.
Miele and T. Wang. Three problems are studied,
in particular the take-off trajectories and the mini-
mization of the maximum deviation of the abso-
lute path inclination from a reference value.
Since we have arrived at the end of this brief ac-
count of the topics in this volume, we will skip the
other problems treated by the two preceding con-
tributors (in particular the question of abort land-
ing trajectories) to conclude that the variety and the
deepness of the subjects treated in this book show
that the take-off of nonsmooth optimization is
successful and its flight is enjoyable.
by R.L. Graham, B.L. Rothschild
and J.H. Spencer
This is the second edition of a successful book first
published 10 years ago. As the authors proudly
state in the introduction, "The response to the first
edition of this volume has been most gratifying.
Before its publication this subject matter had been
generally regarded as a collection of loosely tied
results. Today it is recognized for what it is-a
cohesive subdiscipline of Discrete Mathematics.
We are particularly pleased with the name given
to this subdiscipline: Ramsey Theory!"
The authors updated the first edition by adding
just a few remarks here and there and a single
(major) new result: Shelah's proof of Van der
Waerden theorem, with detail hierarchy consider-
ations. Thus the authors do not attempt to expand
thevolumebyafinemosaic of recent development;
instead they keep the nice style of the first edition
which concentrates on the main stream of the
spectacular development (during the 1980s). The
interested reader may consult Mathematics of
Ramsey Theory (. Nesetril, V. R6dl. eds.), Springer
Verlag, 1990, and forthcoming bookby H.J. Pr6mel
and B. Voigt for a more extensive treatment of the
Handbook of Theoretical
Volume A.Algorithms and
Edited by J. Van Leeuwen
North Holland, 1990
The Handbook of Computer Science is designed to
provide an overview of the major results and re-
cent developments in this wide area. It is divided
into two parts: Volume A on "Algorithms and
Complexity" and Volume B on "Formal Models
and Semantics," thereby reflecting the division
between "algorithm-oriented" and "description-
oriented" research in this field. Is is valuable for far
more than just reference purposes. Each of the two
volumes contains about 20 chapters on different
topics, written by leading experts in the field. The
contributions are intended for students in com-
puter science and related disciplines and also pro-
vide an in-depth view of the subject. There is no
doubt that the Handbook of Computer Science is
one of the most useful and needed publications
in the field.
PAGE EIGHTEEN -
PAGE18 umbe thrty-iveNOVEBER'99
The contents and authors of Volume A follow:
1. "Machine Models and Simulations," P. van
Emde Boas; 2. "A Catalog of Complexity Classes,"
D. S. Johnson; 3. "Machine-Independent Complex-
ity Theory," J.I. Seiferas; 4. "Kolmogorov Com-
plexity and its Applications," M. Li and P.M.B.
VitAnyi; 5. "Algorithms for Finding Patterns in
Strings." A.V. Aho; 6. "Data Structures," K.
Mehlhorn and A. Tsakalidis; 7. "Computational
Geometry," F.F.Yao;8. "Algorithmic Motion Plan-
ning in Robotics," J.T. Schwartz and M. Sharir; 9.
"Average-Case Analysis of Algorithms and Data
Structures," J.S. Vitter and Ph. Flajolet; 10. "Graph
Algorithms," J. van Leeuwen; 11. "AlgebraicCom-
plexity Theory," V. Strassen; 12. "Algorithms in
Number Theory," A.K. Lenstra and H.W. Lenstra,
Jr.; 13. "Cryptography," R.L. Rivest; 14. "The
ComplexityofFinite Functions," R.B. Boppana and
M. Sipser; 15. "Communication Networks," N.
Pippenger; 16. "VLSI Theory," Th. Lengauer; 17.
"Parallel Algorithms for Shared-Memory Ma-
chines," R.M. Karp and V. Ramachandran; 18.
"General Purpose Parallel Architectures," L.G.
VOL 51, N. 2 V01. 1, No. -
M. Gr6tschel and 0. Holland, "Solution of
Large-Scale Symmetric Travelling Salesman
R.M. Freund, "Polynomial-Time Algorithms
for Linear Programming Based only on Primal
Scaling and Projected Gradients of a Potential
E.K. Yang and J.W. Tolle, "A Class of Methods
for Solving Large, Convex Quadratic Programs
Subject to Box Constraints."
H. Tuy, "Normal Conical Algorithm for
Concave Minimization over Polytopes."
F.A. Al-Khayyal, "Necessary and Sufficient
Conditions for the Existence of Complementary
Solutions and Characterizations of the Matrix
Classes Q and Qo'"
L.J. Cromme and I. Diener, "Fixed Point
Theorems for Discontinuous Mapping."
L.S. Zaremba, "Perfect Graphs and Norms."
A. Griewank, H.TH. Jongen and M.K.
Kwong, "The Equivalence of Strict Convexity
and Injectivity of the Gradient in Bounded Level
C.T. Kelley and S.J. Wright, "Sequential
Quadratic Programming for Certain Parameter
P. Alart and B. Lemaire, "Penalization in
Non-Classical Convex Programming via
B. Luderer, "Directional Derivative Estimates
for the Optimal Value Function of a
Quasidifferentiable Programming Problem."
S. van de Geer and L. Stougie, "On Rates of
Convergence and Asymptotic Normality in the
D. Naddef and G. Rinaldi, "The Symmetric
Traveling Salesman Polytope and its Graphical
Relaxation: Composition of Valid Inequalities."
M. Fischetti, "Facets of Two Steiner Arbores-
PAGE 190 numbrhirtNieENOEMER 99
ACHIM BACHEM, having assumed several
new responsibilities at Cologne and in the
German OR societies, has resigned as As-
sociate/Book Review Editor of OPTIMA.
We thank him for 10 years of excellent ser-
vice to the Society and welcome PROFES-
SOR ADOLPHUS J.J. TALMAN (Tilburg)
who is the newAssociateEditor. JMARTIN
GROTSCHEL has moved to Berlin Institute
of Technology as professor of mathematics
and vice-president of the Konrad-Zuse-
Zentrum for Information Technology Ber-
lin. 9PANOS PARDALOS is visiting the ISE
Department, University of Florida. IFAIZ
AL-KHAYYAL (Georgia Tech) is the new
chairman of the MPS Committee on Algo-
rithms (COAL). Deadline for the next
OPTIMA is February 1, 1992.
Books for review should be
sent to the Book Review Editor,
Professor Adolphus J.J. Talman
Department of Econometrics
P.O. Box 90153
5000 LE Tilburg
Journal contents are subject
to change by the publisher.
Donald W. Hearn, EDITOR
A.J.J. Talman, ASSOCIATE EDITOR
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